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 integral kernel through state preparation and postselection. A qubit register represents and 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 squeezed-Fock coefficient-projection error of the ideal kernel state. It decays superalgebraically with cutoff \(N\) for Schwartz-class kernels and at a stretched-exponential rate under stronger joint decay and smoothness assumptions. The finite squeezed-Fock kernel state generically has stellar rank \(N-1\), making \(N\) a discrete measure of the oracle's non-Gaussian resource. For hybrid oscillator--qubit evolution, a \(p\)th-order product formula requires \(\mathcal{O}(t^{1+1/p}N_{\mathrm{Fock}}^{(p+1)/(2p)}\epsilon_t^{-1/p})\) Trotter steps in the worst case, up to generator-dependent commutator factors, to reach error \(\epsilon_t\), where \(N_{\mathrm{Fock}}\) is the oscillator dimension. A perturbation bound separates the total scaled-map error from the physical postselection probability. We benchmark Law--Eberly synthesis and assess a variational SNAP+\(\mathcal D\) route at the state-preparation level on discretized heat-equation instances. For full circuit-level maps of one-dimensional heat and non-normal advection--diffusion instances up to \(D=32\), the fixed-scale map error is at most \(1.48\%\) and the conditional infidelity at most \(4.68\times10^{-4}\) over all computational-basis inputs. At kernel parameters selected on the one-dimensional family, a \(4\times4\) two-dimensional stress case reaches \(7.40\%\) fixed-scale error under a reference norm shrunk by the stronger two-dimensional damping, with worst-input conditional infidelity \(6.20\times10^{-3}\). At the prescribed DV sizing, the hybrid CV--DV route has smaller fixed-scale error in all ten instances. The DV route accepts with fewer repetitions in every instance. These results provide a block-by-block finite-size resource account of when a single continuous qumode can replace a discretized ancilla register.