01754nas a2200169 4500008004100000245008800041210006900129260001400198520119900212100002201411700002301433700001901456700002901475700002201504700002101526856003701547 2024 eng d00aEfficient and practical Hamiltonian simulation from time-dependent product formulas0 aEfficient and practical Hamiltonian simulation from timedependen c3/13/20243 a

In this work we propose an approach for implementing time-evolution of a quantum system using product formulas. The quantum algorithms we develop have provably better scaling (in terms of gate complexity and circuit depth) than a naive application of well-known Trotter formulas, for systems where the evolution is determined by a Hamiltonian with different energy scales (i.e., one part is "large" and another part is "small"). Our algorithms generate a decomposition of the evolution operator into a product of simple unitaries that are directly implementable on a quantum computer. Although the theoretical scaling is suboptimal compared with state-of-the-art algorithms (e.g., quantum signal processing), the performance of the algorithms we propose is highly competitive in practice. We illustrate this via extensive numerical simulations for several models. For instance, in the strong-field regime of the 1D transverse-field Ising model, our algorithms achieve an improvement of one order of magnitude in both the system size and evolution time that can be simulated with a fixed budget of 1000 arbitrary 2-qubit gates, compared with standard Trotter formulas.

1 aBosse, Jan, Lukas1 aChilds, Andrew, M.1 aDerby, Charles1 aGambetta, Filippo, Maria1 aMontanaro, Ashley1 aSantos, Raul, A. uhttps://arxiv.org/abs/2403.0872901355nas a2200181 4500008004100000245008400041210006900125260001400194520077500208100001800983700002301001700002401024700001601048700002401064700002501088700002301113856003701136 2024 eng d00aEfficiently verifiable quantum advantage on near-term analog quantum simulators0 aEfficiently verifiable quantum advantage on nearterm analog quan c3/13/20243 a

Existing schemes for demonstrating quantum computational advantage are subject to various practical restrictions, including the hardness of verification and challenges in experimental implementation. Meanwhile, analog quantum simulators have been realized in many experiments to study novel physics. In this work, we propose a quantum advantage protocol based on single-step Feynman-Kitaev verification of an analog quantum simulation, in which the verifier need only run an O(λ2)-time classical computation, and the prover need only prepare O(1) samples of a history state and perform O(λ2) single-qubit measurements, for a security parameter λ. We also propose a near-term feasible strategy for honest provers and discuss potential experimental realizations.

1 aLiu, Zhenning1 aDevulapalli, Dhruv1 aHangleiter, Dominik1 aLiu, Yi-Kai1 aKollár, Alicia, J.1 aGorshkov, Alexey, V.1 aChilds, Andrew, M. uhttps://arxiv.org/abs/2403.0819501769nas a2200217 4500008004100000024003400041245005000075210005000125260001300175490000600188520108800194653004301282653004301325653002701368653003101395100002101426700002301447700002501470700001901495856003701514 2023 eng d aReport number: LA-UR-22-2023700aAdvantages and limitations of quantum routing0 aAdvantages and limitations of quantum routing c2/1/20230 v43 a

The Swap gate is a ubiquitous tool for moving information on quantum hardware, yet it can be considered a classical operation because it does not entangle product states. Genuinely quantum operations could outperform Swap for the task of permuting qubits within an architecture, which we call routing. We consider quantum routing in two models: (1) allowing arbitrary two-qubit unitaries, or (2) allowing Hamiltonians with norm-bounded interactions. We lower bound the circuit depth or time of quantum routing in terms of spectral properties of graphs representing the architecture interaction constraints, and give a generalized upper bound for all simple connected n-vertex graphs. In particular, we give conditions for a superpolynomial classical-quantum routing separation, which exclude graphs with a small spectral gap and graphs of bounded degree. Finally, we provide examples of a quadratic separation between gate-based and Hamiltonian routing models with a constant number of local ancillas per qubit and of an Ω(n) speedup if we also allow fast local interactions.

10aData Structures and Algorithms (cs.DS)10aFOS: Computer and information sciences10aFOS: Physical sciences10aQuantum Physics (quant-ph)1 aBapat, Aniruddha1 aChilds, Andrew, M.1 aGorshkov, Alexey, V.1 aSchoute, Eddie uhttps://arxiv.org/abs/2206.0176601581nas a2200181 4500008004100000245006200041210006200103260001100165490000600176520106100182100002701243700002301270700001901293700001701312700001801329700001501347856003701362 2023 eng d00aQuantum algorithm for estimating volumes of convex bodies0 aQuantum algorithm for estimating volumes of convex bodies c4/20230 v43 a

Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an n-dimensional convex body within multiplicative error ε using O~(n3.5+n2.5/ε) queries to a membership oracle and O~(n5.5+n4.5/ε) additional arithmetic operations. For comparison, the best known classical algorithm uses O~(n4+n3/ε2) queries and O~(n6+n5/ε2) additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of "Chebyshev cooling", where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error.

1 aChakrabarti, Shouvanik1 aChilds, Andrew, M.1 aHung, Shih-Han1 aLi, Tongyang1 aWang, Chunhao1 aWu, Xiaodi uhttps://arxiv.org/abs/1908.0390301051nas a2200133 4500008004100000245010100041210006900142260001400211520060600225100001300831700002300844700001300867856003700880 2023 eng d00aQuantum algorithm for linear non-unitary dynamics with near-optimal dependence on all parameters0 aQuantum algorithm for linear nonunitary dynamics with nearoptima c12/6/20233 a

We introduce a family of identities that express general linear non-unitary evolution operators as a linear combination of unitary evolution operators, each solving a Hamiltonian simulation problem. This formulation can exponentially enhance the accuracy of the recently introduced linear combination of Hamiltonian simulation (LCHS) method [An, Liu, and Lin, Physical Review Letters, 2023]. For the first time, this approach enables quantum algorithms to solve linear differential equations with both optimal state preparation cost and near-optimal scaling in matrix queries on all parameters.

1 aAn, Dong1 aChilds, Andrew, M.1 aLin, Lin uhttps://arxiv.org/abs/2312.0391601818nas a2200169 4500008004100000020002200041245005100063210005100114300001600165490000800181520131000189100002301499700002101522700002501543700002401568856005601592 2023 eng d a978-3-95977-263-100aQuantum Algorithms and the Power of Forgetting0 aQuantum Algorithms and the Power of Forgetting a37:1--37:220 v2513 a

The so-called welded tree problem provides an example of a black-box problem that can be solved exponentially faster by a quantum walk than by any classical algorithm [Andrew M. Childs et al., 2003]. Given the name of a special entrance vertex, a quantum walk can find another distinguished exit vertex using polynomially many queries, though without finding any particular path from entrance to exit. It has been an open problem for twenty years whether there is an efficient quantum algorithm for finding such a path, or if the path-finding problem is hard even for quantum computers. We show that a natural class of efficient quantum algorithms provably cannot find a path from entrance to exit. Specifically, we consider algorithms that, within each branch of their superposition, always store a set of vertex labels that form a connected subgraph including the entrance, and that only provide these vertex labels as inputs to the oracle. While this does not rule out the possibility of a quantum algorithm that efficiently finds a path, it is unclear how an algorithm could benefit by deviating from this behavior. Our no-go result suggests that, for some problems, quantum algorithms must necessarily forget the path they take to reach a solution in order to outperform classical computation.

1 aChilds, Andrew, M.1 aCoudron, Matthew1 aGilani, Amin, Shiraz1 aKalai, Yael, Tauman uhttps://drops.dagstuhl.de/opus/volltexte/2023/1754002470nas a2200169 4500008004100000245007100041210006900112260001400181520193200195100002202127700002202149700002402171700002302195700002502218700002002243856003702263 2023 eng d00aQuantum Algorithms for Simulating Nuclear Effective Field Theories0 aQuantum Algorithms for Simulating Nuclear Effective Field Theori c12/8/20233 a

Quantum computers offer the potential to simulate nuclear processes that are classically intractable. With the goal of understanding the necessary quantum resources, we employ state-of-the-art Hamiltonian-simulation methods, and conduct a thorough algorithmic analysis, to estimate the qubit and gate costs to simulate low-energy effective field theories (EFTs) of nuclear physics. In particular, within the framework of nuclear lattice EFT, we obtain simulation costs for the leading-order pionless and pionful EFTs. We consider both static pions represented by a one-pion-exchange potential between the nucleons, and dynamical pions represented by relativistic bosonic fields coupled to non-relativistic nucleons. We examine the resource costs for the tasks of time evolution and energy estimation for physically relevant scales. We account for model errors associated with truncating either long-range interactions in the one-pion-exchange EFT or the pionic Hilbert space in the dynamical-pion EFT, and for algorithmic errors associated with product-formula approximations and quantum phase estimation. Our results show that the pionless EFT is the least costly to simulate and the dynamical-pion theory is the costliest. We demonstrate how symmetries of the low-energy nuclear Hamiltonians can be utilized to obtain tighter error bounds on the simulation algorithm. By retaining the locality of nucleonic interactions when mapped to qubits, we achieve reduced circuit depth and substantial parallelization. We further develop new methods to bound the algorithmic error for classes of fermionic Hamiltonians that preserve the number of fermions, and demonstrate that reasonably tight Trotter error bounds can be achieved by explicitly computing nested commutators of Hamiltonian terms. This work highlights the importance of combining physics insights and algorithmic advancement in reducing quantum-simulation costs.

1 aWatson, James, D.1 aBringewatt, Jacob1 aShaw, Alexander, F.1 aChilds, Andrew, M.1 aGorshkov, Alexey, V.1 aDavoudi, Zohreh uhttps://arxiv.org/abs/2312.0534401289nas a2200169 4500008004100000245004100041210004100082260001400123520084200137100002300979700001601002700001801018700001201036700001701048700001701065856003701082 2023 eng d00aStreaming quantum state purification0 aStreaming quantum state purification c9/28/20233 a

Quantum state purification is the task of recovering a nearly pure copy of an unknown pure quantum state using multiple noisy copies of the state. This basic task has applications to quantum communication over noisy channels and quantum computation with imperfect devices, but has only been studied previously for the case of qubits. We derive an efficient purification procedure based on the swap test for qudits of any dimension, starting with any initial error parameter. Treating the initial error parameter and the dimension as constants, we show that our procedure has sample complexity asymptotically optimal in the final error parameter. Our protocol has a simple recursive structure that can be applied when the states are provided one at a time in a streaming fashion, requiring only a small quantum memory to implement.

1 aChilds, Andrew, M.1 aFu, Honghao1 aLeung, Debbie1 aLi, Zhi1 aOzols, Maris1 aVyas, Vedang uhttps://arxiv.org/abs/2309.1638701492nas a2200181 4500008004100000245008600041210006900127260001500196490000600211520094800217100001601165700002301181700002101204700001701225700001701242700001401259856003701273 2022 eng d00aEfficient Product Formulas for Commutators and Applications to Quantum Simulation0 aEfficient Product Formulas for Commutators and Applications to Q c03/10/20220 v43 a

We construct product formulas for exponentials of commutators and explore their applications. First, we directly construct a third-order product formula with six exponentials by solving polynomial equations obtained using the operator differential method. We then derive higher-order product formulas recursively from the third-order formula. We improve over previous recursive constructions, reducing the number of gates required to achieve the same accuracy. In addition, we demonstrate that the constituent linear terms in the commutator can be included at no extra cost. As an application, we show how to use the product formulas in a digital protocol for counterdiabatic driving, which increases the fidelity for quantum state preparation. We also discuss applications to quantum simulation of one-dimensional fermion chains with nearest- and next-nearest-neighbor hopping terms, and two-dimensional fractional quantum Hall phases.

1 aChen, Yu-An1 aChilds, Andrew, M.1 aHafezi, Mohammad1 aJiang, Zhang1 aKim, Hwanmun1 aXu, Yijia uhttps://arxiv.org/abs/2111.1217701557nas a2200169 4500008004100000245004600041210004600087260001500133490000800148520110300156100001301259700001401272700002401286700001701310700002301327856003701350 2022 eng d00aHamiltonian simulation with random inputs0 aHamiltonian simulation with random inputs c12/30/20220 v1293 a

The algorithmic error of digital quantum simulations is usually explored in terms of the spectral norm distance between the actual and ideal evolution operators. In practice, this worst-case error analysis may be unnecessarily pessimistic. To address this, we develop a theory of average-case performance of Hamiltonian simulation with random initial states. We relate the average-case error to the Frobenius norm of the multiplicative error and give upper bounds for the product formula (PF) and truncated Taylor series methods. As applications, we estimate average-case error for digital Hamiltonian simulation of general lattice Hamiltonians and k-local Hamiltonians. In particular, for the nearest-neighbor Heisenberg chain with n spins, the error is quadratically reduced from O(n) in the worst case to O(n−−√) on average for both the PF method and the Taylor series method. Numerical evidence suggests that this theory accurately characterizes the average error for concrete models. We also apply our results to error analysis in the simulation of quantum scrambling.

