Optimal Hamiltonian simulation for time-periodic systems

Kaoru Mizuta1 and Keisuke Fujii2,3,1,4

1RIKEN Center for Quantum Computing (RQC), Hirosawa 2-1, Wako, Saitama 351-0198, Japan
2Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.
3Center for Quantum Information and Quantum Biology, Osaka University, Japan.
4Fujitsu Quantum Computing Joint Research Division at QIQB, Osaka University, 1-2 Machikaneyama, Toyonaka 560-0043, Japan

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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 $\varepsilon$. 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.

Simulating quantum materials has been an essential task of quantum computers since their beginning. We establish an optimal/nearly-optimal protocol for accurately simulating time-periodic Hamiltonians by fundamental notions of quantum physics such as Floquet theory and Lieb-Robinson bound. Significantly, its efficiency can reach the theoretically best one for time-independent systems, despite the difficulty of time-dependency. Not only our result gives insights into relation between quantum dynamics and quantum computation from the viewpoint of computational complexity, but also it builds versatile technology of quantum computers toward nonequilibrium phenomena in condensed matter physics and quantum chemistry, such as light-irradiated materials.

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[2] Tatsuhiko N. Ikeda, Asir Abrar, Isaac L. Chuang, and Sho Sugiura, "Minimum Trotterization Formulas for a Time-Dependent Hamiltonian", Quantum 7, 1168 (2023).

[3] Marco Lewis, Paolo Zuliani, and Sadegh Soudjani, Lecture Notes in Computer Science 14287, 346 (2023) ISBN:978-3-031-43834-9.

[4] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[5] Yue Wang and Qi Zhao, "Faster Quantum Algorithms with "Fractional''-Truncated Series", arXiv:2402.05595, (2024).

[6] Xiao-Ming Zhang, Zixuan Huo, Kecheng Liu, Ying Li, and Xiao Yuan, "Unbiased random circuit compiler for time-dependent Hamiltonian simulation", arXiv:2212.09445, (2022).

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