Time-Dependent Hamiltonian Simulation with Optimal Query Complexity

2026/08/06 by Boyang Chen, Minbo Gao, Xinzhao Wang +1
Physics and Astronomy · Computer Science · #quant-ph #cs.DS

paper · pdf

35 pages, 1 figure, 1 table

arxiv created 2026/08/06 · arxiv updated 2026/08/07

Abstract

We give a query-optimal algorithm for simulating a general n-qubit time-dependent Hamiltonian H(t) on [0,T], assuming that H is Lipschitz continuous and ‖H(t)‖≤α. In the standard HAM-T access model, the algorithm approximates the time-ordered propagator U_H(T) to error ε using O( αT+\fraclog(1/ε) log(e+log(1/ε)/(αT)) ) HAM-T queries. This matches the known query lower bound for time-independent Hamiltonians, showing that time dependence incurs no asymptotic query overhead. Our method first constructs a one-query transducer that, given an auxiliary state, implements an approximation to U_H(T) and returns the state unchanged. A weighted combination of circuits that apply the transducer different numbers of times makes the error caused by omitting this state decay factorially, yielding the stated optimal precision dependence. For time-independent Hamiltonians, the same method also gives a query-optimal alternative to qubitization.

Citations

• Optimal Lower Bounds for Hamiltonian Simulation
• High-order Magnus Expansion for Hamiltonian Simulation
• Time-Dependent Hamiltonian Simulation via Time-Independent Dynamics in a Larger Space
• Discrete Superconvergence Analysis for Quantum Magnus Algorithms of Unbounded Hamiltonian Simulation
• Unifying framework for quantum simulation algorithms for time-dependent Hamiltonian dynamics
• Explicit error bounds with commutator scaling for time-dependent product and multi-product formulas
• One-Way Ticket to Las Vegas and the Quantum Adversary
• Time Dependent Hamiltonian Simulation Using Discrete Clock Constructions
• Time-dependent unbounded Hamiltonian simulation with vector norm scaling
• Time-dependent Hamiltonian simulation with L¹-norm scaling
• Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
• Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series
• Hamiltonian Simulation in the Interaction Picture
• Hamiltonian Simulation by Qubitization
• Atomic quantum gases in periodically driven optical lattices
• Optimal Hamiltonian Simulation by Quantum Signal Processing
• Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters
• Simulating Hamiltonian Dynamics with a Truncated Taylor Series
• Hamiltonian Simulation Using Linear Combinations of Unitary Operations
• Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
• Control of quantum phenomena: Past, present, and future
• Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer
• Higher Order Decompositions of Ordered Operator Exponentials
• Quantum algorithm for solving linear systems of equations
• Simulated Quantum Computation of Molecular Energies
• Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
• Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation
• Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation
• Adiabatic Quantum State Generation and Statistical Zero Knowledge
• Simulating physics with computers