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.
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.
This thesis develops a theoretical framework for hybrid continuous-variable (CV) and discrete-variable (DV) quantum systems, with emphasis on quantum control, state preparation, and error correction. A central contribution is non-abelian quantum signal processing (NA-QSP), a generalization of quantum signal processing to settings where control parameters are non-commuting operators. Within this framework, we introduce the Gaussian-Controlled-Rotation (GCR) protocol, which enables high-fidelity control of CV states using DV ancillae. This approach allows for deterministic preparation of squeezed, cat, and Gottesman-Kitaev-Preskill (GKP) states without numerical optimization. Two previously unpublished contributions are included: (i) Chapter 2.3 introduces the Gaussian hierarchy, a classification of CV operations analogous to the Clifford hierarchy, offering a new lens for understanding CV gate sets; (ii) Chapter 5 presents an analytical framework for correcting photon loss in finite-energy GKP codes, introducing the notion of probabilistic error correction, providing insight into recent GKP experiments surpassing break-even thresholds. Overall, this work lays the groundwork for scalable, fault-tolerant quantum computation in hybrid CV-DV architectures, with applications to logical gate synthesis, readout, and hybrid algorithm design using ancilla oscillators.
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.
The continuous-variable (CV) Gaussian no-go theorem fundamentally limits the suppression of Gaussian displacement errors using only Gaussian gates and states. Prior studies have employed Gottesman-Kitaev-Preskill (GKP) states as ancillary qumodes to suppress small Gaussian displacement errors, but when the displacement magnitude becomes large, lattice-crossing events arise beyond the correctable range of the GKP state. To address this issue, we concatenate a Gaussian-noise-suppression circuit with an outer analog Steane code that corrects such occasional lattice-crossing events as well as other abrupt displacement errors. Unlike conventional concatenation, which primarily aims to reduce logical error rates, the Steane-GKP duality in encoding provides complementary protection against both large and small displacement errors, enabling CV error correction within the continuous encoding space and contrasting with earlier approaches that concatenate GKP states with repetition codes for discrete qubit or qudit encodings. Analytical results show that, under infinite squeezing, the concatenated code suppresses the variance of Gaussian displacement errors across all qumodes by up to 50 percent while enabling unbiased correction of lattice-crossing events, with a success probability determined by the ratio between the residual Gaussian error standard deviation and the lattice-crossing magnitude. Even with finite squeezing, the proposed architecture continues to provide Gaussian-error suppression together with lattice-crossing correction, and the presence of the outer analog Steane code relaxes the squeezing requirement of the inner GKP states, indicating near-term experimental feasibility. This work establishes a viable route toward fault-tolerant continuous-variable quantum computation and provides new insight into the design of concatenated CV error-correcting architectures.
Bosonic quantum error correction represents a route towards significant hardware overhead reduction for fault-tolerant quantum computation. However, controlling quantum harmonic oscillators that encode the quantum information comes with its own challenges. In many platforms such as superconducting circuits and trapped ions, the bosonic degree of freedom is controlled by an ancillary qubit whose decay during a qubit-oscillator gate propagates to the oscillator as a continuous error. Such errors generally lie beyond the scope of errors that the bosonic code is capable of correcting. We introduce the continuous-variable flag: an auxiliary oscillator coupled to the ancilla whose phase-space position encodes the continuous error corrupting the storage mode. After the gate, a heterodyne measurement of the flag estimates this continuous-variable error and a subsequent feedback control undoes it up to a bosonic code stabilizer. The scheme requires no new ingredients, as the flag is driven by the same dispersive coupling, displacements and rotations that implement the gates it is supposed to protect. We construct sequences protecting conditional rotations to all orders in ancilla decay with one flag, and conditional displacements to first order with one flag and to all orders with two. Simulations with realistic circuit-QED parameters predict more than three orders of magnitude fewer logical errors for the parity measurement of a four-component cat code, and a finite-energy GKP logical lifetime near its noiseless-ancilla bound. By discretizing continuous-variable errors, the flags introduced in this work are a useful tool for any hybrid oscillator-qubit processor, and open the route towards fault-tolerant simulation and computation with bosonic modes.
To implement fault-tolerant quantum computation with continuous variables, Gottesman-Kitaev-Preskill (GKP) qubits have been recognized as an important technological element. However, the analog outcome of GKP qubits, which includes beneficial information to improve the error tolerance, has been wasted, because the GKP qubits have been treated as only discrete variables. In this Letter, we propose a hybrid quantum error correction approach that combines digital information with the analog information of the GKP qubits using a maximum-likelihood method. As an example, we demonstrate that the three-qubit bit-flip code can correct double errors, whereas the conventional method based on majority voting on the binary measurement outcome can correct only a single error. As another example, we show that a concatenated code known as Knill's C_{4}/C_{6} code can achieve the hashing bound for the quantum capacity of the Gaussian quantum channel (GQC). To the best of our knowledge, this approach is the first attempt to draw both digital and analog information to improve quantum error correction performance and achieve the hashing bound for the quantum capacity of the GQC.
In order to solve practical problems on a quantum computer, it is necessary to use fault-tolerant quantum error correction schemes to overcome noise: the errors arising from imperfections in physical components. The Gottesman-Kitaev-Preskill (GKP) code aims at achieving this in a hardware efficient manner by encoding finite-dimensional logical subspaces in the Hilbert space of one or more continuous variable modes. In near term implementations, however, it is not feasible to eliminate errors entirely, so it is natural to also employ alternative error mitigation techniques together with error correction. In this work, we study a quantum error mitigation method known as probabilistic error cancellation in the context of the GKP code. We compare Steane-type and teleportation-based GKP error correction, and calculate the sampling overheads associated with the technique for square and hexagonal GKP codes. We employ the stabilizer subsystem decomposition for GKP codes to obtain an effective logical channel for noisy operations that is used to express the ideal one of a target logical unitary. We consider noise from finite squeezing on the data and the two ancilla modes, needed for error correction, and examine the relationship between the sampling overhead and the noise for different decoding methods. Our results are calculated for single- and two-qubit GKP Clifford gates and one round of error correction, and show the nontrivial combination of error correction and mitigation in continuous variable codes.