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arXiv:2103.15334 · PRX Quantum

Quantum Algorithm for Time-Dependent Hamiltonian Simulation by Permutation Expansion

Yi‐Hsiang Chen, Amir Kalev, Itay Hen

Published Sep 9, 2021• 31 citations
arXivHTMLPDF

Abstract

We present a quantum algorithm for the dynamical simulation of time-dependent Hamiltonians.Our method involves expanding the interaction-picture Hamiltonian as a sum of generalized permutations, which leads to an integral-free Dyson series of the time-evolution operator.Under this representation, we perform a quantum simulation for the time-evolution operator by means of the linear combination of unitaries technique.We optimize the time steps of the evolution based on the Hamiltonian's dynamical characteristics, leading to a gate count that scales with an L 1 -norm-like scaling with respect only to the norm of the interaction Hamiltonian, rather than that of the total Hamiltonian.We demonstrate that the cost of the algorithm is independent of the Hamiltonian's frequencies, implying its advantage for systems with highly oscillating components, and for time-decaying systems the cost does not scale with the total evolution time asymptotically.In addition, our algorithm retains the near optimal log(1/ )/ log log(1/ ) scaling with simulation error .

DOI
10.1103/prxquantum.2.030342

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We present a quantum algorithm for the dynamical simulation of time-dependent Hamiltonians.Our method involves expanding the interaction-picture Hamiltonian as a sum of generalized permutations, which leads to an integral-free Dyson series of the time-evolution operator.Under this representation, we perform a quantum simulation for the time-evolution operator by means of the linear combination of unitaries technique.We optimize the time steps of the evolution based on the Hamiltonian's dynamical characteristics, leading to a gate count that scales with an L 1 -norm-like scaling with respect only to the norm of the interaction Hamiltonian, rather than that of the total Hamiltonian.We demonstrate that the cost of the algorithm is independent of the Hamiltonian's frequencies, implying its advantage for systems with highly oscillating components, and for time-decaying systems the cost does not scale with the total evolution time asymptotically.In addition, our algorithm retains the near optimal log(1/ )/ log log(1/ ) scaling with simulation error .
We propose an efficient quantum algorithm for simulating the dynamics of general Hamiltonian systems. Our technique is based on a power series expansion of the time-evolution operator in its off-diagonal terms. The expansion decouples the dynamics due to the diagonal component of the Hamiltonian from the dynamics generated by its off-diagonal part, which we encode using the linear combination of unitaries technique. Our method has an optimal dependence on the desired precision and, as we illustrate, generally requires considerably fewer resources than the current state-of-the-art. We provide an analysis of resource costs for several sample models.
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 sparse Hamiltonian simulation, the gate complexity scales with the L 1 norm ∫ 0 t d τ ‖ H ( τ ) ‖ max , whereas the best previous results scale with t max τ ∈ [ 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. These algorithms could potentially be applied to semi-classical simulations of scattering processes in quantum chemistry.
We propose an efficient quantum algorithm for simulating the dynamics of general Hamiltonian systems. Our technique is based on a power series expansion of the time-evolution operator in its off-diagonal terms. The expansion decouples the dynamics due to the diagonal component of the Hamiltonian from the dynamics generated by its off-diagonal part, which we encode using the linear combination of unitaries technique. Our method has an optimal dependence on the desired precision and, as we illustrate, generally requires considerably fewer resources than the current state-of-the-art. We provide an analysis of resource costs for several sample models.
We provide a general method for efficiently simulating time-dependent Hamiltonian dynamics on a circuit-model-based quantum computer. Our approach is based on approximating the truncated Dyson series of the evolution operator, extending the earlier proposal by Berry et al. [Phys. Rev. Lett. 114, 090502 (2015)] to evolution generated by explicitly time-dependent Hamiltonians. Two alternative strategies are proposed to implement time ordering while exploiting the superposition principle for sampling the Hamiltonian at different times. The resource cost of our simulation algorithm retains the optimal logarithmic dependence on the inverse of the desired precision.
We present a low-space overhead simulation algorithm based on the truncated Dyson series for time-dependent quantum dynamics. This algorithm is applied to simulating time-independent Hamiltonians by transitioning to the interaction picture, where some portions are made time-dependent. This can provide a favorable complexity trade-off as the algorithm scales exponentially better with derivatives of the time-dependent component than the original Hamiltonian. We show that this leads to an exponential improvement in gate complexity for simulating some classes of diagonally dominant Hamiltonian. Additionally we show that this can reduce the gate-complexity scaling for simulating NNN-site Hubbard models for time ttt with arbitrary long-range interactions as well as reduce the cost of quantum chemistry simulations within a similar-sized plane-wave basis to O~(N2t)\widetilde{\mathcal{O}}(N^2t)O(N2t) from O~(N11/3t)\widetilde{\mathcal{O}}(N^{11/3}t)O(N11/3t). We also show a quadratic improvement in query complexity for simulating sparse time-dependent Hamiltonians, which may be of independent interest.
We present an efficient quantum algorithm for simulating the dynamics of Markovian open quantum systems. The performance of our algorithm is similar to the previous state-of-the-art quantum algorithm, i.e., it scales linearly in evolution time and poly-logarithmically in inverse precision. However, our algorithm is conceptually cleaner, and it only uses simple quantum primitives without compressed encoding. Our approach is based on a novel mathematical treatment of the evolution map, which involves a higher-order series expansion based on Duhamel's principle and approximating multiple integrals using scaled Gaussian quadrature. Our method easily generalizes to simulating quantum dynamics with time-dependent Lindbladians. Furthermore, our method of approximating multiple integrals using scaled Gaussian quadrature could potentially be used to produce a more efficient approximation of time-ordered integrals, and therefore can simplify existing quantum algorithms for simulating time-dependent Hamiltonians based on a truncated Dyson series.