Circuit Depth Reduction of One-Ancilla Quantum Differential Equation Solver via Extrapolation

Abstract

Solving linear differential equations is a fundamental task in scientific computing and an important primitive for quantum computing. A recent one-ancilla quantum differential equation solver provides a hardware-friendly and locality-preserving approach with provable performance guarantees, making it highly suitable for the early fault-tolerant and near-term regimes. Its simple circuit structure comes with a natural trade-off: the maximum single-run circuit depth scales as $O (1/ε)$ in the target accuracy $ε$. In this work, we reduce this depth by combining the solver with classical step-size postprocessing. By running the one-ancilla solver at a logarithmic number of finite time step sizes and using classical post-processing to cancel leading discretization errors, we reduce the maximum single-run circuit depth to $O(\mathrm{polylog}(1/ε))$ without adding quantum ancillae or sacrificing locality. Technically, extending extrapolation ideas beyond Hamiltonian and Lindbladian dynamics requires regularity estimates for observable maps under nonunitary evolution, which we obtain through a holomorphic extension of the adjoint evolution. Numerical experiments on the Hatano-Nelson model (ODE) and the convection-diffusion equation (PDE) demonstrate the effectiveness of the approach.