Quantum Differential Equation Solver via Hybrid Oscillator-Qubit Linear Combination of Hamiltonian Simulations

TL;DR

This paper presents a hybrid oscillator-qubit approach to solve linear differential equations efficiently, reducing ancilla overhead and providing analytical error bounds, with demonstrated high fidelity in heat-equation benchmarks.

Contribution

It introduces a continuous-variable ancillary mode in LCHS, eliminating explicit qubit overhead, and derives error bounds showing superalgebraic convergence and resource efficiency.

Findings

Achieves at least 99.90% fidelity in heat-equation benchmarks.

Reduces circuit cost compared to matrix-product-state-based implementations.

Provides analytical bounds on error and resource requirements for the hybrid approach.

Abstract

We introduce a hybrid oscillator-qubit formulation of linear combination of Hamiltonian simulation (LCHS) for solving linear ordinary differential equations. Instead of representing the quadrature rule with a discrete-variable (DV) ancilla register in qubit-only LCHS, the method encodes the LCHS kernel in a continuous-variable (CV) ancillary mode, thereby eliminating the explicit $O(\log M_a)$ ancilla-qubit overhead, where $M_a$ is the number of discretized integral terms in the DV quadrature rule. We derive analytical error bounds for two main approximation mechanisms for the ideal kernel state preparation, showing superalgebraic convergence for Schwartz-class kernels in the truncation cutoff $N$. The required CV non-Gaussianity is captured by the finite squeezed-Fock kernel state, which generically has stellar rank $N-1$, identifying the truncation cutoff as a discrete measure of the oracle's non-Gaussian resource. For the hybrid oscillator-qubit evolution, we also obtain a product-formula bound showing that a $p$th-order formula requires $O(t^{1+1/p}(\Gamma_{p,N}/\epsilon_t)^{1/p})$ Trotter steps to reach error $\epsilon_t$, where $\Gamma_{p,N}$ collects Pauli commutator terms weighted by powers of the truncated position-operator norm $\|\hat{x}\|_N$. We further derive a perturbation bound for the probability of obtaining the required oscillator measurement outcome, showing that an $\epsilon$-close implementation of the ideal LCHS oracle in operator norm induces only an $O(\epsilon)$ perturbation in the postselection probability. In the heat-equation benchmarks, the Law--Eberly protocol achieves end-to-end solution fidelity at least 99.90%. A comparison with a matrix-product-state-based DV LCHS implementation further shows that, the hybrid construction uses a substantially more compact oracle description with reduce circuit cost.