Gate-Level Quantum Simulation of Nonunitary Linear Dynamics with Hybrid Oscillator-Qubit Architecture

Abstract

We translate the one-mode unitary-dilation framework for nonunitary linear dynamics into a gate-level hybrid oscillator--qubit architecture. An ancillary oscillator encodes the kernel through state preparation and postselection, while a qubit register simulates the discretized system operator. The construction applies to time-independent dynamics $\dot{u}=-(L+iH)u$, including discretized partial differential equations, and removes the $\mathcal{O}(\log M_a)$ ancilla-qubit overhead of a discrete-variable (DV) $M_a$-term quadrature register. We bound the ideal state's squeezed-Fock coefficient-projection error, which decays superalgebraically in cutoff $N$ for Schwartz-class kernels and at a stretched-exponential rate under stronger joint decay and smoothness assumptions. The finite squeezed-Fock state generically has stellar rank $N-1$, making $N$ a measure of the oracle's non-Gaussian resource. For oscillator dimension $N_{\mathrm{Fock}}$, a $p$th-order product formula reaches error ${\epsilon }_t$ in $\mathcal{O}(t^{1+1/p}{N}_{\mathrm{Fock}}^{(p+1)/(2p)}{\epsilon }_t^{-1/p})$ Trotter steps, up to generator-dependent commutator factors. A perturbation bound separates total scaled-map error from postselection probability. We benchmark Law--Eberly synthesis and assess variational SNAP+$\mathcal{D}$ state preparation on heat and advection--diffusion equations. At the prescribed DV sizing, the hybrid CV--DV route has smaller fixed-scale error in all ten instances, while the DV route accepts with fewer repetitions in every instance. These results quantify when a continuous qumode can replace a discretized ancilla register.