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/2403.0819501492nas 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/2111.1217702034nas 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/2011.0318501386nas 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/1611.0554302001nas 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/1312.1414v201395nas 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/1304.4642v100984nas 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/0703015v301113nas 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/0209131v201212nas 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/0103020v1