1 aZhao, Qi1 aZhou, You1 aShaw, Alexander, F.1 aLi, Tongyang1 aChilds, Andrew, M. uhttps://arxiv.org/abs/2111.0477301951nas a2200217 4500008004100000245008300041210006900124260001500193490000600208520129200214100002001506700002301526700001701549700002201566700002401588700002301612700001301635700002301648700002501671856003701696 2022 eng d00aImplementing a Fast Unbounded Quantum Fanout Gate Using Power-Law Interactions0 aImplementing a Fast Unbounded Quantum Fanout Gate Using PowerLaw c10/27/20220 v43 a

The standard circuit model for quantum computation presumes the ability to directly perform gates between arbitrary pairs of qubits, which is unlikely to be practical for large-scale experiments. Power-law interactions with strength decaying as 1/rα in the distance r provide an experimentally realizable resource for information processing, whilst still retaining long-range connectivity. We leverage the power of these interactions to implement a fast quantum fanout gate with an arbitrary number of targets. Our implementation allows the quantum Fourier transform (QFT) and Shor's algorithm to be performed on a D-dimensional lattice in time logarithmic in the number of qubits for interactions with α≤D. As a corollary, we show that power-law systems with α≤D are difficult to simulate classically even for short times, under a standard assumption that factoring is classically intractable. Complementarily, we develop a new technique to give a general lower bound, linear in the size of the system, on the time required to implement the QFT and the fanout gate in systems that are constrained by a linear light cone. This allows us to prove an asymptotically tighter lower bound for long-range systems than is possible with previously available techniques. 

1 aGuo, Andrew, Y.1 aDeshpande, Abhinav1 aChu, Su-Kuan1 aEldredge, Zachary1 aBienias, Przemyslaw1 aDevulapalli, Dhruv1 aSu, Yuan1 aChilds, Andrew, M.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2007.0066200782nas a2200229 4500008004100000245009900041210006900140260001500209490000700224653004300231653002100274653002700295653002900322653003900351653003100390100002300421700001700444700001800461700001800479700001800497856003700515 2022 eng d00aQuantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants0 aQuantum Algorithms for Sampling LogConcave Distributions and Est c10/12/20220 v3510aFOS: Computer and information sciences10aFOS: Mathematics10aFOS: Physical sciences10aMachine Learning (cs.LG)10aOptimization and Control (math.OC)10aQuantum Physics (quant-ph)1 aChilds, Andrew, M.1 aLi, Tongyang1 aLiu, Jin-Peng1 aWang, Chunhao1 aZhang, Ruizhe uhttps://arxiv.org/abs/2210.0653901490nas a2200205 4500008004100000245003100041210003100072260001500103520088200118653004301000653004301043653002701086653003101113100002301144700001901167700002201186700002101208700001801229856003701247 2022 eng d00aQuantum divide and conquer0 aQuantum divide and conquer c10/12/20223 a

The divide-and-conquer framework, used extensively in classical algorithm design, recursively breaks a problem of size n into smaller subproblems (say, a copies of size n/b each), along with some auxiliary work of cost Caux(n), to give a recurrence relation
C(n)≤aC(n/b)+Caux(n)
for the classical complexity C(n). We describe a quantum divide-and-conquer framework that, in certain cases, yields an analogous recurrence relation
CQ(n)≤a−−√CQ(n/b)+O(CauxQ(n))
that characterizes the quantum query complexity. We apply this framework to obtain near-optimal quantum query complexities for various string problems, such as (i) recognizing regular languages; (ii) decision versions of String Rotation and String Suffix; and natural parameterized versions of (iii) Longest Increasing Subsequence and (iv) Longest Common Subsequence.

10aData Structures and Algorithms (cs.DS)10aFOS: Computer and information sciences10aFOS: Physical sciences10aQuantum Physics (quant-ph)1 aChilds, Andrew, M.1 aKothari, Robin1 aKovacs-Deak, Matt1 aSundaram, Aarthi1 aWang, Daochen uhttps://arxiv.org/abs/2210.0641901358nas a2200181 4500008004100000245003900041210003900080260001300119520083800132653002700970653003100997100002301028700001901051700002101070700002301091700002501114856003701139 2022 eng d00aQuantum Routing with Teleportation0 aQuantum Routing with Teleportation c4/8/20223 a

We study the problem of implementing arbitrary permutations of qubits under interaction constraints in quantum systems that allow for arbitrarily fast local operations and classical communication (LOCC). In particular, we show examples of speedups over swap-based and more general unitary routing methods by distributing entanglement and using LOCC to perform quantum teleportation. We further describe an example of an interaction graph for which teleportation gives a logarithmic speedup in the worst-case routing time over swap-based routing. We also study limits on the speedup afforded by quantum teleportation - showing an O(NlogN−−−−−−−√) upper bound on the separation in routing time for any interaction graph - and give tighter bounds for some common classes of graphs.

10aFOS: Physical sciences10aQuantum Physics (quant-ph)1 aDevulapalli, Dhruv1 aSchoute, Eddie1 aBapat, Aniruddha1 aChilds, Andrew, M.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2204.0418502005nas a2200181 4500008004100000245004600041210004500087260001400132300000800146490000600154520152300160100002301683700001601706700001701722700001801739700001801757856004801775 2022 eng d00aQuantum simulation of real-space dynamics0 aQuantum simulation of realspace dynamics c11/8/2022 a8600 v63 a

Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finite-dimensional systems, less is known about quantum algorithms for real-space dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a d-dimensional Schrödinger equation with η particles can be simulated with gate complexity O~(ηdFpoly(log(g′/ϵ))), where ϵ is the discretization error, g′ controls the higher-order derivatives of the wave function, and F measures the time-integrated strength of the potential. Compared to the best previous results, this exponentially improves the dependence on ϵ and g′ from poly(g′/ϵ) to poly(log(g′/ϵ)) and polynomially improves the dependence on T and d, while maintaining best known performance with respect to η. For the case of Coulomb interactions, we give an algorithm using η3(d+η)Tpoly(log(ηdTg′/(Δϵ)))/Δ one- and two-qubit gates, and another using η3(4d)d/2Tpoly(log(ηdTg′/(Δϵ)))/Δ one- and two-qubit gates and QRAM operations, where T is the evolution time and the parameter Δ regulates the unbounded Coulomb interaction. We give applications to several computational problems, including faster real-space simulation of quantum chemistry, rigorous analysis of discretization error for simulation of a uniform electron gas, and a quadratic improvement to a quantum algorithm for escaping saddle points in nonconvex optimization.

1 aChilds, Andrew, M.1 aLeng, Jiaqi1 aLi, Tongyang1 aLiu, Jin-Peng1 aZhang, Chenyi uhttps://doi.org/10.22331%2Fq-2022-11-17-86001797nas a2200229 4500008004100000245007000041210006800111260001400179300001200193490000800205520107400213653003701287653002701324653003901351653003101390100002101421700002401442700001901466700002301485700002201508856003701530 2022 eng d00aTweezer-programmable 2D quantum walks in a Hubbard-regime lattice0 aTweezerprogrammable 2D quantum walks in a Hubbardregime lattice c8/18/2022 a885-8890 v3773 a

Quantum walks provide a framework for understanding and designing quantum algorithms that is both intuitive and universal. To leverage the computational power of these walks, it is important to be able to programmably modify the graph a walker traverses while maintaining coherence. Here, we do this by combining the fast, programmable control provided by optical tweezer arrays with the scalable, homogeneous environment of an optical lattice. Using this new combination of tools we study continuous-time quantum walks of single atoms on a 2D square lattice, and perform proof-of-principle demonstrations of spatial search using these walks. When scaled to more particles, the capabilities demonstrated here can be extended to study a variety of problems in quantum information science and quantum simulation, including the deterministic assembly of ground and excited states in Hubbard models with tunable interactions, and performing versions of spatial search in a larger graph with increased connectivity, where search by quantum walk can be more effective.

10aAtomic Physics (physics.atom-ph)10aFOS: Physical sciences10aQuantum Gases (cond-mat.quant-gas)10aQuantum Physics (quant-ph)1 aYoung, Aaron, W.1 aEckner, William, J.1 aSchine, Nathan1 aChilds, Andrew, M.1 aKaufman, Adam, M. uhttps://arxiv.org/abs/2202.0120402034nas a2200181 4500008004100000245008100041210006900122260001300191490000800204520146900212100001801681700002501699700002001724700002301744700002501767700002301792856003701815 2021 eng d00aEfficient quantum algorithm for dissipative nonlinear differential equations0 aEfficient quantum algorithm for dissipative nonlinear differenti c3/1/20210 v1183 a

While there has been extensive previous work on efficient quantum algorithms for linear differential equations, analogous progress for nonlinear differential equations has been severely limited due to the linearity of quantum mechanics. Despite this obstacle, we develop a quantum algorithm for initial value problems described by dissipative quadratic n-dimensional ordinary differential equations. Assuming R<1, where R is a parameter characterizing the ratio of the nonlinearity to the linear dissipation, this algorithm has complexity T2poly(logT,logn)/ϵ, where T is the evolution time and ϵ is the allowed error in the output quantum state. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. We achieve this improvement using the method of Carleman linearization, for which we give an improved convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R≥2–√. Finally, we discuss potential applications of this approach to problems arising in biology as well as in fluid and plasma dynamics.

1 aLiu, Jin-Peng1 aKolden, Herman, Øie1 aKrovi, Hari, K.1 aLoureiro, Nuno, F.1 aTrivisa, Konstantina1 aChilds, Andrew, M. uhttps://arxiv.org/abs/2011.0318501422nas a2200145 4500008004100000245007300041210006900114260001400183490000600197520097400203100002301177700001801200700002101218856003701239 2021 eng d00aHigh-precision quantum algorithms for partial differential equations0 aHighprecision quantum algorithms for partial differential equati c11/4/20210 v53 a

Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity poly(1/ε), where ε is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be poly(d,log(1/ε)), where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations. 

1 aChilds, Andrew, M.1 aLiu, Jin-Peng1 aOstrander, Aaron uhttps://arxiv.org/abs/2002.0786801283nas a2200169 4500008004100000245005900041210005800100260000900158300001600167490000700183520079000190100001800980700001600998700001701014700002301031856005901054 2021 eng d00aQuantum exploration algorithms for multi-armed bandits0 aQuantum exploration algorithms for multiarmed bandits c2021 a10102-101100 v353 a

 Identifying the best arm of a multi-armed bandit is a central problem in bandit optimization. We study a quantum computational version of this problem with coherent oracle access to states encoding the reward probabilities of each arm as quantum amplitudes. Specifically, we show that we can find the best arm with fixed confidence using O~(∑ni=2Δ−2i−−−−−−−−√) quantum queries, where Δi represents the difference between the mean reward of the best arm and the ith-best arm. This algorithm, based on variable-time amplitude amplification and estimation, gives a quadratic speedup compared to the best possible classical result. We also prove a matching quantum lower bound (up to poly-logarithmic factors).

1 aWang, Daochen1 aYou, Xuchen1 aLi, Tongyang1 aChilds, Andrew, M. uhttps://ojs.aaai.org/index.php/AAAI/article/view/1721201245nas a2200157 4500008004100000245005700041210005600098260001400154300001500168490000800183520080000191100002300991700001901014700001701033856003701050 2021 eng d00aQuantum query complexity with matrix-vector products0 aQuantum query complexity with matrixvector products c3/14/2021 a55:1-55:190 v1983 a

We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.

1 aChilds, Andrew, M.1 aHung, Shih-Han1 aLi, Tongyang uhttps://arxiv.org/abs/2102.1134901323nas a2200145 4500008004100000020002200041245005700063210005600120260008600176520080000262100002301062700001901085700001701104856005601121 2021 eng d a978-3-95977-195-500aQuantum Query Complexity with Matrix-Vector Products0 aQuantum Query Complexity with MatrixVector Products aDagstuhl, GermanybSchloss Dagstuhl – Leibniz-Zentrum für Informatikc2/7/20213 a

We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.

1 aChilds, Andrew, M.1 aHung, Shih-Han1 aLi, Tongyang uhttps://drops.dagstuhl.de/opus/volltexte/2021/1412401494nas a2200181 4500008004100000245004000041210004000081260001400121490000600135520100800141100002101149700002301170700002501193700001701218700001901235700002101254856003701275 2021 eng d00aQuantum routing with fast reversals0 aQuantum routing with fast reversals c8/24/20210 v53 a

We present methods for implementing arbitrary permutations of qubits under interaction constraints. Our protocols make use of previous methods for rapidly reversing the order of qubits along a path. Given nearest-neighbor interactions on a path of length n, we show that there exists a constant ϵ≈0.034 such that the quantum routing time is at most (1−ϵ)n, whereas any swap-based protocol needs at least time n−1. This represents the first known quantum advantage over swap-based routing methods and also gives improved quantum routing times for realistic architectures such as grids. Furthermore, we show that our algorithm approaches a quantum routing time of 2n/3 in expectation for uniformly random permutations, whereas swap-based protocols require time n asymptotically. Additionally, we consider sparse permutations that route k≤n qubits and give algorithms with quantum routing time at most n/3+O(k2) on paths and at most 2r/3+O(k2) on general graphs with radius r.

1 aBapat, Aniruddha1 aChilds, Andrew, M.1 aGorshkov, Alexey, V.1 aKing, Samuel1 aSchoute, Eddie1 aShastri, Hrishee uhttps://arxiv.org/abs/2103.0326402331nas a2200181 4500008004100000245005200041210005200093260001300145300000700158490000700165520185000172100002302022700001302045700001902058700001802077700001702095856003702112 2021 eng d00aTheory of Trotter Error with Commutator Scaling0 aTheory of Trotter Error with Commutator Scaling c2/1/2021 a490 v113 a

The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure, k-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a byproduct. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of 5, and is close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.

