Multi-Error-Correcting Amplitude Damping Codes

We construct new families of multi-error-correcting quantum codes for the amplitude damping channel. Our key observation is that, with proper encoding, two uses of the amplitude damping channel simulate a quantum erasure channel. This allows us to use concatenated codes with quantum erasure-correcting codes as outer codes for correcting multiple amplitude damping errors. Our new codes are degenerate stabilizer codes and have parameters which are better than the amplitude damping codes obtained by any previously known construction.Comment: 5 pages. Submitted to ISIT 201

Concatenated Codes for Amplitude Damping

We discuss a method to construct quantum codes correcting amplitude damping errors via code concatenation. The inner codes are chosen as asymmetric Calderbank-Shor-Steane (CSS) codes. By concatenating with outer codes correcting symmetric errors, many new codes with good parameters are found, which are better than the amplitude damping codes obtained by any previously known construction.Comment: 5 page

Codeword Stabilized Quantum Codes for Asymmetric Channels

We discuss a method to adapt the codeword stabilized (CWS) quantum code framework to the problem of finding asymmetric quantum codes. We focus on the corresponding Pauli error models for amplitude damping noise and phase damping noise. In particular, we look at codes for Pauli error models that correct one or two amplitude damping errors. Applying local Clifford operations on graph states, we are able to exhaustively search for all possible codes up to length $9$. With a similar method, we also look at codes for the Pauli error model that detect a single amplitude error and detect multiple phase damping errors. Many new codes with good parameters are found, including nonadditive codes and degenerate codes.Comment: 5 page

Reliable channel-adapted error correction: Bacon-Shor code recovery from amplitude damping

We construct two simple error correction schemes adapted to amplitude damping noise for Bacon-Shor codes and investigate their prospects for fault-tolerant implementation. Both consist solely of Clifford gates and require far fewer qubits, relative to the standard method, to achieve correction to a desired order in the damping rate. The first, employing one-bit teleportation and single-qubit measurements, needs only one fourth as many physical qubits, while the second, using just stabilizer measurements and Pauli corrections, needs only half. We show that existing fault-tolerance methods can be employed for the latter, while the former can be made to avoid potential catastrophic errors and can easily cope with damping faults in ancilla qubits.Comment: 8 pages, 1 figur

Performance and structure of single-mode bosonic 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 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.Comment: 34 pages, 11 figures, 4 tables. v3: published version. See related talk at https://absuploads.aps.org/presentation.cfm?pid=1351

Approximate quantum error correction for generalized amplitude damping errors

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 non-zero environmental temperature. The recovery scheme employed throughout our work applies, in principle, to arbitrary quantum codes and is the analogue 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.Comment: 44 pages, 8 figures, improved v

Entanglement properties of optical coherent states under amplitude damping

Through concurrence, we characterize the entanglement properties of optical coherent-state qubits subject to an amplitude damping channel. We investigate the distillation capabilities of known error correcting codes and obtain upper bounds on the entanglement depending on the non-orthogonality of the coherent states and the channel damping parameter. This work provides a first, full quantitative analysis of these photon-loss codes which are naturally reminiscent of the standard qubit codes against Pauli errors.Comment: 7 pages, 6 figures. Revised version with small corrections; main results remain unaltere

A simple comparative analysis of exact and approximate quantum error correction

We present a comparative analysis of exact and approximate quantum error correction by means of simple unabridged analytical computations. For the sake of clarity, using primitive quantum codes, we study the exact and approximate error correction of the two simplest unital (Pauli errors) and nonunital (non-Pauli errors) noise models, respectively. The similarities and differences between the two scenarios are stressed. In addition, the performances of quantum codes quantified by means of the entanglement fidelity for different recovery schemes are taken into consideration in the approximate case. Finally, the role of self-complementarity in approximate quantum error correction is briefly addressed.Comment: 29 pages, 1 figure, improved v2; accepted for publication in Open Systems and Information Dynamics (2014