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 $σ$ to $\tilde O(σ^2)$. 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--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 σ to 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.
Robust continuous-variable (CV) quantum information processing requires correcting realistic errors in bosonic systems, but all existing schemes rely on auxiliary Gottesman-Kitaev-Preskill (GKP) states which the preparation and operation are demanding in many platforms. In this work, we propose a novel CV quantum error correction (QEC) scheme that utilizes a broadly accessible resource: discrete-variable (DV) ancilla. Our scheme extracts information about CV displacement to the DV ancilla, measuring that allows counteracting the unwanted displacement error. We show that a simple single-qubit ancilla can already suppress CV infidelity by more than 20%. By concatenating with DV QEC codes, our scheme is robust against the physical errors in hybrid CV-DV systems, and yields a new class of oscillator-in-oscillator code that does not involve GKP states. Our work facilitates the implementation of CV QEC on realistic platforms.
Physical-level noise on traveling bosonic modes remains a critical bottleneck for scalable quantum information processing. We show that for any single-mode bosonic code (qumode) corrupted by thermal or Gaussian displacement noise at loss rate μ and amplification G, a hybrid continuous-discrete-variable (CV-DV) interferometer using a single qubit ancilla and two controlled-Fourier (CF) gates sandwiching the noise channel suppresses its effects from linear to quadratic scaling. This is achieved without active error correction or destructive measurements of the encoded state, maintaining high success probabilities ≥0.5 when μG≤0.5. When supplemented with multiple ancillas, the protocol converts photon loss into coherent Fock-damping, and thermal or displacement noise into a mixture of Fock-diagonal noise. The protocol is entirely code-agnostic. For the special case of 2K-fold rotation-symmetric bosonic codes, it simplifies to conventional error detection and projection with K ancillas. Suppression with simple gates and few ancillas demonstrates a clear hardware-efficient advantage over previously proposed ``bypass'' schemes, where quantum information transferred to the DV ancillas is readily corrupted by ancilla noise. Finally, we extend the protocol to a qutrit DV ancilla. This demonstrates resilience to both CV noise and composite DV damping noise, achieving a truly hybrid noise suppression scheme that operates effectively even on CV encodings lacking a well-defined photon-number parity syndrome.
Noise suppression is of paramount importance for reliable quantum information processing and computation. We show that for any single-mode bosonic code (qumode) corrupted by thermal~noise at rate~η and mean \mbox{excitation}~nˉ, a hybrid continuous-discrete-variable~(CV-DV) interferometer using only a single qubit ancilla~(DV) and two controlled~Fourier~(CF) gates sandwiching the noise channel suppresses its effects to O(η2) \emph{without} any active error correction or destructive measurements of the encoded state and with high success probabilities~>0.5 if~η(1+nˉ)<0.5. This suppression scheme works by conditionally monitoring the photon-number parities after the interferometer. Bosonic codes with two logical states of the same photon-number parity (like-parity codes) are \emph{completely resilient} to DV amplitude- and phase-damping ancilla noise. For such codes, the interferometer simplifies to the use of a qumode rotation gate and a \emph{single} CF~gate. This presents a clear advantage of our CF-gate-based error suppression scheme over previously-proposed ``bypass'' protocols, where qubit information transferred to the DV mode is readily corrupted by damping~noise. Finally, we present a simple extension to direct communication of qumode states between two parties over a noisy channel using a preshared DV entangled state, by implementing a CF gate in the first laboratory and its inverse in the other. Such a communication protocol achieves a similar fidelity performance at the same success rate as the single-party case, but with greater resilience to the ancilla noise than DV~teleportation. Resource-efficient multi-qubit codes that depend on a few essential long-range interactions can benefit from it.
This tutorial offers a pedagogical guide to hybrid quantum processors that integrate discrete-variable (DV) qubits and continuous-variable (CV) oscillators. Aimed at computer scientists, engineers, and physicists, it provides an overview of the experimental, algorithmic, and architectural aspects of this novel and rapidly developing hardware model. Experimental realizations of this model include superconducting, trapped-ion, and neutral-atom platforms. By combining DV and CV components, hybrid oscillator-qubit processors enable a powerful new paradigm that offers complementary strengths for quantum control, error correction, computation, and simulation. Working toward the goal of a full-stack system connecting applications to CV-DV hardware, we define and formulate abstract machine models and instruction set architectures. These essential abstractions enable codesign of hardware and software, and resource estimation for exploring the potential of current and future hardware for computational and simulation tasks. Using these abstractions, we present both new and existing examples that illustrate the benefits of hybrid CV-DV processors relative to traditional DV-only hardware in computation as well as quantum simulation of physical models. Examples include algorithms for transferring states between DV and CV systems, performing the quantum Fourier transform, and simulation of lattice gauge theories. Relative to qubit-only hardware, the bosonic degrees of freedom natively available in hybrid architectures can substantially reduce the circuit complexity of simulations for physical models containing bosons. A key technique is the extension of quantum signal processing ideas to CV-DV systems. This work is intended to serve as a timely and comprehensive guide to this relatively unexplored yet promising approach to quantum computation and to provide a road map to guide future development.
Hybrid quantum systems combining discrete-variable (DV) and continuous-variable (CV) subsystems arise across a broad range of physical platforms and enable quantum information processing beyond purely qubit-based designs. Their classical simulation, however, is challenging since computational costs of both CV and DV systems exponentially increase. In this paper, we introduce a simulation framework for CV--DV hybrid systems based on linear combination of Gaussian (LCoG) decomposition. We decompose the hybrid density matrix into blocks in a DV basis and represent each block as an LCoG expansion. Within the representation, Gaussian operations on the bosonic subsystem independently act on each Gaussian function. We derive analytic formulas describing how complex Gaussian functions transform under controlled Gaussian operations. We further extend the framework to controlled Gaussian channels with state-dependent Gaussian noise. The method avoids a Fock-space truncation for the CV subsystem, and the cost of updating each Gaussian term scales polynomially with the number of modes. The framework provides a general computational tool for analyzing hybrid quantum algorithms, quantum simulation protocols, and non-Gaussian state generation schemes. As an example, we simulate Gottesman--Kitaev--Preskill (GKP) state generation based on a cavity-QED system.
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, including discretized partial differential equations, and removes the O(logMa) ancilla-qubit overhead of a discrete-variable (DV) Ma-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 NFock, a pth-order product formula reaches error εt in 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 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.