Approximate quantum error correction for generalized amplitude damping errors

TL;DR

This paper analytically evaluates the effectiveness of various approximate quantum error correction codes, including stabilizer and nonadditive codes, for the generalized amplitude damping channel, highlighting the superiority of nonadditive codes.

Contribution

It introduces an analytical framework for assessing approximate quantum error correction schemes for GAD errors, comparing stabilizer and nonadditive codes.

Findings

Nonadditive codes outperform stabilizer codes in entanglement fidelity.

Degenerate stabilizer and self-complementary nonadditive codes are particularly effective.

Analytical results recover known numerical findings for zero-temperature environments.

Abstract

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.