Minimum Trotterization Formulas for a Time-Dependent Hamiltonian

Tatsuhiko N. Ikeda1,2,3, Asir Abrar4, Isaac L. Chuang5, and Sho Sugiura4,6

1RIKEN Center for Quantum Computing, Wako, Saitama 351-0198, Japan
2Department of Physics, Boston University, Boston, Massachusetts 02215, USA
3Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
4Physics and Informatics Laboratory, NTT Research, Inc.,940 Stewart Dr., Sunnyvale, California, 94085, USA
5Department of Physics, Department of Electrical Engineering and Computer Science, and Co-Design Center for Quantum Advantage, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
6Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, 02139, MA, USA

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Abstract

When a time propagator $e^{\delta t A}$ for duration $\delta t$ consists of two noncommuting parts $A=X+Y$, Trotterization approximately decomposes the propagator into a product of exponentials of $X$ and $Y$. Various Trotterization formulas have been utilized in quantum and classical computers, but much less is known for the Trotterization with the time-dependent generator $A(t)$. Here, for $A(t)$ given by the sum of two operators $X$ and $Y$ with time-dependent coefficients $A(t) = x(t) X + y(t) Y$, we develop a systematic approach to derive high-order Trotterization formulas with minimum possible exponentials. In particular, we obtain fourth-order and sixth-order Trotterization formulas involving seven and fifteen exponentials, respectively, which are no more than those for time-independent generators. We also construct another fourth-order formula consisting of nine exponentials having a smaller error coefficient. Finally, we numerically benchmark the fourth-order formulas in a Hamiltonian simulation for a quantum Ising chain, showing that the 9-exponential formula accompanies smaller errors per local quantum gate than the well-known Suzuki formula.

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[1] Luis Hidalgo and Patrick Draper, "Quantum simulations for strong-field QED", Physical Review D 109 7, 076004 (2024).

[2] Hongzheng Zhao, Marin Bukov, Markus Heyl, and Roderich Moessner, "Adaptive Trotterization for time-dependent Hamiltonian quantum dynamics using instantaneous conservation laws", arXiv:2307.10327, (2023).

[3] Pooja Siwach, Kaytlin Harrison, and A. Baha Balantekin, "Collective neutrino oscillations on a quantum computer with hybrid quantum-classical algorithm", Physical Review D 108 8, 083039 (2023).

[4] Tim Möbus, "On Strong Bounds for Trotter and Zeno Product Formulas with Bosonic Applications", arXiv:2404.01422, (2024).

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