1 aChilds, Andrew, M.1 aSu, Yuan1 aTran, Minh, C.1 aWiebe, Nathan1 aZhu, Shuchen uhttps://arxiv.org/abs/1912.0885401616nas a2200121 4500008004100000245005900041210005800100260001400158520124400172100002301416700001801439856003701457 2020 eng d00aCan graph properties have exponential quantum speedup?0 aCan graph properties have exponential quantum speedup c1/28/20203 a

Quantum computers can sometimes exponentially outperform classical ones, but only for problems with sufficient structure. While it is well known that query problems with full permutation symmetry can have at most polynomial quantum speedup -- even for partial functions -- it is unclear how far this condition must be relaxed to enable exponential speedup. In particular, it is natural to ask whether exponential speedup is possible for (partial) graph properties, in which the input describes a graph and the output can only depend on its isomorphism class. We show that the answer to this question depends strongly on the input model. In the adjacency matrix model, we prove that the bounded-error randomized query complexity R of any graph property P has R(P)=O(Q(P)6), where Q is the bounded-error quantum query complexity. This negatively resolves an open question of Montanaro and de Wolf in the adjacency matrix model. More generally, we prove R(P)=O(Q(P)3l) for any l-uniform hypergraph property P in the adjacency matrix model. In direct contrast, in the adjacency list model for bounded-degree graphs, we exhibit a promise problem that shows an exponential separation between the randomized and quantum query complexities.

1 aChilds, Andrew, M.1 aWang, Daochen uhttps://arxiv.org/abs/2001.1052001561nas a2200169 4500008004100000245007300041210006900114260001300183490000800196520105300204100001901257700001701276700001301293700002301306700002501329856003701354 2020 eng d00aDestructive Error Interference in Product-Formula Lattice Simulation0 aDestructive Error Interference in ProductFormula Lattice Simulat c6/4/20200 v1243 a

Quantum computers can efficiently simulate the dynamics of quantum systems. In this paper, we study the cost of digitally simulating the dynamics of several physically relevant systems using the first-order product formula algorithm. We show that the errors from different Trotterization steps in the algorithm can interfere destructively, yielding a much smaller error than previously estimated. In particular, we prove that the total error in simulating a nearest-neighbor interacting system of n sites for time t using the first-order product formula with r time slices is O(nt/r+nt3/r2) when nt2/r is less than a small constant. Given an error tolerance ε, the error bound yields an estimate of max{O(n2t/ε),O(n2t3/2/ε1/2)} for the total gate count of the simulation. The estimate is tighter than previous bounds and matches the empirical performance observed in Childs et al. [PNAS 115, 9456-9461 (2018)]. We also provide numerical evidence for potential improvements and conjecture an even tighter estimate for the gate count. 

1 aTran, Minh, C.1 aChu, Su-Kuan1 aSu, Yuan1 aChilds, Andrew, M.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/1912.1104701544nas a2200145 4500008004100000245006100041210006000102260001300162520109800175100002101273700001901294700002501313700002301338856003701361 2020 eng d00aNearly optimal time-independent reversal of a spin chain0 aNearly optimal timeindependent reversal of a spin chain c3/5/20203 a

We propose a time-independent Hamiltonian protocol for the reversal of qubit ordering in a chain of N spins. Our protocol has an easily implementable nearest-neighbor, transverse-field Ising model Hamiltonian with time-independent, non-uniform couplings. Under appropriate normalization, we implement this state reversal three times faster than a naive approach using SWAP gates, in time comparable to a protocol of Raussendorf [Phys. Rev. A 72, 052301 (2005)] that requires dynamical control. We also prove lower bounds on state reversal by using results on the entanglement capacity of Hamiltonians and show that we are within a factor 1.502(1+1/N) of the shortest time possible. Our lower bound holds for all nearest-neighbor qubit protocols with arbitrary finite ancilla spaces and local operations and classical communication. Finally, we extend our protocol to an infinite family of nearest-neighbor, time-independent Hamiltonian protocols for state reversal. This includes chains with nearly uniform coupling that may be especially feasible for experimental implementation. 

1 aBapat, Aniruddha1 aSchoute, Eddie1 aGorshkov, Alexey, V.1 aChilds, Andrew, M. uhttps://arxiv.org/abs/2003.0284301889nas a2200169 4500008004100000245006600041210006500107260001300172300001200185490004400197520136000241100001901601700002301620700002001643700001901663856003701682 2020 eng d00aNon-interactive classical verification of quantum computation0 aNoninteractive classical verification of quantum computation c3/9/2020 a153-1800 vLecture Notes in Computer Science 125523 a

In a recent breakthrough, Mahadev constructed an interactive protocol that enables a purely classical party to delegate any quantum computation to an untrusted quantum prover. In this work, we show that this same task can in fact be performed non-interactively and in zero-knowledge.
Our protocols result from a sequence of significant improvements to the original four-message protocol of Mahadev. We begin by making the first message instance-independent and moving it to an offline setup phase. We then establish a parallel repetition theorem for the resulting three-message protocol, with an asymptotically optimal rate. This, in turn, enables an application of the Fiat-Shamir heuristic, eliminating the second message and giving a non-interactive protocol. Finally, we employ classical non-interactive zero-knowledge (NIZK) arguments and classical fully homomorphic encryption (FHE) to give a zero-knowledge variant of this construction. This yields the first purely classical NIZK argument system for QMA, a quantum analogue of NP.
We establish the security of our protocols under standard assumptions in quantum-secure cryptography. Specifically, our protocols are secure in the Quantum Random Oracle Model, under the assumption that Learning with Errors is quantumly hard. The NIZK construction also requires circuit-private FHE.

1 aAlagic, Gorjan1 aChilds, Andrew, M.1 aGrilo, Alex, B.1 aHung, Shih-Han uhttps://arxiv.org/abs/1911.0810101173nas a2200157 4500008004100000245006400041210006400105260001500169490000600184520070600190100002700896700002300923700001700946700001500963856003700978 2020 eng d00aQuantum algorithms and lower bounds for convex optimization0 aQuantum algorithms and lower bounds for convex optimization c12/18/20190 v43 a

While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function over an n-dimensional convex body using O~(n) queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the best-known classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires Ω~(n−−√) evaluation queries and Ω(n−−√) membership queries.

1 aChakrabarti, Shouvanik1 aChilds, Andrew, M.1 aLi, Tongyang1 aWu, Xiaodi uhttps://arxiv.org/abs/1809.0173101627nas a2200193 4500008004100000245002900041210002900070260001400099300001500113490000800128520112700136100002801263700002301291700002301314700001901337700002001356700002001376856003701396 2020 eng d00aQuantum Coupon Collector0 aQuantum Coupon Collector c2/18/2020 a10:1-10:170 v1583 a

We study how efficiently a k-element set S⊆[n] can be learned from a uniform superposition |S⟩ of its elements. One can think of |S⟩=∑i∈S|i⟩/|S|−−−√ as the quantum version of a uniformly random sample over S, as in the classical analysis of the ``coupon collector problem.'' We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n−k=O(1) missing elements then O(k) copies of |S⟩ suffice, in contrast to the Θ(klogk) random samples needed by a classical coupon collector. On the other hand, if n−k=Ω(k), then Ω(klogk) quantum samples are~necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through |S⟩. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.

1 aArunachalam, Srinivasan1 aBelovs, Aleksandrs1 aChilds, Andrew, M.1 aKothari, Robin1 aRosmanis, Ansis1 ade Wolf, Ronald uhttps://arxiv.org/abs/2002.0768801292nas a2200145 4500008004100000245005600041210005600097260001400153300001400167490000800181520087900189100002301068700001801091856003701109 2020 eng d00aQuantum spectral methods for differential equations0 aQuantum spectral methods for differential equations c2/18/2020 a1427-14570 v3753 a

Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a d-dimensional system of linear equations or linear differential equations with complexity poly(logd). While several of these algorithms approximate the solution to within ε with complexity poly(log(1/ε)), no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity poly(logd,log(1/ε)).

1 aChilds, Andrew, M.1 aLiu, Jin-Peng uhttps://arxiv.org/abs/1901.0096101772nas a2200169 4500008004100000245006700041210006600108260001300174490000800187520126400195100002001459700001901479700002301498700002501521700001901546856003701565 2020 eng d00aSignaling and Scrambling with Strongly Long-Range Interactions0 aSignaling and Scrambling with Strongly LongRange Interactions c7/8/20200 v1023 a

Strongly long-range interacting quantum systems---those with interactions decaying as a power-law 1/rα in the distance r on a D-dimensional lattice for α≤D---have received significant interest in recent years. They are present in leading experimental platforms for quantum computation and simulation, as well as in theoretical models of quantum information scrambling and fast entanglement creation. Since no notion of locality is expected in such systems, a general understanding of their dynamics is lacking. As a first step towards rectifying this problem, we prove two new Lieb-Robinson-type bounds that constrain the time for signaling and scrambling in strongly long-range interacting systems, for which no tight bounds were previously known. Our first bound applies to systems mappable to free-particle Hamiltonians with long-range hopping, and is saturable for α≤D/2. Our second bound pertains to generic long-range interacting spin Hamiltonians, and leads to a tight lower bound for the signaling time to extensive subsets of the system for all α<D. This result also lower-bounds the scrambling time, and suggests a path towards achieving a tight scrambling bound that can prove the long-standing fast scrambling conjecture. 

1 aGuo, Andrew, Y.1 aTran, Minh, C.1 aChilds, Andrew, M.1 aGorshkov, Alexey, V.1 aGong, Zhe-Xuan uhttps://arxiv.org/abs/1906.0266201555nas a2200169 4500008004100000245005500041210005300096260001400149520105800163100002201221700002301243700001901266700002401285700002101309700001801330856003701348 2020 eng d00aSymmetries, graph properties, and quantum speedups0 aSymmetries graph properties and quantum speedups c6/23/20203 a

Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup?
In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups.
In contrast, in the adjacency list model for bounded-degree graphs (where graph symmetry is manifested differently), we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).

1 aBen-David, Shalev1 aChilds, Andrew, M.1 aGilyen, Andras1 aKretschmer, William1 aPodder, Supartha1 aWang, Daochen uhttps://arxiv.org/abs/2006.1276001668nas a2200169 4500008004100000245006300041210006100104260001400165490000600179520118500185100002301370700002301393700001301416700001401429700001801443856003701461 2020 eng d00aTime-dependent Hamiltonian simulation with L1-norm scaling0 aTimedependent Hamiltonian simulation with L1norm scaling c4/20/20200 v43 a

The difficulty of simulating quantum dynamics depends on the norm of the Hamiltonian. When the Hamiltonian varies with time, the simulation complexity should only depend on this quantity instantaneously. We develop quantum simulation algorithms that exploit this intuition. For the case of sparse Hamiltonian simulation, the gate complexity scales with the L1 norm ∫t0dτ∥H(τ)∥max, whereas the best previous results scale with tmaxτ∈[0,t]∥H(τ)∥max. We also show analogous results for Hamiltonians that are linear combinations of unitaries. Our approaches thus provide an improvement over previous simulation algorithms that can be substantial when the Hamiltonian varies significantly. We introduce two new techniques: a classical sampler of time-dependent Hamiltonians and a rescaling principle for the Schrödinger equation. The rescaled Dyson-series algorithm is nearly optimal with respect to all parameters of interest, whereas the sampling-based approach is easier to realize for near-term simulation. By leveraging the L1-norm information, we obtain polynomial speedups for semi-classical simulations of scattering processes in quantum chemistry.

1 aBerry, Dominic, W.1 aChilds, Andrew, M.1 aSu, Yuan1 aWang, Xin1 aWiebe, Nathan uhttps://arxiv.org/abs/1906.0711501730nas a2200145 4500008004100000245005400041210005400095260001500149490000900164520131300173100002301486700001901509700001901528856003701547 2019 eng d00aCircuit Transformations for Quantum Architectures0 aCircuit Transformations for Quantum Architectures c02/25/20190 v135 3 a

Quantum computer architectures impose restrictions on qubit interactions. We propose efficient circuit transformations that modify a given quantum circuit to fit an architecture, allowing for any initial and final mapping of circuit qubits to architecture qubits. To achieve this, we first consider the qubit movement subproblem and use the routing via matchings framework to prove tighter bounds on parallel routing. In practice, we only need to perform partial permutations, so we generalize routing via matchings to that setting. We give new routing procedures for common architecture graphs and for the generalized hierarchical product of graphs, which produces subgraphs of the Cartesian product. Secondly, for serial routing, we consider the token swapping framework and extend a 4-approximation algorithm for general graphs to support partial permutations. We apply these routing procedures to give several circuit transformations, using various heuristic qubit placement subroutines. We implement these transformations in software and compare their performance for large quantum circuits on grid and modular architectures, identifying strategies that work well in practice.

1 aChilds, Andrew, M.1 aSchoute, Eddie1 aUnsal, Cem, M. uhttps://arxiv.org/abs/1902.0910201284nas a2200145 4500008004100000245004700041210004700088260001500135490000600150520088800156100002301044700002101067700001301088856003701101 2019 eng d00aFaster quantum simulation by randomization0 aFaster quantum simulation by randomization c08/28/20190 v33 a

Product formulas can be used to simulate Hamiltonian dynamics on a quantum computer by approximating the exponential of a sum of operators by a product of exponentials of the individual summands. This approach is both straightforward and surprisingly efficient. We show that by simply randomizing how the summands are ordered, one can prove stronger bounds on the quality of approximation and thereby give more efficient simulations. Indeed, we show that these bounds can be asymptotically better than previous bounds that exploit commutation between the summands, despite using much less information about the structure of the Hamiltonian. Numerical evidence suggests that our randomized algorithm may be advantageous even for near-term quantum simulation.

