Time-Dependent Hamiltonian Simulation with Optimal Query Complexity

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

Quantum Physics (quant-ph); Data Structures and Algorithms (cs.DS)

arXiv:2608.06094 [quant-ph]

(or arXiv:2608.06094v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.06094 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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