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arXiv:2605.10708 · Quantum Science and Technology

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

Elin Ranjan Das, Muqing Zheng, Rishab Dutta, Ang Li, Timothy Stavenger, Yuan Liu

quant-phSubmitted May 11, 2026• Announced May 12, 2026• 0 citations• Revised Aug 21, 2026 (v3)
v1 · May 11, 2026v2 · Jul 27, 2026v3 · Aug 21, 2026
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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 $ε_t$ in $\mathcal{O}(t^{1+1/p}N_{\mathrm{Fock}}^{(p+1)/(2p)}ε_t^{-1/p})$ 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.

DOI
10.1088/2058-9565/aea2c6

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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 u˙=−(L+iH)u\dot u=-(L+iH)uu˙=−(L+iH)u, including discretized partial differential equations, and removes the O(log⁡Ma)\mathcal{O}(\log M_a)O(logMa​) ancilla-qubit overhead of a discrete-variable (DV) MaM_aMa​-term quadrature register. We bound the ideal state's squeezed-Fock coefficient-projection error, which decays superalgebraically in cutoff NNN 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−1N-1N−1, making NNN a measure of the oracle's non-Gaussian resource. For oscillator dimension NFockN_{\mathrm{Fock}}NFock​, a pppth-order product formula reaches error εtε_tεt​ in O(t1+1/pNFock(p+1)/(2p)εt−1/p)\mathcal{O}(t^{1+1/p}N_{\mathrm{Fock}}^{(p+1)/(2p)}ε_t^{-1/p})O(t1+1/pNFock(p+1)/(2p)​εt−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+D\mathcal DD 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.
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We develop a hybrid qubit--qumode algorithm for nonlinear ordinary differential equations of the form x˙=f(x)\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})x˙=f(x) with drift of polynomial degree~LLL. Following the Fokker--Planck route of Tennie and Magri, the algorithm propagates the state density and returns the deterministic trajectory as the peak of that density in the small-noise limit. The discretised generator is carried into a parametrised family of Schrödinger equations by the warped-phase transformation of Jin, Liu, and Yu, and the Fourier-mode parameter of that family is placed on a single continuous-variable qumode. Our central structural result is that the Hermitian parts H1H_{1}H1​ and H2H_{2}H2​ of the discretised generator admit a bipartite Pauli decomposition that sorts the non-zero Pauli strings into O(log⁡N)\mathcal{O}(\log N)O(logN) mutually commuting families and factorises each family into a diagonal of degree at most LLL tensored with a fixed rank-two bond operator. The factorisation renders each family exponential an exact product of O(nL)\mathcal{O}(n^{L})O(nL) monomial-controlled momentum displacements, with no intra-family Trotter error. On a ddd-dimensional grid of N=2nN=2^{n}N=2n points per axis the circuit costs O(dL+1nL+2)\mathcal{O}(d^{L+1}n^{L+2})O(dL+1nL+2) gates per Trotter step. No sparse-access oracle and no block encoding is invoked: every gate is fixed in closed form by the polynomial coefficients of the drift. We also prove a bound on the numerical abscissa λmax⁡(H1)λ_{\max}(H_{1})λmax​(H1​) that fixes the recovery domain of the warped-phase transform and the post-selection cost. A classical simulation on two nonlinear benchmarks confirms the structural theorems, the shifted recovery, and the accuracy-per-resource advantage of the continuous-variable coupling over a discretised mode register.
Hybrid continuous-variable--discrete-variable (CV--DV) architectures process quantum information in bosonic modes and qubits, but noise limits their performance. To reduce the noise, existing DV error correction must be complemented by CV noise reduction. Existing CV noise-reduction schemes -- such as GKP-stabilizer codes -- can reduce CV noise, but only for Gaussian gates. Therefore, no current noise-reduction scheme can correct arbitrary CV--DV gates, including non-Gaussian ones. Here, we develop noise reduction for a universal CV--DV gate set, making it applicable to arbitrary CV--DV gates. We do so by introducing an ancilla qubit into a GKP-stabilizer code, allowing us to reduce the standard deviation of Gaussian displacement noise from σσσ to O~(σ2)\tilde O(σ^2)O~(σ2). To demonstrate the scheme, we show that it significantly reduces noise and improves fidelity in the preparation of non-Gaussian cat and Fock states.
