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arXiv:2209.05048 · Quantum

Optimal Hamiltonian simulation for time-periodic systems

Kaoru Mizuta, Keisuke Fujii

Published Mar 28, 2023• 15 citations
arXivHTMLPDF

Abstract

The implementation of time-evolution operators U(t) , called Hamiltonian simulation, is one of the most promising usage of quantum computers. For time-independent Hamiltonians, qubitization has recently established efficient realization of time-evolution U(t)=e−iHt , with achieving the optimal computational resource both in time t and an allowable error ε . In contrast, those for time-dependent systems require larger cost due to the difficulty of handling time-dependency. In this paper, we establish optimal/nearly-optimal Hamiltonian simulation for generic time-dependent systems with time-periodicity, known as Floquet systems. By using a so-called Floquet-Hilbert space equipped with auxiliary states labeling Fourier indices, we develop a way to certainly obtain the target time-evolved state without relying on either time-ordered product or Dyson-series expansion. Consequently, the query complexity, which measures the cost for implementing the time-evolution, has optimal and nearly-optimal dependency respectively in time t and inverse error ε , and becomes sufficiently close to that of qubitization. Thus, our protocol tells us that, among generic time-dependent systems, time-periodic systems provides a class accessible as efficiently as time-independent systems despite the existence of time-dependency. As we also provide applications to simulation of nonequilibrium phenomena and adiabatic state preparation, our results will shed light on nonequilibrium phenomena in condensed matter physics and quantum chemistry, and quantum tasks yielding time-dependency in quantum computation.