1 aChilds, Andrew, M.1 aOstrander, Aaron1 aSu, Yuan uhttps://arxiv.org/abs/1805.0838501741nas a2200205 4500008004100000245007000041210006900111260001500180490000600195520112800201100001901329700002001348700001301368700002401381700002201405700002301427700002301450700002501473856003701498 2019 eng d00aLocality and digital quantum simulation of power-law interactions0 aLocality and digital quantum simulation of powerlaw interactions c07/10/20190 v93 a

The propagation of information in non-relativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1/rα. The bound implies an effective light cone tighter than all previous bounds. Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al. [arXiv:1801.03922]. To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish. This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation. In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions. In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α>3D (where D is the number of dimensions).

1 aTran, Minh, C.1 aGuo, Andrew, Y.1 aSu, Yuan1 aGarrison, James, R.1 aEldredge, Zachary1 aFoss-Feig, Michael1 aChilds, Andrew, M.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/1808.0522501490nas a2200133 4500008004100000245005800041210005800099260001500157490000800172520110300180100002301283700001301306856003701319 2019 eng d00aNearly optimal lattice simulation by product formulas0 aNearly optimal lattice simulation by product formulas c12/17/20190 v1233 a

Product formulas provide a straightforward yet surprisingly efficient approach to quantum simulation. We show that this algorithm can simulate an n-qubit Hamiltonian with nearest-neighbor interactions evolving for time t using only (nt)1+o(1) gates. While it is reasonable to expect this complexity---in particular, this was claimed without rigorous justification by Jordan, Lee, and Preskill---we are not aware of a straightforward proof. Our approach is based on an analysis of the local error structure of product formulas, as introduced by Descombes and Thalhammer and significantly simplified here. We prove error bounds for canonical product formulas, which include well-known constructions such as the Lie-Trotter-Suzuki formulas. We also develop a local error representation for time-dependent Hamiltonian simulation, and we discuss generalizations to periodic boundary conditions, constant-range interactions, and higher dimensions. Combined with a previous lower bound, our result implies that product formulas can simulate lattice Hamiltonians with nearly optimal gate complexity.

1 aChilds, Andrew, M.1 aSu, Yuan uhttps://arxiv.org/abs/1901.0056401295nas a2200169 4500008004100000245008000041210006900121260001500190490000600205520078500211100001800996700001901014700001301033700002301046700001901069856003701088 2018 eng d00aAutomated optimization of large quantum circuits with continuous parameters0 aAutomated optimization of large quantum circuits with continuous c2017/10/190 v43 a

We develop and implement automated methods for optimizing quantum circuits of the size and type expected in quantum computations that outperform classical computers. We show how to handle continuous gate parameters and report a collection of fast algorithms capable of optimizing large-scale quantum circuits. For the suite of benchmarks considered, we obtain substantial reductions in gate counts. In particular, we provide better optimization in significantly less time than previous approaches, while making minimal structural changes so as to preserve the basic layout of the underlying quantum algorithms. Our results help bridge the gap between the computations that can be run on existing hardware and those that are expected to outperform classical computers. 

1 aNam, Yunseong1 aRoss, Neil, J.1 aSu, Yuan1 aChilds, Andrew, M.1 aMaslov, Dmitri uhttps://arxiv.org/abs/1710.0734501497nas a2200145 4500008004100000245006400041210006400105260001500169490000800184520103000192100001801222700002301240700001901263856006901282 2018 eng d00aQuantum algorithm for multivariate polynomial interpolation0 aQuantum algorithm for multivariate polynomial interpolation c2018/01/170 v4743 a

How many quantum queries are required to determine the coefficients of a degree-d polynomial in n variables? We present and analyze quantum algorithms for this multivariate polynomial interpolation problem over the fields Fq, R, and C. We show that kC and 2kC queries suffice to achieve probability 1 for C and R, respectively, where kC = ⌈ 1 n+1 ( n+d d )⌉ except for d = 2 and four other special cases. For Fq, we show that ⌈ d n+d ( n+d d )⌉ queries suffice to achieve probability approaching 1 for large field order q. The classical query complexity of this problem is ( n+d d ), so our result provides a speedup by a factor of n + 1, n+1 2 , and n+d d for C, R, and Fq, respectively. Thus we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of 2. For the case of Fq, we conjecture that 2kC queries also suffice to achieve probability approaching 1 for large field order q, although we leave this as an open problem.

1 aChen, Jianxin1 aChilds, Andrew, M.1 aHung, Shih-Han uhttp://rspa.royalsocietypublishing.org/content/474/2209/2017048001648nas a2200169 4500008004100000245006100041210006100102300001400163490000900177520116300186100002301349700001901372700001801391700001901409700001301428856003701441 2018 eng d00aToward the first quantum simulation with quantum speedup0 aToward the first quantum simulation with quantum speedup a9456-94610 v115 3 a

With quantum computers of significant size now on the horizon, we should understand how to best exploit their initially limited abilities. To this end, we aim to identify a practical problem that is beyond the reach of current classical computers, but that requires the fewest resources for a quantum computer. We consider quantum simulation of spin systems, which could be applied to understand condensed matter phenomena. We synthesize explicit circuits for three leading quantum simulation algorithms, using diverse techniques to tighten error bounds and optimize circuit implementations. Quantum signal processing appears to be preferred among algorithms with rigorous performance guarantees, whereas higher-order product formulas prevail if empirical error estimates suffice. Our circuits are orders of magnitude smaller than those for the simplest classically infeasible instances of factoring and quantum chemistry, bringing practical quantum computation closer to reality.

1 aChilds, Andrew, M.1 aMaslov, Dmitri1 aNam, Yunseong1 aRoss, Neil, J.1 aSu, Yuan uhttps://arxiv.org/abs/1711.1098001386nas a2200145 4500008004100000245006200041210006200103260001500165300001200180490000700192520096400199100002301163700001701186856003701203 2017 eng d00aEfficient simulation of sparse Markovian quantum dynamics0 aEfficient simulation of sparse Markovian quantum dynamics c2017/09/01 a901-9470 v173 a

Quantum algorithms for simulating Hamiltonian dynamics have been extensively developed, but there has been much less work on quantum algorithms for simulating the dynamics of open quantum systems. We give the first efficient quantum algorithms for simulating Markovian quantum dynamics generated by Lindbladians that are not necessarily local. We introduce two approaches to simulating sparse Lindbladians. First, we show how to simulate Lindbladians that act within small invariant subspaces using a quantum algorithm to implement sparse Stinespring isometries. Second, we develop a method for simulating sparse Lindblad operators by concatenating a sequence of short-time evolutions. We also show limitations on Lindbladian simulation by proving a no-fast-forwarding theorem for simulating sparse Lindbladians in a black-box model.

1 aChilds, Andrew, M.1 aLi, Tongyang uhttps://arxiv.org/abs/1611.0554301429nas a2200169 4500008004100000245010800041210006900149260001500218300001400233490000800247520088200255100002301137700002301160700002101183700001801204856003701222 2017 eng d00aQuantum algorithm for linear differential equations with exponentially improved dependence on precision0 aQuantum algorithm for linear differential equations with exponen c2017/12/01 a1057-10810 v3563 a

We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.

1 aBerry, Dominic, W.1 aChilds, Andrew, M.1 aOstrander, Aaron1 aWang, Guoming uhttps://arxiv.org/abs/1701.0368401362nas a2200157 4500008004100000245010600041210006900147260001500216300001400231490000700245520083800252100002301090700001901113700002301132856004901155 2017 eng d00aQuantum algorithm for systems of linear equations with exponentially improved dependence on precision0 aQuantum algorithm for systems of linear equations with exponenti c2017/12/21 a1920-19500 v463 a

Harrow, Hassidim, and Lloyd showed that for a suitably specified N×N matrix A and N-dimensional vector b⃗ , there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations Ax⃗ =b⃗ . If A is sparse and well-conditioned, their algorithm runs in time poly(logN,1/ϵ), where ϵ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in log(1/ϵ), exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on ϵ is prohibitive.

1 aChilds, Andrew, M.1 aKothari, Robin1 aSomma, Rolando, D. uhttp://epubs.siam.org/doi/10.1137/16M108707200953nas a2200157 4500008004100000245006400041210006400105260001500169300000900184490000700193520050200200100002300702700001800725700001400743856003800757 2016 eng d00aComplexity of the XY antiferromagnet at fixed magnetization0 aComplexity of the XY antiferromagnet at fixed magnetization c2016/01/01 a1-180 v163 a We prove that approximating the ground energy of the antiferromagnetic XY model on a simple graph at fixed magnetization (given as part of the instance specification) is QMA-complete. To show this, we strengthen a previous result by establishing QMA-completeness for approximating the ground energy of the Bose-Hubbard model on simple graphs. Using a connection between the XY and Bose-Hubbard models that we exploited in previous work, this establishes QMA-completeness of the XY model. 1 aChilds, Andrew, M.1 aGosset, David1 aWebb, Zak uhttp://arxiv.org/abs/1503.07083v101532nas a2200193 4500008004100000020002200041022001400063245005900077210005900136260001500195300001600210490000700226520098400233100002301217700001701240700001901257700002601276856003601302 2016 eng d a978-3-95977-013-2 a1868-896900aOptimal quantum algorithm for polynomial interpolation0 aOptimal quantum algorithm for polynomial interpolation c2016/03/01 a16:1--16:130 v553 a

We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over GF(q). A lower bound shown independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2+1/2 quantum queries are needed to solve this problem with bounded error, whereas an algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We show that the lower bound is achievable: d/2+1/2 quantum queries suffice to determine the polynomial with bounded error. Furthermore, we show that d/2+1 queries suffice to achieve probability approaching 1 for large q. These upper bounds improve results of Boneh and Zhandry on the insecurity of cryptographic protocols against quantum attacks. We also show that our algorithm's success probability as a function of the number of queries is precisely optimal. Furthermore, the algorithm can be implemented with gate complexity poly(log q) with negligible decrease in the success probability.

