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 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.

Figure 1

Effect of the nonzero environmental temperature on quantum coding. The truncated series expansion of the entanglement fidelity $\mathcal{F}\left(\gamma \right)$ vs the AD parameter $\gamma$ with $0\le \gamma \le {10}^{-1}$ for the five-qubit code: $\varepsilon \left(t\right)=0$ (dashed line); $\varepsilon \left(t\right)={10}^{-1}\gamma$ (thin solid line); $\varepsilon \left(t\right)=3\times {10}^{-1}\gamma$ (thick solid line).

Figure 2

Robustness against nonzero environmental temperature of degenerate and nondegenerate codes. The truncated series expansions of the entanglement fidelity $\mathcal{F}\left(t\right)$ vs the environmental temperature $T$ for $\hslash =\omega =k_B=1$ and $\gamma =10\varepsilon$: the Shor nine-qubit code (dashed line) and the CSS seven-qubit code (thin solid line).

Figure 3

Ranking additive codes. The truncated series expansions of the entanglement fidelity $\mathcal{F}\left(\gamma \right)$ vs the AD parameter $\gamma$ for $\varepsilon =0$ and $0\le \gamma \le {10}^{-1}$: the Shor nine-qubit code (dotted line), the degenerate six-qubit code (dashed line), the five-qubit code (thin solid line), and the CSS seven-qubit code (thick solid line).

Figure 4

Additive vs nonadditive codes of length nine. The truncated series expansions of the normalized entanglement fidelity $\mathcal{F}\left(\gamma \right)$ vs the AD parameter $\gamma$ for $\varepsilon =0$ and $0\le \gamma \le {10}^{-1}$: the Shor nine-qubit code (thin solid line) and the $\left(\right(9,12,3\left)\right)$ code (dashed line).

Figure 5

Additive vs nonadditive codes of length eight. The truncated series expansions of the normalized entanglement fidelity $\mathcal{F}\left(\gamma \right)$ vs the AD parameter $\gamma$ for $\varepsilon =0$ and $0\le \gamma \le {10}^{-1}$: Gottesman's $\left[\left[8,3,3\right]\right]$ code (thin solid line) and the $\left(\left(8,12\right)\right)$ code (dashed line).

Figure 6

Ranking nonadditive codes. The truncated series expansions of the normalized entanglement fidelity $\mathcal{F}\left(\gamma \right)$ vs the AD parameter $\gamma$ for $\varepsilon =0$ and $0\le \gamma \le {10}^{-1}$: the $\left(\left(6,5\right)\right)$-code (dotted line), the $\left(\left(8,12\right)\right)$ code (dashed line), the $\left(\left(9,12,3\right)\right)$ code (thin solid line), and the $\left(\left(11,2,3\right)\right)$ code (thick solid line).

Figure 7

Performance of the CSS seven-qubit code. The nontruncated expression of the entanglement fidelity $\mathcal{F}\left(\gamma \right)$ vs the AD parameter $\gamma$ in the presence (thin solid line) and absence (dashed line) of QEC.

Figure 8

Performance of the Shor nine-qubit code. The nontruncated expression of the entanglement fidelity $\mathcal{F}\left(\gamma \right)$ vs the AD parameter $\gamma$ in the presence (thin solid line) and absence (dashed line) of QEC.