Efficient Qubit Simulation of Hybrid Oscillator-Qubit Quantum Computation

TL;DR

This work establishes that hybrid oscillator-qubit algorithms can be implemented on qubit processors with polynomial overhead, providing new insights into the computational power trade-offs between discrete-variable and hybrid continuous-discrete-variable quantum computing.

Abstract

We introduce a framework for simulating hybrid oscillator-qubit quantum processors on qubit-only systems through position encoding. By encoding continuous-variable position and momentum wave functions into qubit amplitudes, our method efficiently simulates all Gaussian and conditional Gaussian operations -- encompassing the phase-space instruction set (beam splitter, single-qubit rotation, conditional displacement) and extending to squeezing, conditional squeezing, conditional rotation, and conditional beam splitter -- using $O\!\left(\log^2\!\left(Γ+ \log(1/ε)\right)\right)$ qubit gates per hybrid gate, where $Γ$ is the Fock-level bound and $ε$ is the target precision. This polylogarithmic per-gate complexity represents an exponential improvement over Fock basis encoding approaches, which require exponential quantum or classical resources in the number of qubits per mode. We provide rigorous numerical characterization of quantum Fourier transform errors for Fock-bounded states, enabling precise resource estimation for practical implementations. This work establishes that hybrid oscillator-qubit algorithms can be implemented on qubit processors with polynomial overhead, providing new insights into the computational power trade-offs between discrete-variable and hybrid continuous-discrete-variable quantum computing.

Figures (8)

• Figure 1: Per-Fock QFT error $\epsilon_F(\ket{k}_{F})$ as a function of Fock level $k$ for different grid sizes $N = 2^n$. The linear relationship on the log scale confirms the per-Fock scaling $\log\epsilon_F(\ket{k}_{F}) \lesssim ak + b$. The dashed lines show the fitted formula in \ref{['eq:qft-err-numerical']} with coefficients in \ref{['eq:qft-coeff-scaling']}.
• Figure 2: Per-Fock gate discretization error $\epsilon_U(\ket{k}_{F})$ as a function of Fock level $k$ for different grid sizes $N = 2^n$. Each point gives the error when the gate is applied to the individual Fock state $\ket{k}_{F}$. (a) Displacement with $\alpha=2$. (b) Rotation with $\theta=\pi/4$. (c) Squeezing with $r=1$. (d) Beam splitter with $\theta=\pi/2$ applied to $\ket{k}_{F}\ket{0}_{F}$.
• Figure 3: Qubit circuit to simulate the elementary displacement operator $e^{it\hat{q}}$ on position-encoding qubits.
• Figure 4: Qubit circuits to simulate elementary quadratic Gaussian operators.
• Figure 5: The (a) $\widetilde{\mathcal{SWAP}}$, (b) $\widetilde{\mathcal{P}}$, and (c) $\widetilde{\mathcal{CP}}$ gates, each corresponding to the discretization of their continuous counterparts, $\mathcal{SWAP}$, $\mathcal{P}$, and $\mathcal{CP}$, respectively.

Theorems & Definitions (3)

• Lemma 1: Per-Fock QFT Error
• Corollary 1: General-State QFT Error
• Theorem 1: Elementary Decomposition