Minimum Trotterization Formulas for a Time-Dependent Hamiltonian

Abstract

When a time propagator $e^{\delta tA}$ 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.

15 pages, 5 figure

References in corpus (5)

• The Magnus expansion and some of its applications
• Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
• Optimised Trotter Decompositions for Classical and Quantum Computing
• Optimal Hamiltonian simulation for time-periodic systems
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• On Strong Bounds for Trotter and Zeno Product Formulas with Bosonic Applications
• Quantum simulation of time-dependent Hamiltonians via commutator-free quasi-Magnus operators
• Unifying framework for quantum simulation algorithms for time-dependent Hamiltonian dynamics
• Multi-target quantum compilation algorithm