A unifying framework for quantum algorithms for time-dependent non-unitary dynamics

Abstract

We present an improved autonomization method for quantum simulation of time-dependent homogeneous dissipative linear systems, combining a clock-variable reformulation with Schr\"odingerization to obtain a time-independent Hamiltonian system. To control both discretization error and recovery probability, we construct the clock profile from a compactly supported window function and a normalized Dirichlet kernel: the former imposes endpoint regularity, while the latter concentrates the sampled mass and keeps the discrete normalization bounded. For the subsequent Schr\"odingerization step, we use an exactly periodic version of an error-function initial profile, whose explicit Fourier coefficients allow direct error estimates and simplify the analysis of smooth initialization. By combining Fourier projection with weighted recovery, we retain a single logarithmic precision factor in the full algorithm. Under finite-order derivative bounds, sampled matrix access, and exact state-preparation access, we prepare the normalized solution at time $T$ to accuracy $\varepsilon$ with success probability $\Theta(1)$ using $\mathcal O\!\bigl(g\alpha_AT\log(g\alpha_AT/\varepsilon)\bigr)$ matrix queries and $\mathcal O(g)$ queries to each state-preparation oracle, where $g=\|\boldsymbol{x}_0\|/\|\boldsymbol{x}(T)\|$ and $\alpha_A$ is the matrix-oracle normalization. We illustrate the construction and its recovery probabilities through a numerical experiment on a time-dependent two-cavity system.