02331nas 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/1912.0885401946nas a2200157 4500008004100000245006600041210006600107260001300173490000600186520147300192100002401665700002101689700002301710700001801733856003701751 2020 eng d00aQuantum Algorithms for Simulating the Lattice Schwinger Model0 aQuantum Algorithms for Simulating the Lattice Schwinger Model c8/5/20200 v43 a

The Schwinger model (quantum electrodynamics in 1+1 dimensions) is a testbed for the study of quantum gauge field theories. We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In particular, we perform a tight analysis of low-order Trotter formula simulations of the Schwinger model, using recently derived commutator bounds, and give upper bounds on the resources needed for simulations in both scenarios. In lattice units, we find a Schwinger model on N/2 physical sites with coupling constant x−1/2 and electric field cutoff x−1/2Λ can be simulated on a quantum computer for time 2xT using a number of T-gates or CNOTs in O˜(N3/2T3/2x−−√Λ) for fixed operator error. This scaling with the truncation Λ is better than that expected from algorithms such as qubitization or QDRIFT. Furthermore, we give scalable measurement schemes and algorithms to estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable---the mean pair density. Finally, we bound the root-mean-square error in estimating this observable via simulation as a function of the diamond distance between the ideal and actual CNOT channels. This work provides a rigorous analysis of simulating the Schwinger model, while also providing benchmarks against which subsequent simulation algorithms can be tested. 

1 aShaw, Alexander, F.1 aLougovski, Pavel1 aStryker, Jesse, R.1 aWiebe, Nathan uhttps://arxiv.org/abs/2002.1114601668nas 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/1906.0711502587nas a2200157 4500008004100000245011000041210006900151260001500220300001200235520207500247100001902322700001302341700002002354700001802374856003702392 2018 eng d00aQuantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics0 aQuantum singular value transformation and beyond exponential imp c2018/06/05 a193-2043 a

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.

1 aGilyen, Andras1 aSu, Yuan1 aLow, Guang, Hao1 aWiebe, Nathan uhttps://arxiv.org/abs/1806.0183801289nas 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/1211.4945v201023nas 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/1202.5822v1