Hybrid continuous-variable (CV)-discrete-variable (DV) quantum systems present a promising direction for quantum computing by combining the high dimensional encoding capabilities of qumodes with the control offered by DV qubits on the coupled qumodes. There have been exciting recent progresses on hybrid CV-DV quantum computing, including variational algorithms, error correction, compiler-level optimizations for Hamiltonian simulation, etc. However, there is a lack of a standardized CV-DV benchmark suite for assessing various emerging hardware platforms and evaluating software optimizations on hybrid CV-DV circuits. In this work, we introduce a simulation and benchmarking framework for hybrid CV-DV circuits, implemented using Bosonic Qiskit-a tool specifically designed to model CV-DV systems, along with QuTip for functional correctness verification. We construct and characterize representative CV-DV benchmarks, including cat state generation, GKP state generation, CV-DV state transfers, hybrid quantum Fourier transform, variational quantum algorithms, Hamiltonian simulation, and Shor's algorithm. To assess circuit complexity and scalability, we define a feature map organized into two categories: general features (e.g., qubit/qumode count, gate counts) and CV-DV-specific features (e.g., Wigner negativity, energy, truncation cost). These metrics enable evaluation of both classical simulability and hardware resource requirements. Our results, including one benchmark on real hardware, demonstrate that hybrid CV-DV architectures are not only viable but well-suited for a range of computational tasks, from optimization to Hamiltonian simulation. This framework lays the groundwork for systematic evaluation and future development of hybrid quantum systems.
Quantum simulation has primarily focused on unitary dynamics, while many physical and engineering systems can be modeled by linear ordinary differential equations whose generators include non-Hermitian terms. Recent studies have shown that such equations, which give rise to nonunitary dynamics, can be embedded into a larger unitary framework via dilation techniques. However, their concrete realization on quantum circuits remains underexplored. In this paper we present a concrete pipeline that connects the dilation formalism with explicit quantum circuit constructions. On the analytical side, building on the recent dilation framework, we introduce a discretization of the continuous dilation operator that is tailored for quantum implementation. This construction ensures an exactly skew-Hermitian ancillary generator, which allows the moment conditions to be satisfied without imposing artificial constraints. We prove that the resulting scheme achieves a global error bound of order O(M−3/2)O(M^{-3/2})O(M−3/2), up to exponentially small boundary effects. This error can be suppressed by refining the discretization, where MMM denotes the discretization parameter. On the algorithmic side, we demonstrate that the dilation triple (Fh,∣rh⟩,⟨lh∣)(F_h, |r_h\rangle, \langle l_h|)(Fh​,∣rh​⟩,⟨lh​∣) can be efficiently implemented on quantum circuits. Using linear combinations of unitaries, QFT-adder operators, and quantum singular value transformation, the framework requires resources ranging from O(log⁡M)O(\log M)O(logM) to O((log⁡M)2)O((\log M)^2)O((logM)2), depending on the stage of the pipeline.
The realization of universal control in hybrid oscillator-qubit quantum processors enables the systematic design and implementation of quantum algorithms. However, the algorithmic development for such platforms remains at an early stage. While the Quantum Approximate Optimization Algorithm (QAOA) has been extensively studied in both continuous-variable (CV) and discrete-variable (DV) quantum systems, its development in the hybrid CV-DV setting remains limited. In this paper, we propose a hardware-native non-Abelian mixer for QAOA on hybrid CV-DV quantum processors and develop a corresponding hybrid ansatz for the Max-Cut problem. We evaluate the proposed ansatz on unweighted Erdős-Rényi graphs and benchmark it against the standard transverse-field mixer using the approximation ratio and optimal-solution probability. Across all graph sizes and Fock cutoffs in our simulations, the proposed non-Abelian mixer consistently improves both expected solution quality and the probability of sampling an optimal solution relative to the transverse-field mixer. These results indicate that the proposed non-Abelian mixer is a promising building block for QAOA on hybrid oscillator-qubit platforms.