Optimal quantum simulation of linear non-unitary dynamics

• URL: http://arxiv.org/abs/2508.19238v2
• Date: Fri, 12 Sep 2025 00:08:05 GMT
• Title: Optimal quantum simulation of linear non-unitary dynamics
• Authors: Guang Hao Low, Rolando D. Somma,
• Abstract summary: We present a quantum algorithm for simulating the time evolution generated by any bounded, time-dependent operator $-A$.<n>Our method generalizes the recent Linear-Combination-of-Hamiltonian-Simulation (LCHS) framework.
• Score: 0.31439717339537293
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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