Optimal quantum simulation of linear non-unitary dynamics

TL;DR

This paper introduces an optimal quantum algorithm for simulating linear non-unitary dynamics, extending Hamiltonian simulation techniques to a broader class of operators with improved efficiency and error scaling.

Contribution

It generalizes the Linear-Combination-of-Hamiltonian-Simulation framework to non-unitary operators, providing optimal complexity bounds and improved quantum circuit designs for time-dependent cases.

Findings

Optimal query complexity for time-independent operators

Exponential convergence with trapezoidal rule for time-dependent operators

Improved gate complexity over previous nonuniform quadrature methods

Abstract

We present a quantum algorithm for simulating the time evolution generated by any bounded, time-dependent operator $-A$ with non-positive logarithmic norm, thereby serving as a natural generalization of the Hamiltonian simulation problem. Our method generalizes the recent Linear-Combination-of-Hamiltonian-Simulation (LCHS) framework. In instances where $A$ is time-independent, we provide a block-encoding of the evolution operator $e^{-At}$ with $\mathcal{O}\big(t\log\frac{1}{\epsilon})$ queries to the block-encoding oracle for $A$. We also show how the normalized evolved state can be prepared with $\mathcal{O}(1/\|e^{-At}|{\vec{u}_0}\rangle\|)$ queries to the oracle that prepares the normalized initial state $|{\vec{u}_0}\rangle$. These complexities are optimal in all parameters and improve the error scaling over prior results. Furthermore, we show that any improvement of our approach exceeding a constant factor of approximately 3 is infeasible. For general time-dependent operators $A$, we also prove that a uniform trapezoidal rule on our LCHS construction yields exponential convergence, leading to simplified quantum circuits with improved gate complexity compared to prior nonuniform-quadrature methods.