Reducing Circuit Depth in Lindblad Simulation via Step-Size Extrapolation

We study \emph{algorithmic error mitigation} via Richardson-style extrapolation for quantum simulations of open quantum systems governed by the Lindblad equation. Focusing on two first-order primitives (a Kraus-form channel update and a Stinespring-dilation step), we conduct a backward-error analysis that yields a step-size expansion of observables with explicit, Gevrey-class derivative bounds. These smoothness estimates justify extrapolation and allow a complete bias-variance tradeoff: we prove bias decays exponentially in the number of nodes \(n\), while the sampling variance remains at \(\mathcal{O}(1/\varepsilon^{2})\) when using perturbed Chebyshev nodes. For evolutions with generator norm \(\ell\) and time \(T\), our main theorem shows that an \(n=\Omega(\log(1/\varepsilon))\)-point extrapolator reduces the \emph{maximum circuit depth} required for accuracy \(\varepsilon\) from \(\mathcal{O}((\ell T)^{2}/\varepsilon)\) to \(\mathcal{O}\!\big((\ell T)^{2}\log(\ell T)\log^{2}(1/\varepsilon)\big)\)—an exponential improvement in \(1/\varepsilon\)—while preserving sampling complexity \(\mathcal{O}(1/\varepsilon^{2})\). Numerical experiments with both primitives corroborate the theory and indicate practical viability on near-term devices.