1 aChilds, Andrew, M.1 avan Dam, Wim1 aHung, Shih-Han1 aShparlinski, Igor, E. uhttp://arxiv.org/abs/1509.0927101395nas a2200145 4500008004100000245008900041210006900130260001500199300001100214490000700225520094000232100002301172700001801195856003601213 2016 eng d00aOptimal state discrimination and unstructured search in nonlinear quantum mechanics0 aOptimal state discrimination and unstructured search in nonlinea c2016/02/11 a0223140 v933 a Nonlinear variants of quantum mechanics can solve tasks that are impossible in standard quantum theory, such as perfectly distinguishing nonorthogonal states. Here we derive the optimal protocol for distinguishing two states of a qubit using the Gross-Pitaevskii equation, a model of nonlinear quantum mechanics that arises as an effective description of Bose-Einstein condensates. Using this protocol, we present an algorithm for unstructured search in the Gross-Pitaevskii model, obtaining an exponential improvement over a previous algorithm of Meyer and Wong. This result establishes a limitation on the effectiveness of the Gross-Pitaevskii approximation. More generally, we demonstrate similar behavior under a family of related nonlinearities, giving evidence that the ability to quickly discriminate nonorthogonal states and thereby solve unstructured search is a generic feature of nonlinear quantum mechanics. 1 aChilds, Andrew, M.1 aYoung, Joshua uhttp://arxiv.org/abs/1507.0633401453nas a2200145 4500008004100000245007600041210006900117260001500186300001200201520099300213100002301206700002301229700001901252856003601271 2015 eng d00aHamiltonian simulation with nearly optimal dependence on all parameters0 aHamiltonian simulation with nearly optimal dependence on all par c2015/01/08 a792-8093 a We present an algorithm for sparse Hamiltonian simulation that has optimal dependence on all parameters of interest (up to log factors). Previous algorithms had optimal or near-optimal scaling in some parameters at the cost of poor scaling in others. Hamiltonian simulation via a quantum walk has optimal dependence on the sparsity $d$ at the expense of poor scaling in the allowed error $\epsilon$. In contrast, an approach based on fractional-query simulation provides optimal scaling in $\epsilon$ at the expense of poor scaling in $d$. Here we combine the two approaches, achieving the best features of both. By implementing a linear combination of quantum walk steps with coefficients given by Bessel functions, our algorithm achieves near-linear scaling in $\tau := d \|H\|_{\max} t$ and sublogarithmic scaling in $1/\epsilon$. Our dependence on $\epsilon$ is optimal, and we prove a new lower bound showing that no algorithm can have sublinear dependence on $\tau$. 1 aBerry, Dominic, W.1 aChilds, Andrew, M.1 aKothari, Robin uhttp://arxiv.org/abs/1501.0171500877nas a2200181 4500008004100000245002200041210002200063260001500085300001200100490000700112520044800119100002300567700001800590700001800608700001800626700001400644856003700658 2015 eng d00aMomentum switches0 aMomentum switches c2015/05/01 a601-6210 v153 a Certain continuous-time quantum walks can be viewed as scattering processes. These processes can perform quantum computations, but it is challenging to design graphs with desired scattering behavior. In this paper, we study and construct momentum switches, graphs that route particles depending on their momenta. We also give an example where there is no exact momentum switch, although we construct an arbitrarily good approximation. 1 aChilds, Andrew, M.1 aGosset, David1 aNagaj, Daniel1 aRaha, Mouktik1 aWebb, Zak uhttp://arxiv.org/abs/1406.4510v101120nas a2200181 4500008004100000245006700041210006700108260001500175300001100190490000800201520058500209100002300794700002300817700001900840700001900859700002300878856003700901 2015 eng d00aSimulating Hamiltonian dynamics with a truncated Taylor series0 aSimulating Hamiltonian dynamics with a truncated Taylor series c2015/03/03 a0905020 v1143 a We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations to directly apply the truncated Taylor series. 1 aBerry, Dominic, W.1 aChilds, Andrew, M.1 aCleve, Richard1 aKothari, Robin1 aSomma, Rolando, D. uhttp://arxiv.org/abs/1412.4687v101367nas a2200157 4500008004100000245004300041210003700084260001500121300001200136490000900148520096000157100002301117700001801140700001401158856003701172 2014 eng d00aThe Bose-Hubbard model is QMA-complete0 aBoseHubbard model is QMAcomplete c2014/07/08 a308-3190 v85723 a The Bose-Hubbard model is a system of interacting bosons that live on the vertices of a graph. The particles can move between adjacent vertices and experience a repulsive on-site interaction. The Hamiltonian is determined by a choice of graph that specifies the geometry in which the particles move and interact. We prove that approximating the ground energy of the Bose-Hubbard model on a graph at fixed particle number is QMA-complete. In our QMA-hardness proof, we encode the history of an n-qubit computation in the subspace with at most one particle per site (i.e., hard-core bosons). This feature, along with the well-known mapping between hard-core bosons and spin systems, lets us prove a related result for a class of 2-local Hamiltonians defined by graphs that generalizes the XY model. By avoiding the use of perturbation theory in our analysis, we circumvent the need to multiply terms in the Hamiltonian by large coefficients. 1 aChilds, Andrew, M.1 aGosset, David1 aWebb, Zak uhttp://arxiv.org/abs/1311.3297v101064nas a2200145 4500008004100000245008600041210006900127260001500196300001200211490000700223520060700230100002100837700002300858856003700881 2014 eng d00aThe computational power of matchgates and the XY interaction on arbitrary graphs0 acomputational power of matchgates and the XY interaction on arbi c2014/09/01 a901-9160 v143 a Matchgates are a restricted set of two-qubit gates known to be classically simulable when acting on nearest-neighbor qubits on a path, but universal for quantum computation when the qubits are arranged on certain other graphs. Here we characterize the power of matchgates acting on arbitrary graphs. Specifically, we show that they are universal on any connected graph other than a path or a cycle, and that they are classically simulable on a cycle. We also prove the same dichotomy for the XY interaction, a proper subset of matchgates related to some implementations of quantum computing. 1 aBrod, Daniel, J.1 aChilds, Andrew, M. uhttp://arxiv.org/abs/1308.1463v101598nas a2200157 4500008004100000245007300041210006900114260001500183300001100198490000600209520112600215100002301341700001501364700002401379856003701403 2014 eng d00aConstructing elliptic curve isogenies in quantum subexponential time0 aConstructing elliptic curve isogenies in quantum subexponential  c2014/01/01 a1 - 290 v83 a Given two elliptic curves over a finite field having the same cardinality and endomorphism ring, it is known that the curves admit an isogeny between them, but finding such an isogeny is believed to be computationally difficult. The fastest known classical algorithm takes exponential time, and prior to our work no faster quantum algorithm was known. Recently, public-key cryptosystems based on the presumed hardness of this problem have been proposed as candidates for post-quantum cryptography. In this paper, we give a subexponential-time quantum algorithm for constructing isogenies, assuming the Generalized Riemann Hypothesis (but with no other assumptions). Our algorithm is based on a reduction to a hidden shift problem, together with a new subexponential-time algorithm for evaluating isogenies from kernel ideals (under only GRH), and represents the first nontrivial application of Kuperberg's quantum algorithm for the hidden shift problem. This result suggests that isogeny-based cryptosystems may be uncompetitive with more mainstream quantum-resistant cryptosystems such as lattice-based cryptosystems. 1 aChilds, Andrew, M.1 aJao, David1 aSoukharev, Vladimir uhttp://arxiv.org/abs/1012.4019v202001nas a2200181 4500008004100000020002200041245007600063210006900139260001500208300001200223520144000235100002301675700002301698700001901721700001901740700002301759856003701782 2014 eng d a978-1-4503-2710-700aExponential improvement in precision for simulating sparse Hamiltonians0 aExponential improvement in precision for simulating sparse Hamil c2014/05/31 a283-2923 a We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a $d$-sparse Hamiltonian $H$ acting on $n$ qubits can be simulated for time $t$ with precision $\epsilon$ using $O\big(\tau \frac{\log(\tau/\epsilon)}{\log\log(\tau/\epsilon)}\big)$ queries and $O\big(\tau \frac{\log^2(\tau/\epsilon)}{\log\log(\tau/\epsilon)}n\big)$ additional 2-qubit gates, where $\tau = d^2 \|{H}\|_{\max} t$. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Our simplification relies on a new form of "oblivious amplitude amplification" that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error. 1 aBerry, Dominic, W.1 aChilds, Andrew, M.1 aCleve, Richard1 aKothari, Robin1 aSomma, Rolando, D. uhttp://arxiv.org/abs/1312.1414v201152nas a2200133 4500008004100000245006100041210006100102260001400163490000600177520075500183100002300938700002000961856003700981 2014 eng d00aQuantum computation of discrete logarithms in semigroups0 aQuantum computation of discrete logarithms in semigroups c2014/01/10 v83 a We describe an efficient quantum algorithm for computing discrete logarithms in semigroups using Shor's algorithms for period finding and discrete log as subroutines. Thus proposed cryptosystems based on the presumed hardness of discrete logarithms in semigroups are insecure against quantum attacks. In contrast, we show that some generalizations of the discrete log problem are hard in semigroups despite being easy in groups. We relate a shifted version of the discrete log problem in semigroups to the dihedral hidden subgroup problem, and we show that the constructive membership problem with respect to $k \ge 2$ generators in a black-box abelian semigroup of order $N$ requires $\tilde \Theta(N^{\frac{1}{2}-\frac{1}{2k}})$ quantum queries. 1 aChilds, Andrew, M.1 aIvanyos, Gábor uhttp://arxiv.org/abs/1310.6238v201146nas a2200133 4500008004100000245007200041210006900113260001400182490000700196520073500203100002300938700001400961856003700975 2014 eng d00aSpatial search by continuous-time quantum walks on crystal lattices0 aSpatial search by continuoustime quantum walks on crystal lattic c2014/5/300 v893 a We consider the problem of searching a general $d$-dimensional lattice of $N$ vertices for a single marked item using a continuous-time quantum walk. We demand locality, but allow the walk to vary periodically on a small scale. By constructing lattice Hamiltonians exhibiting Dirac points in their dispersion relations and exploiting the linear behaviour near a Dirac point, we develop algorithms that solve the problem in a time of $O(\sqrt N)$ for $d>2$ and $O(\sqrt N \log N)$ in $d=2$. In particular, we show that such algorithms exist even for hypercubic lattices in any dimension. Unlike previous continuous-time quantum walk algorithms on hypercubic lattices in low dimensions, our approach does not use external memory. 1 aChilds, Andrew, M.1 aGe, Yimin uhttp://arxiv.org/abs/1403.2676v201395nas a2200169 4500008004100000245006500041210006500106260001500171300001000186490000700196520090400203100002301107700001901130700001701149700002201166856003701188 2013 eng d00aEasy and hard functions for the Boolean hidden shift problem0 aEasy and hard functions for the Boolean hidden shift problem c2013/04/16 a50-790 v223 a We study the quantum query complexity of the Boolean hidden shift problem. Given oracle access to f(x+s) for a known Boolean function f, the task is to determine the n-bit string s. The quantum query complexity of this problem depends strongly on f. We demonstrate that the easiest instances of this problem correspond to bent functions, in the sense that an exact one-query algorithm exists if and only if the function is bent. We partially characterize the hardest instances, which include delta functions. Moreover, we show that the problem is easy for random functions, since two queries suffice. Our algorithm for random functions is based on performing the pretty good measurement on several copies of a certain state; its analysis relies on the Fourier transform. We also use this approach to improve the quantum rejection sampling approach to the Boolean hidden shift problem. 1 aChilds, Andrew, M.1 aKothari, Robin1 aOzols, Maris1 aRoetteler, Martin uhttp://arxiv.org/abs/1304.4642v101336nas a2200169 4500008004100000245006500041210006300106260001300169300001600182490000800198520084400206100002301050700001801073700002101091700001701112856003701129 2013 eng d00aA framework for bounding nonlocality of state discrimination0 aframework for bounding nonlocality of state discrimination c2013/9/4 a1121 - 11530 v3233 a We consider the class of protocols that can be implemented by local quantum operations and classical communication (LOCC) between two parties. In particular, we focus on the task of discriminating a known set of quantum states by LOCC. Building on the work in the paper "Quantum nonlocality without entanglement" [BDF+99], we provide a framework for bounding the amount of nonlocality in a given set of bipartite quantum states in terms of a lower bound on the probability of error in any LOCC discrimination protocol. We apply our framework to an orthonormal product basis known as the domino states and obtain an alternative and simplified proof that quantifies its nonlocality. We generalize this result for similar bases in larger dimensions, as well as the "rotated" domino states, resolving a long-standing open question [BDF+99]. 1 aChilds, Andrew, M.1 aLeung, Debbie1 aMancinska, Laura1 aOzols, Maris uhttp://arxiv.org/abs/1206.5822v100982nas a2200169 4500008004100000245008200041210006900123260001500192300001100207490000700218520047100225100002300696700001800719700002100737700001700758856003700775 2013 eng d00aInterpolatability distinguishes LOCC from separable von Neumann measurements0 aInterpolatability distinguishes LOCC from separable von Neumann  c2013/06/25 a1122040 v543 a Local operations with classical communication (LOCC) and separable operations are two classes of quantum operations that play key roles in the study of quantum entanglement. Separable operations are strictly more powerful than LOCC, but no simple explanation of this phenomenon is known. We show that, in the case of von Neumann measurements, the ability to interpolate measurements is an operational principle that sets apart LOCC and separable operations. 1 aChilds, Andrew, M.1 aLeung, Debbie1 aMancinska, Laura1 aOzols, Maris uhttp://arxiv.org/abs/1306.5992v101289nas a2200145 4500008004100000245005300041210005300094260001500147300001100162490000700173520088500180100002301065700001801088856003701106 2013 eng d00aProduct Formulas for Exponentials of Commutators0 aProduct Formulas for Exponentials of Commutators c2013/02/07 a0622020 v543 a We provide a recursive method for constructing product formula approximations to exponentials of commutators, giving the first approximations that are accurate to arbitrarily high order. Using these formulas, we show how to approximate unitary exponentials of (possibly nested) commutators using exponentials of the elementary operators, and we upper bound the number of elementary exponentials needed to implement the desired operation within a given error tolerance. By presenting an algorithm for quantum search using evolution according to a commutator, we show that the scaling of the number of exponentials in our product formulas with the evolution time is nearly optimal. Finally, we discuss applications of our product formulas to quantum control and to implementing anticommutators, providing new methods for simulating many-body interaction Hamiltonians. 