Trotter error time scaling separation via commutant decomposition

Abstract

Suppressing the Trotter error in dynamical quantum simulation typically requires running deeper circuits, posing a great challenge for noisy near-term quantum devices. Studies have shown that the empirical error is usually much smaller than the one suggested by existing bounds, implying the actual circuit cost required is much less than the ones based on those bounds. Here, we improve the estimate of the Trotter error over existing bounds, by introducing a general framework of commutant decomposition that separates disjoint error components that have fundamentally different scaling with time. In particular we identify two error components that each scale as $O({\tau }^{p}t)$ and $O({\tau }^p)$ for a $p$th-order product formula evolving to time $t$ using a fixed step size $\tau$, it implies one would scale linearly with time $t$ and the other would be constant of $t$. We show that this formalism not only straightforwardly reproduces previous results but also provides a better error estimate for higher-order product formulas. We demonstrate the improvement both analytically and numerically. We also apply the analysis to observable error relating to the heating in Floquet dynamics and thermalization, which is of independent interest.

References in corpus (17)

• Optimal Hamiltonian Simulation by Quantum Signal Processing
• Simulating Hamiltonian dynamics with a truncated Taylor series
• Toward the first quantum simulation with quantum speedup
• A Theory of Trotter Error
• Floquet-Magnus Theory and Generic Transient Dynamics in Periodically Driven Many-Body Quantum Systems
• Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems
• Nearly optimal lattice simulation by product formulas
• Faster quantum simulation by randomization
• Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization
• Floquet prethermalization and regimes of heating in a periodically driven, interacting quantum system
• Quantum algorithm for simulating real time evolution of lattice Hamiltonians
• Quantum localization bounds Trotter errors in digital quantum simulation
• Classically optimized Hamiltonian simulation
• Quantum and classical Floquet prethermalization
• First-Order Trotter Error from a Second-Order Perspective
• Destructive Error Interference in Product-Formula Lattice Simulation
• Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation

Cited by in corpus (1)

• Robustness of near-thermal dynamics on digital quantum computers