Optimal Hamiltonian simulation for time-periodic systems

TL;DR

This paper introduces an optimal quantum simulation method for time-periodic (Floquet) systems, achieving near-optimal resource efficiency and overcoming challenges posed by time-dependency in quantum evolution.

Contribution

The authors develop a novel Floquet-Hilbert space approach enabling efficient Hamiltonian simulation of time-periodic systems without time-ordering complexities, matching the efficiency of time-independent methods.

Findings

Query complexity is optimal in time and nearly optimal in error dependence.

Simulation efficiency for Floquet systems is comparable to that of time-independent Hamiltonians.

Applications include simulating nonequilibrium phenomena and adiabatic state preparation.

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 $\varepsilon$. 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 $\varepsilon$, 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.