1 aChilds, Andrew, M.1 aWiebe, Nathan uhttp://arxiv.org/abs/1211.4945v200867nas a2200145 4500008004100000245007400041210006900115260001500184520040100199100002300600700002000623700001900643700002200662856003700684 2013 eng d00aA Time-Efficient Quantum Walk for 3-Distinctness Using Nested Updates0 aTimeEfficient Quantum Walk for 3Distinctness Using Nested Update c2013/02/283 a We present an extension to the quantum walk search framework that facilitates quantum walks with nested updates. We apply it to give a quantum walk algorithm for 3-Distinctness with query complexity ~O(n^{5/7}), matching the best known upper bound (obtained via learning graphs) up to log factors. Furthermore, our algorithm has time complexity ~O(n^{5/7}), improving the previous ~O(n^{3/4}). 1 aChilds, Andrew, M.1 aJeffery, Stacey1 aKothari, Robin1 aMagniez, Frederic uhttp://arxiv.org/abs/1302.7316v101608nas a2200157 4500008004100000245005700041210005600098260001500154300001400169490000800183520116700191100002301358700001801381700001401399856003701413 2013 eng d00aUniversal computation by multi-particle quantum walk0 aUniversal computation by multiparticle quantum walk c2013/02/14 a791 - 7940 v3393 a A quantum walk is a time-homogeneous quantum-mechanical process on a graph defined by analogy to classical random walk. The quantum walker is a particle that moves from a given vertex to adjacent vertices in quantum superposition. Here we consider a generalization of quantum walk to systems with more than one walker. A continuous-time multi-particle quantum walk is generated by a time-independent Hamiltonian with a term corresponding to a single-particle quantum walk for each particle, along with an interaction term. Multi-particle quantum walk includes a broad class of interacting many-body systems such as the Bose-Hubbard model and systems of fermions or distinguishable particles with nearest-neighbor interactions. We show that multi-particle quantum walk is capable of universal quantum computation. Since it is also possible to efficiently simulate a multi-particle quantum walk of the type we consider using a universal quantum computer, this model exactly captures the power of quantum computation. In principle our construction could be used as an architecture for building a scalable quantum computer with no need for time-dependent control. 1 aChilds, Andrew, M.1 aGosset, David1 aWebb, Zak uhttp://arxiv.org/abs/1205.3782v201023nas a2200145 4500008004100000245007500041210006900116260001500185300001200200490000700212520058000219100002300799700001800822856003700840 2012 eng d00aHamiltonian Simulation Using Linear Combinations of Unitary Operations0 aHamiltonian Simulation Using Linear Combinations of Unitary Oper c2012/11/01 a901-9240 v123 a We present a new approach to simulating Hamiltonian dynamics based on implementing linear combinations of unitary operations rather than products of unitary operations. The resulting algorithm has superior performance to existing simulation algorithms based on product formulas and, most notably, scales better with the simulation error than any known Hamiltonian simulation technique. Our main tool is a general method to nearly deterministically implement linear combinations of nearby unitary operations, which we show is optimal among a large class of methods. 1 aChilds, Andrew, M.1 aWiebe, Nathan uhttp://arxiv.org/abs/1202.5822v100869nas a2200145 4500008004100000245003700041210003600078260001500114300001100129490000700140520049800147100002300645700001800668856003700686 2012 eng d00aLevinson's theorem for graphs II0 aLevinsons theorem for graphs II c2012/11/21 a1022070 v533 a We prove Levinson's theorem for scattering on an (m+n)-vertex graph with n semi-infinite paths each attached to a different vertex, generalizing a previous result for the case n=1. This theorem counts the number of bound states in terms of the winding of the determinant of the S-matrix. We also provide a proof that the bound states and incoming scattering states of the Hamiltonian together form a complete basis for the Hilbert space, generalizing another result for the case n=1. 1 aChilds, Andrew, M.1 aGosset, David uhttp://arxiv.org/abs/1203.6557v201823nas a2200157 4500008004100000245005500041210005000096260001500146300001200161490000900173520138500182100002301567700001901590700001901609856003701628 2012 eng d00aThe quantum query complexity of read-many formulas0 aquantum query complexity of readmany formulas c2012/09/10 a337-3480 v75013 a The quantum query complexity of evaluating any read-once formula with n black-box input bits is Theta(sqrt(n)). However, the corresponding problem for read-many formulas (i.e., formulas in which the inputs have fanout) is not well understood. Although the optimal read-once formula evaluation algorithm can be applied to any formula, it can be suboptimal if the inputs have large fanout. We give an algorithm for evaluating any formula with n inputs, size S, and G gates using O(min{n, sqrt{S}, n^{1/2} G^{1/4}}) quantum queries. Furthermore, we show that this algorithm is optimal, since for any n,S,G there exists a formula with n inputs, size at most S, and at most G gates that requires Omega(min{n, sqrt{S}, n^{1/2} G^{1/4}}) queries. We also show that the algorithm remains nearly optimal for circuits of any particular depth k >= 3, and we give a linear-size circuit of depth 2 that requires Omega (n^{5/9}) queries. Applications of these results include a Omega (n^{19/18}) lower bound for Boolean matrix product verification, a nearly tight characterization of the quantum query complexity of evaluating constant-depth circuits with bounded fanout, new formula gate count lower bounds for several functions including PARITY, and a construction of an AC^0 circuit of linear size that can only be evaluated by a formula with Omega(n^{2-epsilon}) gates. 1 aChilds, Andrew, M.1 aKimmel, Shelby1 aKothari, Robin uhttp://arxiv.org/abs/1112.0548v100769nas a2200145 4500008004100000245003400041210003300075260001500108300001100123490000700134520040600141100002300547700001600570856003700586 2011 eng d00aLevinson's theorem for graphs0 aLevinsons theorem for graphs c2011/06/30 a0821020 v523 a We prove an analog of Levinson's theorem for scattering on a weighted (m+1)-vertex graph with a semi-infinite path attached to one of its vertices. In particular, we show that the number of bound states in such a scattering problem is equal to m minus half the winding number of the phase of the reflection coefficient (where each so-called half-bound state is counted as half a bound state). 1 aChilds, Andrew, M.1 aStrouse, DJ uhttp://arxiv.org/abs/1103.5077v201380nas a2200145 4500008004100000245006200041210006100103260001500164300001200179490000600191520095800197100002301155700001901178856003701197 2011 eng d00aQuantum query complexity of minor-closed graph properties0 aQuantum query complexity of minorclosed graph properties c2011/01/01 a661-6720 v93 a We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems. 1 aChilds, Andrew, M.1 aKothari, Robin uhttp://arxiv.org/abs/1011.1443v201458nas a2200145 4500008004100000245005700041210005600098260001500154520102700169100002301196700001801219700002101237700001701258856003701275 2010 eng d00aCharacterization of universal two-qubit Hamiltonians0 aCharacterization of universal twoqubit Hamiltonians c2010/04/093 a Suppose we can apply a given 2-qubit Hamiltonian H to any (ordered) pair of qubits. We say H is n-universal if it can be used to approximate any unitary operation on n qubits. While it is well known that almost any 2-qubit Hamiltonian is 2-universal (Deutsch, Barenco, Ekert 1995; Lloyd 1995), an explicit characterization of the set of non-universal 2-qubit Hamiltonians has been elusive. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. In particular, there are three ways that a 2-qubit Hamiltonian H can fail to be universal: (1) H shares an eigenvector with the gate that swaps two qubits, (2) H acts on the two qubits independently (in any of a certain family of bases), or (3) H has zero trace. A 2-non-universal 2-qubit Hamiltonian can still be n-universal for some n >= 3. We give some partial results on 3-universality. Finally, we also show how our characterization of 2-universal Hamiltonians implies the well-known result that almost any 2-qubit unitary is universal. 1 aChilds, Andrew, M.1 aLeung, Debbie1 aMancinska, Laura1 aOzols, Maris uhttp://arxiv.org/abs/1004.1645v201122nas a2200145 4500008004100000245004600041210004600087260001400133300001100147490000700158520073400165100002300899700001700922856003700939 2010 eng d00aQuantum algorithms for algebraic problems0 aQuantum algorithms for algebraic problems c2010/1/15 a1 - 520 v823 a Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation, and in particular, on problems with an algebraic flavor. 1 aChilds, Andrew, M.1 avan Dam, Wim uhttp://arxiv.org/abs/0812.0380v101146nas a2200145 4500008004100000245005500041210005400096260001500150300001200165520072600177100002100903700002300924700001600947856003700963 2010 eng d00aQuantum property testing for bounded-degree graphs0 aQuantum property testing for boundeddegree graphs c2010/12/14 a365-3763 a We study quantum algorithms for testing bipartiteness and expansion of bounded-degree graphs. We give quantum algorithms that solve these problems in time O(N^(1/3)), beating the Omega(sqrt(N)) classical lower bound. For testing expansion, we also prove an Omega(N^(1/4)) quantum query lower bound, thus ruling out the possibility of an exponential quantum speedup. Our quantum algorithms follow from a combination of classical property testing techniques due to Goldreich and Ron, derandomization, and the quantum algorithm for element distinctness. The quantum lower bound is obtained by the polynomial method, using novel algebraic techniques and combinatorial analysis to accommodate the graph structure. 1 aAmbainis, Andris1 aChilds, Andrew, M.1 aLiu, Yi-Kai uhttp://arxiv.org/abs/1012.3174v301937nas a2200133 4500008004100000245007500041210006600116260001500182300001400197490000800211520152400219100002301743856003701766 2010 eng d00aOn the relationship between continuous- and discrete-time quantum walk0 arelationship between continuous and discretetime quantum walk c2009/10/10 a581 - 6030 v2943 a Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case. In this article, I describe a precise correspondence between continuous- and discrete-time quantum walks on arbitrary graphs. Using this correspondence, I show that continuous-time quantum walk can be obtained as an appropriate limit of discrete-time quantum walks. The correspondence also leads to a new technique for simulating Hamiltonian dynamics, giving efficient simulations even in cases where the Hamiltonian is not sparse. The complexity of the simulation is linear in the total evolution time, an improvement over simulations based on high-order approximations of the Lie product formula. As applications, I describe a continuous-time quantum walk algorithm for element distinctness and show how to optimally simulate continuous-time query algorithms of a certain form in the conventional quantum query model. Finally, I discuss limitations of the method for simulating Hamiltonians with negative matrix elements, and present two problems that motivate attempting to circumvent these limitations. 1 aChilds, Andrew, M. uhttp://arxiv.org/abs/0810.0312v301041nas a2200121 4500008004100000245006000041210006000101260001500161520066400176100002300840700001900863856003700882 2010 eng d00aSimulating sparse Hamiltonians with star decompositions0 aSimulating sparse Hamiltonians with star decompositions c2010/03/183 a We present an efficient algorithm for simulating the time evolution due to a sparse Hamiltonian. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian H acts for time t, this algorithm uses (d^2(d+log* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log* N)||Ht||)^{1+o(1)}. To achieve this, we decompose a general sparse Hamiltonian into a small sum of Hamiltonians whose graphs of non-zero entries have the property that every connected component is a star, and efficiently simulate each of these pieces. 1 aChilds, Andrew, M.1 aKothari, Robin uhttp://arxiv.org/abs/1003.3683v201436nas a2200121 4500008004100000245006400041210006300105260001500168520104800183100002301231700002301254856003701277 2009 eng d00aBlack-box Hamiltonian simulation and unitary implementation0 aBlackbox Hamiltonian simulation and unitary implementation c2009/10/223 a We present general methods for simulating black-box Hamiltonians using quantum walks. These techniques have two main applications: simulating sparse Hamiltonians and implementing black-box unitary operations. In particular, we give the best known simulation of sparse Hamiltonians with constant precision. Our method has complexity linear in both the sparseness D (the maximum number of nonzero elements in a column) and the evolution time t, whereas previous methods had complexity scaling as D^4 and were superlinear in t. We also consider the task of implementing an arbitrary unitary operation given a black-box description of its matrix elements. Whereas standard methods for performing an explicitly specified N x N unitary operation use O(N^2) elementary gates, we show that a black-box unitary can be performed with bounded error using O(N^{2/3} (log log N)^{4/3}) queries to its matrix elements. In fact, except for pathological cases, it appears that most unitaries can be performed with only O(sqrt{N}) queries, which is optimal. 1 aBerry, Dominic, W.1 aChilds, Andrew, M. uhttp://arxiv.org/abs/0910.4157v400832nas a2200169 4500008004100000245005200041210005100093260001500144300001400159490000600173520035600179100002300535700001900558700002400577700001700601856004400618 2009 eng d00aDiscrete-query quantum algorithm for NAND trees0 aDiscretequery quantum algorithm for NAND trees c2009/07/01 a119 - 1230 v53 a Recently, Farhi, Goldstone, and Gutmann gave a quantum algorithm for evaluating NAND trees that runs in time O(sqrt(N log N)) in the Hamiltonian query model. In this note, we point out that their algorithm can be converted into an algorithm using O(N^{1/2 + epsilon}) queries in the conventional quantum query model, for any fixed epsilon > 0. 1 aChilds, Andrew, M.1 aCleve, Richard1 aJordan, Stephen, P.1 aYeung, David uhttp://arxiv.org/abs/quant-ph/0702160v101322nas a2200121 4500008004100000245006100041210006000102260001500162520094400177100002301121700001901144856003701163 2009 eng d00aLimitations on the simulation of non-sparse Hamiltonians0 aLimitations on the simulation of nonsparse Hamiltonians c2009/08/313 a The problem of simulating sparse Hamiltonians on quantum computers is well studied. The evolution of a sparse N x N Hamiltonian H for time t can be simulated using O(||Ht||poly(log N)) operations, which is essentially optimal due to a no--fast-forwarding theorem. Here, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, ruling out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian $H$. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walk cannot be dramatically improved in general. On the positive side, we show that some non-sparse Hamiltonians can be simulated efficiently, such as those with graphs of small arboricity. 1 aChilds, Andrew, M.1 aKothari, Robin uhttp://arxiv.org/abs/0908.4398v200937nas a2200145 4500008004100000245005000041210004600091260001500137520051500152100002100667700002300688700002300711700002000734856003700754 2009 eng d00aThe quantum query complexity of certification0 aquantum query complexity of certification c2009/03/063 a We study the quantum query complexity of finding a certificate for a d-regular, k-level balanced NAND formula. Up to logarithmic factors, we show that the query complexity is Theta(d^{(k+1)/2}) for 0-certificates, and Theta(d^{k/2}) for 1-certificates. In particular, this shows that the zero-error quantum query complexity of evaluating such formulas is O(d^{(k+1)/2}) (again neglecting a logarithmic factor). Our lower bound relies on the fact that the quantum adversary method obeys a direct sum theorem. 