Time-Dependent Hamiltonian Simulation with Optimal Query Complexity

2026-08-06

AI Summary

This work proposes a quantum simulation algorithm for time-dependent Hamiltonians that are Lipschitz continuous, achieving optimal query complexity for approximating the time-ordered evolution operator to precision ε. By introducing the HAM-T access model, a single-query conversion unitary, and a novel framework of weighted circuit linear combinations with factorial error suppression, the authors demonstrate for the first time that time-dependent simulation incurs no additional query overhead compared to the time-independent case. The algorithm attains a query complexity of O(αT + log(1/ε)/log(e + log(1/ε)/(αT))), which matches the known lower bound for simulating time-independent Hamiltonians and is therefore asymptotically optimal.

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 $\|H(t)\|\leqα$. In the standard $\mathrm{HAM\mbox{-}T}$ access model, the algorithm approximates the time-ordered propagator $U_H(T)$ to error $\varepsilon$ using $$ O\left( αT+\frac{\log(1/\varepsilon)} {\log(e+\log(1/\varepsilon)/(αT))} \right) $$ $\mathrm{HAM\mbox{-}T}$ 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.

Problem

• time-dependent Hamiltonian simulation
• query complexity
• quantum simulation
• optimal algorithm

Innovation

• time-dependent Hamiltonian simulation
• query complexity
• optimal algorithm
• transducer
• quantum simulation