The Knill-Laflamme conditions distinguish exact quantum error correction codes, and they have played a critical role in the discovery of state-of-the-art codes. However, the family of exact codes is a very restrictive one and does not necessarily contain the best-performing codes. Therefore, it is desirable to develop a generalized and quantitative performance metric. In this Letter, we derive the near-optimal channel fidelity, a concise and optimization-free metric for arbitrary codes and noise. The metric provides a narrow two-sided bound to the optimal code performance, and it can be evaluated with exactly the same input required by the Knill-Laflamme conditions. We demonstrate the numerical advantage of the near-optimal channel fidelity through multiple qubit code and oscillator code examples. Compared to conventional optimization-based approaches, the reduced computational cost enables us to simulate systems with previously inaccessible sizes, such as oscillators encoding hundreds of average excitations. Moreover, we analytically derive the near-optimal performance for the thermodynamic code and the Gottesman-Kitaev-Preskill code. In particular, the Gottesman-Kitaev-Preskill code's performance under excitation loss improves monotonically with its energy and converges to an asymptotic limit at infinite energy, which is distinct from other oscillator codes.
The Knill-Laflamme conditions distinguish exact quantum error correction codes, and they have played a critical role in the discovery of state-of-the-art codes. However, the family of exact codes is a very restrictive one and does not necessarily contain the best-performing codes. Therefore, it is desirable to develop a generalized and quantitative performance metric. In this Letter, we derive the near-optimal channel fidelity, a concise and optimization-free metric for arbitrary codes and noise. The metric provides a narrow two-sided bound to the optimal code performance, and it can be evaluated with exactly the same input required by the Knill-Laflamme conditions. We demonstrate the numerical advantage of the near-optimal channel fidelity through multiple qubit code and oscillator code examples. Compared to conventional optimization-based approaches, the reduced computational cost enables us to simulate systems with previously inaccessible sizes, such as oscillators encoding hundreds of average excitations. Moreover, we analytically derive the near-optimal performance for the thermodynamic code and the Gottesman-Kitaev-Preskill code. In particular, the Gottesman-Kitaev-Preskill code's performance under excitation loss improves monotonically with its energy and converges to an asymptotic limit at infinite energy, which is distinct from other oscillator codes.
Approximate quantum error correction (AQEC) extends the framework of discrete- and continuous-variable quantum error correction beyond the Knill-Laflamme (KL) conditions, where the recovery performance is quantified by entanglement fidelity. Recent studies have enabled efficient evaluation of near-optimal entanglement fidelity using transpose-channel recovery. Yet, determining the global optimal recovery map and its entanglement fidelity for general codes beyond the KL conditions remains a major computational challenge. Direct optimization becomes prohibitive as the number of noise Kraus operators grows rapidly with system size, and existing approaches lack rigorous guarantees for reducing this optimization to a tractable dimension. Here, we derive an explicit characterization of the optimal environmental state of complement channel, which transforms the optimization over recovery channels into an equivalent optimization over quotient unitaries. For a broader class of codes that satisfy only the orthogonality part of the KL conditions, we show that the optimal recovery map admits an explicit analytical form. Building on this form, we derive novel rigorous lower bounds of entanglement fidelity that strictly improve upon the transpose-recovery bound. We further develop a novel recovery strategy based on principle components, and derive a rigorous bound on the error introduced by noise truncation. Our approach enables efficient searches for approximate recovery maps for AQEC codes, avoiding the need to optimize over the full Kraus-operator space.
We develop a theory of approximate quantum error correction (QEC) based on the error-set model, complemented by general methods for code construction. Exact QEC has a powerful error-set structure: by the Knill-Laflamme conditions, a code correcting a given error set automatically protects against every channel whose Kraus operators lie in their linear span. This linearity gives rise to code distance, the equivalence between erasures and general errors, and a theory of asymptotically good codes. A longstanding view has been that these features do not extend to AQEC, leaving the theory essentially channel-by-channel. We show instead that, although full Knill--Laflamme linearity fails, a restricted form survives and suffices to extend all three structural features to the approximate setting. Specifically, a common error-set criterion governs families of channels whose Kraus operators are linear combinations of a given error set and whose coefficient matrices satisfy a spectral constraint. Using the Bény-Oreshkov worst-case and Petz average-case frameworks, we derive uniform fidelity guarantees for these families in terms of two new code parameters--the \emph{environment-leakage distance}, controlling worst-case performance, and the \emph{Knill-Laflamme Hellinger distance}, characterizing the average-case performance of Petz recovery.
To demonstrate the scope of this model, we develop partition-based constructions across diverse quantum systems and geometries, placing exact and approximate correction on equal footing. These constructions lead to a metric--error alignment hierarchy for Hilbert spaces, metrics, and error families, which in turn characterizes the resulting recovery guarantees. They yield the first known asymptotically good code families for fermionic systems, one-dimensional Rydberg-blockaded systems, and deletion errors, and extend to other physical platforms.
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.
We derive necessary and sufficient conditions for the approximate correctability of a quantum code, generalizing the Knill-Laflamme conditions for exact error correction. Our measure of success of the recovery operation is the worst-case entanglement fidelity. We show that the optimal recovery fidelity can be predicted exactly from a dual optimization problem on the environment causing the noise. We use this result to obtain an estimate of the optimal recovery fidelity as well as a way of constructing a class of near-optimal recovery channels that work within twice the minimal error. In addition to standard subspace codes, our results hold for subsystem codes and hybrid quantum-classical codes.
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.
We present analytic estimates of the performances of various approximate quantum error-correction schemes for the generalized amplitude-damping (GAD) qubit channel. Specifically, we consider both stabilizer and nonadditive quantum codes. The performance of such error-correcting schemes is quantified by means of the entanglement fidelity as a function of the damping probability and the nonzero environmental temperature. The recovery scheme employed throughout our work applies, in principle, to arbitrary quantum codes and is the analog of the perfect Knill-Laflamme recovery scheme adapted to the approximate quantum error-correction framework for the GAD error model. We also analytically recover and/or clarify some previously known numerical results in the limiting case of vanishing temperature of the environment, the well-known traditional amplitude-damping channel. In addition, our study suggests that degenerate stabilizer codes and self-complementary nonadditive codes are especially suitable for the error correction of the GAD noise model. Finally, comparing the properly normalized entanglement fidelities of the best performant stabilizer and nonadditive codes characterized by the same length, we show that nonadditive codes outperform stabilizer codes not only in terms of encoded dimension but also in terms of entanglement fidelity.