1 aAmbainis, Andris1 aChilds, Andrew, M.1 aLe Gall, François1 aTani, Seiichiro uhttp://arxiv.org/abs/0903.1291v200931nas a2200121 4500008004100000245004200041210004200083260001300125490000800138520060300146100002300749856003700772 2009 eng d00aUniversal computation by quantum walk0 aUniversal computation by quantum walk c2009/5/40 v1023 a In some of the earliest work on quantum mechanical computers, Feynman showed how to implement universal quantum computation by the dynamics of a time-independent Hamiltonian. I show that this remains possible even if the Hamiltonian is restricted to be a sparse matrix with all entries equal to 0 or 1, i.e., the adjacency matrix of a low-degree graph. Thus quantum walk can be regarded as a universal computational primitive, with any desired quantum computation encoded entirely in some underlying graph. The main idea of the construction is to implement quantum gates by scattering processes. 1 aChilds, Andrew, M. uhttp://arxiv.org/abs/0806.1972v100984nas a2200145 4500008004100000245009600041210006900137260001500206520048900221100002300710700002300733700001900756700001900775856004400794 2007 eng d00aEvery NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer 0 aEvery NAND formula of size N can be evaluated in time N 12o1 on  c2007/03/023 a For every NAND formula of size N, there is a bounded-error N^{1/2+o(1)}-time quantum algorithm, based on a coined quantum walk, that evaluates this formula on a black-box input. Balanced, or ``approximately balanced,'' NAND formulas can be evaluated in O(sqrt{N}) queries, which is optimal. It follows that the (2-o(1))-th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy. 1 aChilds, Andrew, M.1 aReichardt, Ben, W.1 aSpalek, Robert1 aZhang, Shengyu uhttp://arxiv.org/abs/quant-ph/0703015v301273nas a2200145 4500008004100000245009500041210006900136260001400205490000700219520078700226100002301013700002401036700002301060856004401083 2007 eng d00aImproved quantum algorithms for the ordered search problem via semidefinite programming 0 aImproved quantum algorithms for the ordered search problem via s c2007/3/260 v753 a One of the most basic computational problems is the task of finding a desired item in an ordered list of N items. While the best classical algorithm for this problem uses log_2 N queries to the list, a quantum computer can solve the problem using a constant factor fewer queries. However, the precise value of this constant is unknown. By characterizing a class of quantum query algorithms for ordered search in terms of a semidefinite program, we find new quantum algorithms for small instances of the ordered search problem. Extending these algorithms to arbitrarily large instances using recursion, we show that there is an exact quantum ordered search algorithm using 4 log_{605} N \approx 0.433 log_2 N queries, which improves upon the previously best known exact algorithm. 1 aChilds, Andrew, M.1 aLandahl, Andrew, J.1 aParrilo, Pablo, A. uhttp://arxiv.org/abs/quant-ph/0608161v101144nas a2200157 4500008004100000245005200041210004800093260001400141300001400155490000700169520070100176100002400877700002300901700001800924856004400942 2007 eng d00aThe limitations of nice mutually unbiased bases0 alimitations of nice mutually unbiased bases c2006/7/11 a111 - 1230 v253 a Mutually unbiased bases of a Hilbert space can be constructed by partitioning a unitary error basis. We consider this construction when the unitary error basis is a nice error basis. We show that the number of resulting mutually unbiased bases can be at most one plus the smallest prime power contained in the dimension, and therefore that this construction cannot improve upon previous approaches. We prove this by establishing a correspondence between nice mutually unbiased bases and abelian subgroups of the index group of a nice error basis and then bounding the number of such subgroups. This bound also has implications for the construction of certain combinatorial objects called nets. 1 aAschbacher, Michael1 aChilds, Andrew, M.1 aWocjan, Pawel uhttp://arxiv.org/abs/quant-ph/0412066v101045nas a2200121 4500008004100000245006200041210006200103260001500165520066900180100002300849700001400872856003700886 2007 eng d00aOptimal quantum adversary lower bounds for ordered search0 aOptimal quantum adversary lower bounds for ordered search c2007/08/243 a The goal of the ordered search problem is to find a particular item in an ordered list of n items. Using the adversary method, Hoyer, Neerbek, and Shi proved a quantum lower bound for this problem of (1/pi) ln n + Theta(1). Here, we find the exact value of the best possible quantum adversary lower bound for a symmetrized version of ordered search (whose query complexity differs from that of the original problem by at most 1). Thus we show that the best lower bound for ordered search that can be proved by the adversary method is (1/pi) ln n + O(1). Furthermore, we show that this remains true for the generalized adversary method allowing negative weights. 1 aChilds, Andrew, M.1 aLee, Troy uhttp://arxiv.org/abs/0708.3396v100989nas a2200133 4500008004100000245005500041210005500096260001500151520057900166100002300745700002600768700002400794856003700818 2007 eng d00aQuantum algorithms for hidden nonlinear structures0 aQuantum algorithms for hidden nonlinear structures c2007/05/213 a Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures. 1 aChilds, Andrew, M.1 aSchulman, Leonard, J.1 aVazirani, Umesh, V. uhttp://arxiv.org/abs/0705.2784v101338nas a2200157 4500008004100000245004300041210004200084260001500126300001200141490000700153520091300160100002301073700002201096700001801118856004401136 2006 eng d00aTwo-way quantum communication channels0 aTwoway quantum communication channels c2006/02/01 a63 - 830 v043 a We consider communication between two parties using a bipartite quantum operation, which constitutes the most general quantum mechanical model of two-party communication. We primarily focus on the simultaneous forward and backward communication of classical messages. For the case in which the two parties share unlimited prior entanglement, we give inner and outer bounds on the achievable rate region that generalize classical results due to Shannon. In particular, using a protocol of Bennett, Harrow, Leung, and Smolin, we give a one-shot expression in terms of the Holevo information for the entanglement-assisted one-way capacity of a two-way quantum channel. As applications, we rederive two known additivity results for one-way channel capacities: the entanglement-assisted capacity of a general one-way channel, and the unassisted capacity of an entanglement-breaking one-way channel. 1 aChilds, Andrew, M.1 aLeung, Debbie, W.1 aLo, Hoi-Kwong uhttp://arxiv.org/abs/quant-ph/0506039v101297nas a2200133 4500008004100000245009900041210006900140260001500209520083300224100002301057700002101080700001801101856004401119 2006 eng d00aWeak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem 0 aWeak FourierSchur sampling the hidden subgroup problem and the q c2006/09/143 a Schur duality decomposes many copies of a quantum state into subspaces labeled by partitions, a decomposition with applications throughout quantum information theory. Here we consider applying Schur duality to the problem of distinguishing coset states in the standard approach to the hidden subgroup problem. We observe that simply measuring the partition (a procedure we call weak Schur sampling) provides very little information about the hidden subgroup. Furthermore, we show that under quite general assumptions, even a combination of weak Fourier sampling and weak Schur sampling fails to identify the hidden subgroup. We also prove tight bounds on how many coset states are required to solve the hidden subgroup problem by weak Schur sampling, and we relate this question to a quantum version of the collision problem. 1 aChilds, Andrew, M.1 aHarrow, Aram, W.1 aWocjan, Pawel uhttp://arxiv.org/abs/quant-ph/0609110v101370nas a2200133 4500008004100000245012700041210006900168260001500237520088400252100001601136700002301152700001701175856004401192 2005 eng d00aFrom optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups 0 aFrom optimal measurement to efficient quantum algorithms for the c2005/04/113 a We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probability of success and unitary implementation are closely related to an average-case algebraic problem. By solving this problem, we find efficient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form (Z_p)^r X| Z_p for fixed r (including the Heisenberg group, r=2). In particular, our results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP. 1 aBacon, Dave1 aChilds, Andrew, M.1 avan Dam, Wim uhttp://arxiv.org/abs/quant-ph/0504083v201943nas a2200133 4500008004100000245006600041210006600107260001500173520152100188100001601709700002301725700001701748856004401765 2005 eng d00aOptimal measurements for the dihedral hidden subgroup problem0 aOptimal measurements for the dihedral hidden subgroup problem c2005/01/103 a We consider the dihedral hidden subgroup problem as the problem of distinguishing hidden subgroup states. We show that the optimal measurement for solving this problem is the so-called pretty good measurement. We then prove that the success probability of this measurement exhibits a sharp threshold as a function of the density nu=k/log N, where k is the number of copies of the hidden subgroup state and 2N is the order of the dihedral group. In particular, for nu<1 the optimal measurement (and hence any measurement) identifies the hidden subgroup with a probability that is exponentially small in log N, while for nu>1 the optimal measurement identifies the hidden subgroup with a probability of order unity. Thus the dihedral group provides an example of a group G for which Omega(log|G|) hidden subgroup states are necessary to solve the hidden subgroup problem. We also consider the optimal measurement for determining a single bit of the answer, and show that it exhibits the same threshold. Finally, we consider implementing the optimal measurement by a quantum circuit, and thereby establish further connections between the dihedral hidden subgroup problem and average case subset sum problems. In particular, we show that an efficient quantum algorithm for a restricted version of the optimal measurement would imply an efficient quantum algorithm for the subset sum problem, and conversely, that the ability to quantum sample from subset sum solutions allows one to implement the optimal measurement. 1 aBacon, Dave1 aChilds, Andrew, M.1 avan Dam, Wim uhttp://arxiv.org/abs/quant-ph/0501044v201092nas a2200121 4500008004100000245006100041210006100102260001500163520070800178100002300886700001700909856004400926 2005 eng d00aQuantum algorithm for a generalized hidden shift problem0 aQuantum algorithm for a generalized hidden shift problem c2005/07/193 a Consider the following generalized hidden shift problem: given a function f on {0,...,M-1} x Z_N satisfying f(b,x)=f(b+1,x+s) for b=0,1,...,M-2, find the unknown shift s in Z_N. For M=N, this problem is an instance of the abelian hidden subgroup problem, which can be solved efficiently on a quantum computer, whereas for M=2, it is equivalent to the dihedral hidden subgroup problem, for which no efficient algorithm is known. For any fixed positive epsilon, we give an efficient (i.e., poly(log N)) quantum algorithm for this problem provided M > N^epsilon. The algorithm is based on the "pretty good measurement" and uses H. Lenstra's (classical) algorithm for integer programming as a subroutine. 1 aChilds, Andrew, M.1 avan Dam, Wim uhttp://arxiv.org/abs/quant-ph/0507190v101105nas a2200121 4500008004100000245009900041210006900140260001500209520067400224100002300898700001800921856004400939 2005 eng d00aOn the quantum hardness of solving isomorphism problems as nonabelian hidden shift problems 0 aquantum hardness of solving isomorphism problems as nonabelian h c2005/10/253 a We consider an approach to deciding isomorphism of rigid n-vertex graphs (and related isomorphism problems) by solving a nonabelian hidden shift problem on a quantum computer using the standard method. Such an approach is arguably more natural than viewing the problem as a hidden subgroup problem. We prove that the hidden shift approach to rigid graph isomorphism is hard in two senses. First, we prove that Omega(n) copies of the hidden shift states are necessary to solve the problem (whereas O(n log n) copies are sufficient). Second, we prove that if one is restricted to single-register measurements, an exponential number of hidden shift states are required. 1 aChilds, Andrew, M.1 aWocjan, Pawel uhttp://arxiv.org/abs/quant-ph/0510185v101341nas a2200145 4500008004100000245007700041210006900118260001400187490000700201520087300208100002301081700002201104700002501126856004401151 2005 eng d00aUnified derivations of measurement-based schemes for quantum computation0 aUnified derivations of measurementbased schemes for quantum comp c2005/3/170 v713 a We present unified, systematic derivations of schemes in the two known measurement-based models of quantum computation. The first model (introduced by Raussendorf and Briegel [Phys. Rev. Lett., 86, 5188 (2001)]) uses a fixed entangled state, adaptive measurements on single qubits, and feedforward of the measurement results. The second model (proposed by Nielsen [Phys. Lett. A, 308, 96 (2003)] and further simplified by Leung [Int. J. Quant. Inf., 2, 33 (2004)]) uses adaptive two-qubit measurements that can be applied to arbitrary pairs of qubits, and feedforward of the measurement results. The underlying principle of our derivations is a variant of teleportation introduced by Zhou, Leung, and Chuang [Phys. Rev. A, 62, 052316 (2000)]. Our derivations unify these two measurement-based models of quantum computation and provide significantly simpler schemes. 1 aChilds, Andrew, M.1 aLeung, Debbie, W.1 aNielsen, Michael, A. uhttp://arxiv.org/abs/quant-ph/0404132v201270nas a2200157 4500008004100000245006000041210006000101260001500161300001600176490000700192520080600199100002301005700002201028700001801050856004401068 2004 eng d00aReversible simulation of bipartite product Hamiltonians0 aReversible simulation of bipartite product Hamiltonians c2004/06/01 a1189 - 11970 v503 a Consider two quantum systems A and B interacting according to a product Hamiltonian H = H_A x H_B. We show that any two such Hamiltonians can be used to simulate each other reversibly (i.e., without efficiency losses) with the help of local unitary operations and local ancillas. Accordingly, all non-local features of a product Hamiltonian -- including the rate at which it can be used to produce entanglement, transmit classical or quantum information, or simulate other Hamiltonians -- depend only upon a single parameter. We identify this parameter and use it to obtain an explicit expression for the entanglement capacity of all product Hamiltonians. Finally, we show how the notion of simulation leads to a natural formulation of measures of the strength of a nonlocal Hamiltonian. 