DOI
10.22331/q-2023-03-28-962

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The implementation of time-evolution operators U(t) , called Hamiltonian simulation, is one of the most promising usage of quantum computers. For time-independent Hamiltonians, qubitization has recently established efficient realization of time-evolution U(t)=e−iHt , with achieving the optimal computational resource both in time t and an allowable error ε . In contrast, those for time-dependent systems require larger cost due to the difficulty of handling time-dependency. In this paper, we establish optimal/nearly-optimal Hamiltonian simulation for generic time-dependent systems with time-periodicity, known as Floquet systems. By using a so-called Floquet-Hilbert space equipped with auxiliary states labeling Fourier indices, we develop a way to certainly obtain the target time-evolved state without relying on either time-ordered product or Dyson-series expansion. Consequently, the query complexity, which measures the cost for implementing the time-evolution, has optimal and nearly-optimal dependency respectively in time t and inverse error ε , and becomes sufficiently close to that of qubitization. Thus, our protocol tells us that, among generic time-dependent systems, time-periodic systems provides a class accessible as efficiently as time-independent systems despite the existence of time-dependency. As we also provide applications to simulation of nonequilibrium phenomena and adiabatic state preparation, our results will shed light on nonequilibrium phenomena in condensed matter physics and quantum chemistry, and quantum tasks yielding time-dependency in quantum computation.
Simulating Hamiltonian dynamics is one of the most fundamental and significant tasks for characterising quantum materials. Recently, a series of quantum algorithms employing block-encoding of Hamiltonians have succeeded in providing efficient simulation of time-evolution operators on quantum computers. While time-independent Hamiltonians can be simulated by the quantum eigenvalue transformation (QET) or quantum singularvalue transformation with the optimal query complexity in time ttt and desirable accuracy ε\varepsilonε, generic time-dependent Hamiltonians face at larger query complexity and more complicated oracles due to the difficulty of handling time-dependency. In this paper, we establish a QET-based approach for simulating time-dependent Hamiltonians with multiple time-periodicity. Such time-dependent Hamiltonians involve a variety of nonequilibrium systems such as time-periodic systems (Floquet systems) and time-quasiperiodic systems. Overcoming the difficulty of time-dependency, our protocol can simulate the dynamics under multi-periodic time-dependent Hamiltonians with optimal/nearly-optimal query complexity both in time ttt and desirable accuracy ε\varepsilonε, and simple oracles as well as the optimal algorithm for time-independent cases.
We give a query-optimal algorithm for simulating a general nnn-qubit time-dependent Hamiltonian H(t)H(t)H(t) on [0,T][0,T][0,T], assuming that HHH is Lipschitz continuous and ∥H(t)∥≤α\|H(t)\|\leqα∥H(t)∥≤α. In the standard $\mathrm{HAM\mbox{-}T}$ access model, the algorithm approximates the time-ordered propagator UH(T)U_H(T)UH​(T) to error ε\varepsilonε using O(αT+log⁡(1/ε)log⁡(e+log⁡(1/ε)/(αT))) O\left( αT+\frac{\log(1/\varepsilon)} {\log(e+\log(1/\varepsilon)/(αT))} \right) O(αT+log(e+log(1/ε)/(αT))log(1/ε)​) $\mathrm{HAM\mbox{-}T}$ queries. This matches the known query lower bound for time-independent Hamiltonians, showing that time dependence incurs no asymptotic query overhead. Our method first constructs a one-query transducer that, given an auxiliary state, implements an approximation to UH(T)U_H(T)UH​(T) and returns the state unchanged. A weighted combination of circuits that apply the transducer different numbers of times makes the error caused by omitting this state decay factorially, yielding the stated optimal precision dependence. For time-independent Hamiltonians, the same method also gives a query-optimal alternative to qubitization.
Time-periodic (Floquet) systems are one of the most interesting nonequilibrium systems. As the computation of energy eigenvalues and eigenstates of time-independent Hamiltonians is a central problem in both classical and quantum computation, quasienergy and Floquet eigenstates are the important targets. However, their computation has difficulty of time dependence; the problem can be mapped to a time-independent eigenvalue problem by the Sambe space formalism, but it instead requires additional infinite-dimensional space and seems to yield higher computational cost than the time-independent cases. It is still unclear whether they can be computed with guaranteed accuracy as efficiently as the time-independent cases. We address this issue by rigorously deriving the cutoff of the Sambe space to achieve the desired accuracy and organizing quantum algorithms for computing quasienergy and Floquet eigenstates based on the cutoff. The quantum algorithms return quasienergy and Floquet eigenstates with guaranteed accuracy like quantum phase estimation, which is the optimal algorithm for outputting energy eigenvalues and eigenstates of time-independent Hamiltonians. While the time periodicity provides the additional dimension for the Sambe space and defines two different kinds of eigenstates, the query complexity of the algorithms achieves the near-optimal scaling in allowable errors. In addition, as a by-product of these algorithms, we also organize a quantum algorithm for Floquet eigenstate preparation, in which a preferred gapped Floquet eigenstate can be deterministically implemented with nearly optimal query complexity in the gap. These results show that, despite the difficulty of time dependence, quasienergy and Floquet eigenstates can be computed almost as efficiently as time-independent cases, shedding light on the accurate and fast simulation of nonequilibrium systems on quantum computers.
Simulating Hamiltonian dynamics is one of the most fundamental and significant tasks for characterizing quantum materials. Recently, a series of quantum algorithms employing block encoding of Hamiltonians have succeeded in providing efficient simulation of time-evolution operators on quantum computers. While time-independent Hamiltonians can be simulated by the quantum eigenvalue transformation (QET) or quantum singular value transformation with the optimal query complexity in time ttt and desirable accuracy $\ensuremath{\varepsilon}$, generic time-dependent Hamiltonians face at larger query complexity and more complicated oracles due to the difficulty of handling time-dependency. In this paper, we establish a QET-based approach for simulating time-dependent Hamiltonians with multiple time periodicity. Such time-dependent Hamiltonians involve a variety of nonequilibrium systems such as time-periodic systems (Floquet systems) and time-quasiperiodic systems. Overcoming the difficulty of time dependency, our protocol can simulate the dynamics under multiperiodic time-dependent Hamiltonians with optimal and nearly optimal query complexity both in time ttt and desirable accuracy $\ensuremath{\varepsilon}$, and simple oracles as well as the optimal algorithm for time-independent cases.
We introduce a variational framework for approximating the time-evolution operator U^(t)\hat{U}(t)U^(t) within a physically motivated operator manifold, reformulating quantum dynamics as a tractable problem in operator space using stationary action principle. For periodically driven systems, the resulting approximate evolution operator directly yields an effective Floquet Hamiltonian, offering a non-perturbative alternative to conventional expansion-based methods. The framework is systematically improvable by enlarging the operator pool and naturally incorporates symmetries and physical constraints. When the operator manifold is chosen from the terms of a truncated Magnus expansion, the variational procedure effectively resums the Magnus series within the restricted space, significantly enhancing accuracy. We benchmark the approach on the driven Rabi model, the driven Lipkin-Meshkov-Glick model, and the one-dimensional driven Ising chain, yielding effective Floquet Hamiltonians that are systematically more accurate than low-order Magnus expansions, particularly in regimes where the latter converge poorly, and illustrating applicability to systems with exponentially large Hilbert spaces. Although we focus here on Floquet systems, the formalism applies equally to generic time-dependent Hamiltonians, providing a versatile tool for non-equilibrium quantum dynamics.
We develop a quantum algorithm for slow analytic Hamiltonians H~(t)=H(t/T)\widetilde H(t)=H(t/T)H(t)=H(t/T) with ∥H(s)∥≤α\|H(s)\|\leqα∥H(s)∥≤α that achieves nearly additive query complexity and low gate overhead. Our main technical contribution is a periodic Gevrey extension of H(s)H(s)H(s), together with Fourier component decay and truncation bounds that enable an efficient finite-dimensional simulation. Combined with Floquet embedding and optimal time-independent Hamiltonian simulation technique, this gives query complexity O~ ⁣(αT+log⁡(1/ε))\widetilde{\mathcal O}\!\left(αT+\log(1/\varepsilon)\right)O(αT+log(1/ε)) and additional gate complexity O~ ⁣((αT+log⁡(1/ε))2log⁡(1/ε))\widetilde{\mathcal O}\!\left((αT+\log(1/\varepsilon))^2\log(1/\varepsilon)\right)O((αT+log(1/ε))2log(1/ε)), assuming coherent access to H′(s)H'(s)H′(s) and endpoint derivatives. For slow analytic control Hamiltonians, only block encodings of the time-independent control operators are required, with the same query complexity and lower gate overhead. Our method also extends to Gevrey Hamiltonians and improves the precision dependence for simulating slow analytic semi-dissipative linear differential equations.