Performance and structure of single-mode bosonic codes (1708.05010v3)

Abstract: The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit in an oscillator has recently been followed by cat- and binomial-code proposals. Numerically optimized codes have also been proposed, and we introduce new codes of this type here. These codes have yet to be compared using the same error model; we provide such a comparison by determining the entanglement fidelity of all codes with respect to the bosonic pure-loss channel (i.e., photon loss) after the optimal recovery operation. We then compare achievable communication rates of the combined encoding-error-recovery channel by calculating the channel's hashing bound for each code. Cat and binomial codes perform similarly, with binomial codes outperforming cat codes at small loss rates. Despite not being designed to protect against the pure-loss channel, GKP codes significantly outperform all other codes for most values of the loss rate. We show that the performance of GKP and some binomial codes increases monotonically with increasing average photon number of the codes. In order to corroborate our numerical evidence of the cat/binomial/GKP order of performance occurring at small loss rates, we analytically evaluate the quantum error-correction conditions of those codes. For GKP codes, we find an essential singularity in the entanglement fidelity in the limit of vanishing loss rate. In addition to comparing the codes, we draw parallels between binomial codes and discrete-variable systems. First, we characterize one- and two-mode binomial as well as multi-qubit permutation-invariant codes in terms of spin-coherent states. Such a characterization allows us to introduce check operators and error-correction procedures for binomial codes. Second, we introduce a generalization of spin-coherent states, extending our characterization to qudit binomial codes and yielding a new multi-qudit code.

An Analytical Comparison of Single-Mode Bosonic Quantum Codes for Quantum Error Correction

The paper "Performance and structure of single-mode bosonic codes" by Albert et al. provides a comprehensive investigation into different single-mode bosonic quantum codes for quantum communication and error correction. Specifically, the paper explores cat codes, binomial codes, numerically optimized codes, and Gottesman-Kitaev-Preskill (GKP) codes, comparing their performance against the bosonic pure-loss channel, a model for photon loss in optical and microwave quantum communication systems.

Numerical simulations reveal that GKP codes generally outperform other bosonic codes in a pure-loss channel, despite not being explicitly designed for such environments. This is particularly apparent for larger loss rates, where the GKP codes maintain high channel fidelity due to their unique structure based on lattice points in phase space. The paper quantifies this performance using the entanglement fidelity, which is optimized across various parameters of each code class, subject to constraints on the average photon number.

The authors show that cat and binomial codes perform comparably well for small loss rates, both being able to correct errors corresponding to photon loss and dephasing to some degree. For instance, the binomial codes, which are constructed using carefully chosen superpositions of Fock states, showcase an advantage given the ability to manage losses by utilizing a spacing strategy in the quantum registers. Notably, for small photon loss rates, these codes exhibit a delicate balance of robustness and energy efficiency, but their performance falls short compared to GKP codes as the loss rate increases.

Analytically, the authors provide insights into why GKP codes excel in this setting. They derive quantum error-correction (QEC) criteria, demonstrating that the GKP codes possess an exponential suppression of uncorrectable errors at small loss rates, which is not achievable by other studied codes. This exponential error suppression is attributed to the GKP code's ability to pack lattice points tightly in the oscillator’s phase space, thus allowing it to correct errors that manifest as displacements in this space. This result is significant as it underscores the robustness of GKP codes against a broad spectrum of noise, making them promising candidates for future quantum communication systems.

Furthermore, the paper explores the implications of these findings for achievable communication rates using these codes. The authors calculate the hashing bound of the resulting quantum channel after optimal recovery, which offers a lower bound on the codes' capacity for entanglement distillation and thus provides insights into practical communication scenarios. GKP codes once again show favorable results, bridging the gap to the unconstrained capacity of the pure-loss channel more effectively than others, particularly for higher energy constraints.

The structure of binomial codes is also discussed from a theoretical standpoint, unveiling their relation to spin-coherent states and discrete-variable codes. This characterization extends into multi-qubit and multi-mode regimes, hinting at potential advancements in complex quantum systems’ error correction schemes.

The examination of coherent errors via a Kerr nonlinearity model further highlights practical challenges in quantum system implementations. The authors observe that although all codes show reduced performance under significant coherent errors, these findings are substantial indicators for refining error-correction techniques in real-world quantum computing environments.

In summary, Albert et al.'s paper provides a detailed analysis of different bosonic codes' capabilities under pure loss, with profound implications for both theoretical and practical developments in quantum information science. The work advances our understanding of quantum error correction in single-mode bosonic systems and lays the groundwork for future explorations in more complex quantum communication networks.