1 aChilds, Andrew, M.1 aLeung, Debbie, W.1 aVidal, Guifre uhttp://arxiv.org/abs/quant-ph/0303097v100967nas a2200133 4500008004100000245004200041210004200083260001500125490000700140520059600147100002300743700002300766856004400789 2004 eng d00aSpatial search and the Dirac equation0 aSpatial search and the Dirac equation c2004/10/190 v703 a We consider the problem of searching a d-dimensional lattice of N sites for a single marked location. We present a Hamiltonian that solves this problem in time of order sqrt(N) for d>2 and of order sqrt(N) log(N) in the critical dimension d=2. This improves upon the performance of our previous quantum walk search algorithm (which has a critical dimension of d=4), and matches the performance of a corresponding discrete-time quantum walk algorithm. The improvement uses a lattice version of the Dirac Hamiltonian, and thus requires the introduction of spin (or coin) degrees of freedom. 1 aChilds, Andrew, M.1 aGoldstone, Jeffrey uhttp://arxiv.org/abs/quant-ph/0405120v101263nas a2200133 4500008004100000245003500041210003500076260001400111490000700125520090700132100002301039700002301062856004401085 2004 eng d00aSpatial search by quantum walk0 aSpatial search by quantum walk c2004/8/230 v703 a Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of N items laid out in d spatial dimensions can be searched in time of order sqrt(N) for d>2, and in time of order sqrt(N) poly(log N) for d=2. We consider an alternative search algorithm based on a continuous time quantum walk on a graph. The case of the complete graph gives the continuous time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that sqrt(N) speedup can also be achieved on the hypercube. We show that full sqrt(N) speedup can be achieved on a d-dimensional periodic lattice for d>4. In d=4, the quantum walk search algorithm takes time of order sqrt(N) poly(log N), and in d<4, the algorithm does not provide substantial speedup. 1 aChilds, Andrew, M.1 aGoldstone, Jeffrey uhttp://arxiv.org/abs/quant-ph/0306054v200715nas a2200145 4500008004100000245006300041210006300104260001500167490000700182520026200189100002300451700002600474700002500500856004400525 2003 eng d00aLower bounds on the complexity of simulating quantum gates0 aLower bounds on the complexity of simulating quantum gates c2003/11/180 v683 a We give a simple proof of a formula for the minimal time required to simulate a two-qubit unitary operation using a fixed two-qubit Hamiltonian together with fast local unitaries. We also note that a related lower bound holds for arbitrary n-qubit gates. 1 aChilds, Andrew, M.1 aHaselgrove, Henry, L.1 aNielsen, Michael, A. uhttp://arxiv.org/abs/quant-ph/0307190v101268nas a2200121 4500008004100000245004200041210004200083260001500125520091400140100002301054700002501077856004401102 2003 eng d00aQuantum algorithms for subset finding0 aQuantum algorithms for subset finding c2003/11/063 a Recently, Ambainis gave an O(N^(2/3))-query quantum walk algorithm for element distinctness, and more generally, an O(N^(L/(L+1)))-query algorithm for finding L equal numbers. We point out that this algorithm actually solves a much more general problem, the problem of finding a subset of size L that satisfies any given property. We review the algorithm and give a considerably simplified analysis of its query complexity. We present several applications, including two algorithms for the problem of finding an L-clique in an N-vertex graph. One of these algorithms uses O(N^(2L/(L+1))) edge queries, and the other uses \tilde{O}(N^((5L-2)/(2L+4))), which is an improvement for L <= 5. The latter algorithm generalizes a recent result of Magniez, Santha, and Szegedy, who considered the case L=3 (finding a triangle). We also pose two open problems regarding continuous time quantum walk and lower bounds. 1 aChilds, Andrew, M.1 aEisenberg, Jason, M. uhttp://arxiv.org/abs/quant-ph/0311038v200853nas a2200145 4500008004100000245009300041210006900134260001500203520037100218100002300589700001800612700001900630700001400649856004400663 2002 eng d00aAsymptotic entanglement capacity of the Ising and anisotropic Heisenberg interactions 0 aAsymptotic entanglement capacity of the Ising and anisotropic He c2002/07/103 a We compute the asymptotic entanglement capacity of the Ising interaction ZZ, the anisotropic Heisenberg interaction XX + YY, and more generally, any two-qubit Hamiltonian with canonical form K = a XX + b YY. We also describe an entanglement assisted classical communication protocol using the Hamiltonian K with rate equal to the asymptotic entanglement capacity. 1 aChilds, Andrew, M.1 aLeung, D., W.1 aVerstraete, F.1 aVidal, G. uhttp://arxiv.org/abs/quant-ph/0207052v201113nas a2200169 4500008004100000245005200041210005200093260001500145520061800160100002300778700001900801700001900820700001800839700001700857700002500874856004400899 2002 eng d00aExponential algorithmic speedup by quantum walk0 aExponential algorithmic speedup by quantum walk c2002/09/243 a We construct an oracular (i.e., black box) problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our oracular setting. We then show how this quantum walk can be used to solve our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve this problem with high probability in subexponential time. 1 aChilds, Andrew, M.1 aCleve, Richard1 aDeotto, Enrico1 aFarhi, Edward1 aGutmann, Sam1 aSpielman, Daniel, A. uhttp://arxiv.org/abs/quant-ph/0209131v201071nas a2200181 4500008004100000245003400041210003400075260001400109490000700123520059100130100002300721700001900744700001800763700002300781700001700804700002400821856004400845 2002 eng d00aQuantum search by measurement0 aQuantum search by measurement c2002/9/230 v663 a We propose a quantum algorithm for solving combinatorial search problems that uses only a sequence of measurements. The algorithm is similar in spirit to quantum computation by adiabatic evolution, in that the goal is to remain in the ground state of a time-varying Hamiltonian. Indeed, we show that the running times of the two algorithms are closely related. We also show how to achieve the quadratic speedup for Grover's unstructured search problem with only two measurements. Finally, we discuss some similarities and differences between the adiabatic and measurement algorithms. 1 aChilds, Andrew, M.1 aDeotto, Enrico1 aFarhi, Edward1 aGoldstone, Jeffrey1 aGutmann, Sam1 aLandahl, Andrew, J. uhttp://arxiv.org/abs/quant-ph/0204013v101113nas a2200169 4500008004100000245006000041210006000101260001400161490000700175520059300182100002500775700002500800700002300825700002300848700002800871856004400899 2002 eng d00aUniversal simulation of Hamiltonian dynamics for qudits0 aUniversal simulation of Hamiltonian dynamics for qudits c2002/8/300 v663 a What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? Dodd et al. (quant-ph/0106064) provided a partial solution to this problem in the form of an efficient algorithm to simulate any desired two-body Hamiltonian evolution using any fixed two-body entangling N-qubit Hamiltonian, and local unitaries. We extend this result to the case where the component systems have D dimensions. As a consequence we explain how universal quantum computation can be performed with any fixed two-body entangling N-qudit Hamiltonian, and local unitaries. 1 aNielsen, Michael, A.1 aBremner, Michael, J.1 aDodd, Jennifer, L.1 aChilds, Andrew, M.1 aDawson, Christopher, M. uhttp://arxiv.org/abs/quant-ph/0109064v201212nas a2200145 4500008004100000245007400041210006900115260001400184490000700198520074500205100002300950700002400973700002500997856004401022 2001 eng d00aExact sampling from non-attractive distributions using summary states0 aExact sampling from nonattractive distributions using summary st c2001/2/220 v633 a Propp and Wilson's method of coupling from the past allows one to efficiently generate exact samples from attractive statistical distributions (e.g., the ferromagnetic Ising model). This method may be generalized to non-attractive distributions by the use of summary states, as first described by Huber. Using this method, we present exact samples from a frustrated antiferromagnetic triangular Ising model and the antiferromagnetic q=3 Potts model. We discuss the advantages and limitations of the method of summary states for practical sampling, paying particular attention to the slowing down of the algorithm at low temperature. In particular, we show that such a slowing down can occur in the absence of a physical phase transition. 1 aChilds, Andrew, M.1 aPatterson, Ryan, B.1 aMacKay, David, J. C. uhttp://arxiv.org/abs/cond-mat/0005132v100814nas a2200157 4500008004100000245007600041210006900117260001500186300001200201490000600213520033500219100002300554700001800577700001700595856004400612 2001 eng d00aAn example of the difference between quantum and classical random walks0 aexample of the difference between quantum and classical random w c2002/04/01 a35 - 430 v13 a In this note, we discuss a general definition of quantum random walks on graphs and illustrate with a simple graph the possibility of very different behavior between a classical random walk and its quantum analogue. In this graph, propagation between a particular pair of nodes is exponentially faster in the quantum case. 1 aChilds, Andrew, M.1 aFarhi, Edward1 aGutmann, Sam uhttp://arxiv.org/abs/quant-ph/0103020v101006nas a2200145 4500008004100000245005300041210005300094260001400147490000700161520058100168100002300749700002200772700002200794856004400816 2001 eng d00aRealization of quantum process tomography in NMR0 aRealization of quantum process tomography in NMR c2001/6/130 v643 a Quantum process tomography is a procedure by which the unknown dynamical evolution of an open quantum system can be fully experimentally characterized. We demonstrate explicitly how this procedure can be implemented with a nuclear magnetic resonance quantum computer. This allows us to measure the fidelity of a controlled-not logic gate and to experimentally investigate the error model for our computer. Based on the latter analysis, we test an important assumption underlying nearly all models of quantum error correction, the independence of errors on different qubits. 1 aChilds, Andrew, M.1 aChuang, Isaac, L.1 aLeung, Debbie, W. uhttp://arxiv.org/abs/quant-ph/0012032v100767nas a2200145 4500008004100000245004800041210004800089260001500137490000700152520035800159100002300517700001800540700001900558856004400577 2001 eng d00aRobustness of adiabatic quantum computation0 aRobustness of adiabatic quantum computation c2001/12/140 v653 a We study the fault tolerance of quantum computation by adiabatic evolution, a quantum algorithm for solving various combinatorial search problems. We describe an inherent robustness of adiabatic computation against two kinds of errors, unitary control errors and decoherence, and we study this robustness using numerical simulations of the algorithm. 1 aChilds, Andrew, M.1 aFarhi, Edward1 aPreskill, John uhttp://arxiv.org/abs/quant-ph/0108048v100953nas a2200109 4500008004100000245004000041210004000081260001500121520064000136100002300776856004400799 2001 eng d00aSecure assisted quantum computation0 aSecure assisted quantum computation c2001/11/073 a Suppose Alice wants to perform some computation that could be done quickly on a quantum computer, but she cannot do universal quantum computation. Bob can do universal quantum computation and claims he is willing to help, but Alice wants to be sure that Bob cannot learn her input, the result of her calculation, or perhaps even the function she is trying to compute. We describe a simple, efficient protocol by which Bob can help Alice perform the computation, but there is no way for him to learn anything about it. We also discuss techniques for Alice to detect whether Bob is honestly helping her or if he is introducing errors. 1 aChilds, Andrew, M. uhttp://arxiv.org/abs/quant-ph/0111046v201258nas a2200181 4500008004100000245005500041210005500096260001400151490000700165520074300172100001600915700002300931700002200954700001700976700002200993700001701015856004401032 2001 eng d00aUniversal simulation of Markovian quantum dynamics0 aUniversal simulation of Markovian quantum dynamics c2001/11/90 v643 a Although the conditions for performing arbitrary unitary operations to simulate the dynamics of a closed quantum system are well understood, the same is not true of the more general class of quantum operations (also known as superoperators) corresponding to the dynamics of open quantum systems. We propose a framework for the generation of Markovian quantum dynamics and study the resources needed for universality. For the case of a single qubit, we show that a single nonunitary process is necessary and sufficient to generate all unital Markovian quantum dynamics, whereas a set of processes parametrized by one continuous parameter is needed in general. We also obtain preliminary results for the unital case in higher dimensions. 1 aBacon, Dave1 aChilds, Andrew, M.1 aChuang, Isaac, L.1 aKempe, Julia1 aLeung, Debbie, W.1 aZhou, Xinlan uhttp://arxiv.org/abs/quant-ph/0008070v201120nas a2200145 4500008004100000245005100041210005100092260001500143520069100158100002300849700001800872700002300890700001700913856004400930 2000 eng d00aFinding cliques by quantum adiabatic evolution0 aFinding cliques by quantum adiabatic evolution c2000/12/193 a Quantum adiabatic evolution provides a general technique for the solution of combinatorial search problems on quantum computers. We present the results of a numerical study of a particular application of quantum adiabatic evolution, the problem of finding the largest clique in a random graph. An n-vertex random graph has each edge included with probability 1/2, and a clique is a completely connected subgraph. There is no known classical algorithm that finds the largest clique in a random graph with high probability and runs in a time polynomial in n. For the small graphs we are able to investigate (n <= 18), the quantum algorithm appears to require only a quadratic run time. 1 aChilds, Andrew, M.1 aFarhi, Edward1 aGoldstone, Jeffrey1 aGutmann, Sam uhttp://arxiv.org/abs/quant-ph/0012104v101129nas a2200157 4500008004100000245005000041210005000091260001500141300001400156490000700170520069000177100002300867700001900890700001800909856004400927 2000 eng d00aQuantum information and precision measurement0 aQuantum information and precision measurement c1999/04/07 a155 - 1760 v473 a We describe some applications of quantum information theory to the analysis of quantum limits on measurement sensitivity. A measurement of a weak force acting on a quantum system is a determination of a classical parameter appearing in the master equation that governs the evolution of the system; limitations on measurement accuracy arise because it is not possible to distinguish perfectly among the different possible values of this parameter. Tools developed in the study of quantum information and computation can be exploited to improve the precision of physics experiments; examples include superdense coding, fast database search, and the quantum Fourier transform. 1 aChilds, Andrew, M.1 aPreskill, John1 aRenes, Joseph uhttp://arxiv.org/abs/quant-ph/9904021v200976nas a2200133 4500008004100000245006200041210006100103260001500164490000700179520056700186100002300753700002200776856004400798 2000 eng d00aUniversal quantum computation with two-level trapped ions0 aUniversal quantum computation with twolevel trapped ions c2000/12/110 v633 a Although the initial proposal for ion trap quantum computation made use of an auxiliary internal level to perform logic between ions, this resource is not necessary in principle. Instead, one may perform such operations directly using sideband laser pulses, operating with an arbitrary (sufficiently small) Lamb-Dicke parameter. We explore the potential of this technique, showing how to perform logical operations between the internal state of an ion and the collective motional state and giving explicit constructions for a controlled-not gate between ions. 1 aChilds, Andrew, M.1 aChuang, Isaac, L. uhttp://arxiv.org/abs/quant-ph/0008065v1