Bosonic quantum error-correcting codes encode logical information in a harmonic oscillator, with the Gottesman-Kitaev-Preskill (GKP) and number-phase (NP) codes representing two fundamentally different encoding paradigms. Although both have been extensively studied, it remains unclear under what physical noise conditions (including photon loss and dephasing) one encoding intrinsically outperforms the other. Here we estimate a quantitative performance boundary between GKP and NP codes under general photon loss-dephasing noise. By optimizing code parameters within each encoding family, we identify the noise regimes in which each code exhibits a fundamental advantage. In particular, we find that the crossover occurs when the dephasing strength is approximately two orders of magnitude smaller than the loss strength, revealing a sharp separation between operational regimes. Beyond this specific comparison, our work establishes a practical and extensible methodology for benchmarking bosonic codes and optimizing their parameters, providing concrete guidance for the experimental selection and deployment of bosonic encodings in realistic noise environments.
Bosonic quantum error-correcting codes encode logical information in a harmonic oscillator, with the Gottesman-Kitaev-Preskill (GKP) and number-phase (NP) codes representing two fundamentally different encoding paradigms. Although both have been extensively studied, it remains unclear under what physical noise conditions (including photon loss and dephasing) one encoding intrinsically outperforms the other. Here we estimate a quantitative performance boundary between GKP and NP codes under general photon loss-dephasing noise. By optimizing code parameters within each encoding family, we identify the noise regimes in which each code exhibits a fundamental advantage. In particular, we find that the crossover occurs when the dephasing strength is approximately two orders of magnitude smaller than the loss strength, revealing a sharp separation between operational regimes. Beyond this specific comparison, our work establishes a practical and extensible methodology for benchmarking bosonic codes and optimizing their parameters, providing concrete guidance for the experimental selection and deployment of bosonic encodings in realistic noise environments.
Bosonic quantum computation harnesses the infinite-dimensional Hilbert space of harmonic oscillators, offering scalable architectures with inherent error correction advantages over qubit-based systems. Despite their potential, bosonic modes encounter significant challenges from prevalent noise sources, particularly photon loss and dephasing, which degrade quantum coherence. Among bosonic error-correcting codes, the Gottesman-Kitaev-Preskill (GKP) code stands out for its ability to correct small phase-space displacements, enabling applications in quantum communication, sensing, and computation. While ideal GKP states require infinite squeezing and are unphysical, the finite squeezing approximate GKP states have been experimentally realized, though their performance depend critically on the squeezing level and measurement strategies. This work addresses a central challenge: how to optimize the squeezing parameter of approximate GKP states and the local oscillator (LO) phase in homodyne detection to maximize precision in estimating bosonic loss-dephasing channel parameters. To address this question, we introduce an adaptive optimization framework guided by classical Fisher information (CFI) and quantum Fisher information (QFI). Numerical simulations demonstrate that this adaptive protocol consistently converges to optimal parameter configurations, determining optimized squeezing levels and LO phases. By offering a practical approach to highprecision quantum channel parameter estimation, this framework has broad implications, from quantum metrology to enabling robust quantum communication systems.
The performance of Gottesman-Kitaev-Preskill (GKP) codes, an approach to hardware-efficient quantum error correction, is limited by the finite squeezing capabilities of current experimental platforms. To circumvent this hardware demand, we introduce Energy-Scaled Zero-Noise Extrapolation (ES-ZNE), a quantum error mitigation protocol that uses the mean photon number of the GKP code as a tunable effective noise parameter. The protocol measures logical observables at a series of accessible finite energies and extrapolates the results to the ideal, infinite-energy limit using an ansatz based on the code's asymptotic error scaling. Through simulating a GKP qubit under a pure-loss channel, we demonstrate that ES-ZNE successfully mitigates finite-energy errors, recovering the ideal expectation values (within numerical uncertainty) in the shallow-noise regime. Furthermore, by computationally removing artifacts arising from the finite-energy encoding, our method characterizes the intrinsic performance of the ideal GKP code, revealing a sharp error threshold beyond which the code's corrective power diminishes. These results establish ES-ZNE as a practical, software-based strategy for enhancing the performance of near-term bosonic quantum processors, trading sampling overhead for demanding physical resources like high squeezing.
Quantum error-correction codes protect information from realistic noisy channels and lie at the heart of quantum computation and communication tasks. Understanding the optimal performance and other information-theoretic properties, such as the achievable rates, of a given code is crucial, as these factors determine the fundamental limits imposed by the encoding in conjunction with the noise channel. Here, we use the transpose channel to analytically obtain the near-optimal performance of any Gottesman-Kitaev-Preskill (GKP) code under pure loss and pure amplification. We present rigorous connections between GKP code’s near-optimal performance and its dual lattice geometry and average input energy. With no energy constraint, we show that when | τ / ( 1 − τ ) | is an integer, specific families of GKP codes simultaneously achieve the loss and amplification capacity. τ is the transmissivity (gain) for loss (amplification). Our results establish GKP code as the first structured bosonic code family that achieves the capacity of loss and amplification.
Bosonic quantum error correction (QEC) relies on non-Gaussian bosonic encodings whose preparation cost is a central practical constraint. In this work, we use stellar rank as a resource measure to design and benchmark bosonic codes under finite non-Gaussian resources. For fixed cat and Gottesman--Kitaev--Preskill (GKP) code families, we show that finite stellar rank creates a trade-off among state approximability, energy, and logical protection under photon loss and photon-number dephasing, evaluated with optimal recovery. This trade-off implies that codewords with better ideal error-correction properties need not be optimal once finite-rank preparation constraints are imposed. Going beyond fixed-target codewords, we directly optimize bosonic encodings at fixed stellar rank, revealing noise-adapted code structures and concrete resource thresholds. Grid-like encodings emerge under photon loss, whereas approximately rotation-symmetric encodings arise under dephasing. In the optimized search, stellar rank k=2 suffices to surpass break-even for all dephasing strengths considered, while under photon loss the required rank increases with the loss rate. These results establish stellar rank as an operationally meaningful resource measure for bosonic QEC under practical state-preparation constraints.
The Gottesman-Kitaev-Preskill (GKP) code is a promising bosonic quantum error-correcting code, encoding logical qubits into a bosonic mode in such a way that many physically relevant noise types can be corrected effectively. A particularly relevant noise channel is the pure loss channel, which the GKP code is known to protect against. In particular, it is commonly pointed out that losses can be corrected by the GKP code by transforming the losses into random Gaussian displacements through a quantum-limited amplification channel. However, implementing such amplification in practice is not ideal and could easily introduce an additional overhead of noise from associated experimental imperfections. Here, we analyze the performance of teleportation-based GKP error correction against loss in the absence of an amplification channel. We show that amplification is not required to perform GKP error correction and that performing amplification actually worsens the performance for practically relevant parameter regimes.
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 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 multiqubit 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 multiqudit code.