Recent advancements in multi-mode Gottesman-Kitaev-Preskill (GKP) codes have shown great promise in enhancing the protection of both discrete and analog quantum information. This broadened range of protection brings opportunities beyond quantum computing to benefit quantum sensing by safeguarding squeezing — the essential resource in many quantum metrology protocols. However, the potential for quantum sensing to benefit quantum error correction has been less explored. In this work, we provide a unique example where techniques from quantum sensing can be applied to improve multi-mode GKP codes. Inspired by distributed quantum sensing, we propose the distributed two-mode squeezing (dtms) GKP codes that offer benefits in error correction with minimal active encoding operations. Indeed, the proposed codes rely on a single (active) two-mode squeezing element and an array of beamsplitters that effectively distributes continuous-variable correlations to many GKP ancillae, similar to continuous-variable distributed quantum sensing. Despite this simple construction, the code distance achievable with dtms-GKP qubit codes is comparable to previous results obtained through brute-force numerical search \cite{lin2023closest}. Moreover, these codes enable analog noise suppression beyond that of the best-known two-mode codes \cite{noh2020o2o} without requiring an additional squeezer. We also provide a simple two-stage decoder for the proposed codes, which appears near-optimal for the case of two modes and permits analytical evaluation.
Recent advancements in multi-mode Gottesman-Kitaev-Preskill (GKP) codes have shown great promise in enhancing the protection of both discrete and analog quantum information. This broadened range of protection brings opportunities beyond quantum computing to benefit quantum sensing by safeguarding squeezing — the essential resource in many quantum metrology protocols. However, the potential for quantum sensing to benefit quantum error correction has been less explored. In this work, we provide a unique example where techniques from quantum sensing can be applied to improve multi-mode GKP codes. Inspired by distributed quantum sensing, we propose the distributed two-mode squeezing (dtms) GKP codes that offer benefits in error correction with minimal active encoding operations. Indeed, the proposed codes rely on a single (active) two-mode squeezing element and an array of beamsplitters that effectively distributes continuous-variable correlations to many GKP ancillae, similar to continuous-variable distributed quantum sensing. Despite this simple construction, the code distance achievable with dtms-GKP qubit codes is comparable to previous results obtained through brute-force numerical search \cite{lin2023closest}. Moreover, these codes enable analog noise suppression beyond that of the best-known two-mode codes \cite{noh2020o2o} without requiring an additional squeezer. We also provide a simple two-stage decoder for the proposed codes, which appears near-optimal for the case of two modes and permits analytical evaluation.
Entanglement has shown promise in enhancing information processing tasks in a sensor network via distributed quantum sensing protocols. As noise is ubiquitous in sensor networks, error correction schemes based on Gottesman-Kitaev-Preskill (GKP) states are required to enhance the performance, as shown in [Q. Zhuang, J. Preskill, and L. Jiang, New J. Phys. 22, 022001 (2020)] assuming homogeneous noise among sensors and perfect GKP states. Here, we extend the analyses of performance enhancement to finite-squeezed GKP states in a heterogeneous noise model. To begin with, we study different concatenation schemes of GKP-two-mode-squeezing codes. While traditional sequential concatenation schemes in previous works do improve the suppression of noise, we propose a balanced concatenation scheme that outperforms the sequential schemes in the presence of finite GKP squeezing. We then apply these results to two specific tasks in distributed quantum sensing (parameter estimation and hypothesis testing) to understand the trade-off between imperfect squeezing and performance. For the first task, we consider an energy-constrained scenario and provide an optimal way to distribute the energy of the finite-squeezed GKP states among the sensors. For the later task, we show that the error probability can still be drastically lowered via concatenation of realistic finite-squeezed GKP codes.
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.
Bosonic encoding of quantum information into harmonic oscillators is a hardware efficient approach to battle noise. In this regard, oscillator-to-oscillator codes not only provide an additional opportunity in bosonic encoding, but also extend the applicability of error correction to continuous-variable states ubiquitous in quantum sensing and communication. In this work, we derive the optimal oscillator-to-oscillator codes among the general family of Gottesman-Kitaev-Preskill (GKP)-stablizer codes for homogeneous noise. We prove that an arbitrary GKP-stabilizer code can be reduced to a generalized GKP two-mode-squeezing (TMS) code. The optimal encoding to minimize the geometric mean error can be constructed from GKP-TMS codes with an optimized GKP lattice and TMS gains. For single-mode data and ancilla, this optimal code design problem can be efficiently solved, and we further provide numerical evidence that a hexagonal GKP lattice is optimal and strictly better than the previously adopted square lattice. For the multimode case, general GKP lattice optimization is challenging. In the two-mode data and ancilla case, we identify the D4 lattice—a 4-dimensional dense-packing lattice—to be superior to a product of lower dimensional lattices. As a by-product, the code reduction allows us to prove a universal no-threshold-theorem for arbitrary oscillators-to-oscillators codes based on Gaussian encoding, even when the ancilla are not GKP states.
The Gottesman-Kitaev-Preskill (GKP) code offers the possibility to encode higher-dimensional qudits into individual bosonic modes with, for instance, photonic excitations. Since photons enable the reliable transmission of quantum information over long distances and since GKP states subject to photon loss can be recovered to some extent, the GKP code has found recent applications in theoretical investigations of quantum communication protocols. While previous studies have primarily focused on GKP qubits, the possible practical benefits of higher-dimensional GKP qudits are hitherto widely unexplored. In this paper, we carry out performance analyses for three quantum repeater protocols based on GKP qudits including concatenations with a multi-qudit quantum polynomial code. We find that the potential data transmission gains for qudits are often hampered by their decreased GKP error-correcting capabilities. However, we also identify parameter regimes in which having access to an increased number of quantum levels per mode can enhance the theoretically achievable secret-key rate of the quantum repeater. Some of our protocols share the attractive feature that local processing and complete error syndrome identification are realizable without online squeezing. Provided a supply of suitable multi-mode GKP states is available, this can be realized with a minimal set of passive linear optical operations, even when the logical qudits are composed of many physical qudits.
Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance.In this work, we propose a highly effective use of the surface Gottesman-Kitaev-Preskill (GKP) code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-level qubits.In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum-likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates.Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from 0.87% to 0.36% at a GKP squeezing of 12 dB, compared to the case where the simple closest-integer decoding is used.Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by 9.9 dB under the the assumption that finite squeezing of the GKP states is the dominant noise source.More importantly, we show that a low logical failure rate p L < 10 -7 can be achieved with moderate hardware requirements, e.g., 291 modes and 97 qubits at a GKP squeezing of 12 dB as opposed to 1457 bare qubits for the standard rotated surface code at an equivalent noise level (i.e., p = 0.36%).Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface-code decoder.Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds.We also show that a highly squeezed GKP state of GKP squeezing 12 dB can be experimentally realized by using a dissipative stabilization method, namely, the big-small-big method, with fairly conservative experimental parameters.Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.
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.