Quantifying the Performance of Quantum Codes

Abstract

We study the properties of error correcting codes for noise models in the presence of asymmetries and/or correlations by means of the entanglement fidelity and the code entropy. First, we consider a dephasing Markovian memory channel and characterize the performance of both a repetition code and an error avoiding code in terms of the entanglement fidelity. We also consider the concatenation of such codes and show that it is especially advantageous in the regime of partial correlations. Finally, we characterize the effectiveness of the codes and their concatenation by means of the code entropy and find, in particular, that the effort required for recovering such codes decreases when the error probability decreases and the memory parameter increases. Second, we consider both symmetric and asymmetric depolarizing noisy quantum memory channels and perform quantum error correction via the five qubit stabilizer code. We characterize this code by means of the entanglement fidelity and the code entropy as function of the asymmetric error probabilities and the degree of memory. Specifically, we uncover that while the asymmetry in the depolarizing errors does not affect the entanglement fidelity of the five qubit code, it becomes a relevant feature when the code entropy is used as a performance quantifier.Comment: 21 pages, 10 figure

A Simple Comparative Analysis of Exact and Approximate Quantum Error Correction

“…However, we believe its consideration is suitable for our purposes, since we wish to essentially stress the similarities and differences between exact and approximate error correction schemes avoiding unnecessary complications. The exact error correction analysis for more realistic and truly quantum error models along the lines presented here could be found in previous works of one of the Authors [22][23][24].…”

“…The latter scheme was computed via semidefinite programming methods in [19]. However, no explicit analytical investigation (similar, for instance, to the investigations presented in [22] and [23] for depolarizing and Weyl unitary errors, respectively) is available. Furthermore, as pointed out in [20], numerically computed recovery maps are difficult to describe and understand analytically.…”

Surface Code Design for Asymmetric Error Channels

“…One of the most popular approaches is to concatenate two QECC codes. The simplest codes of this nature are presented in [32], where repetition code is concatenated with CSS code, and in [33], where a concatenation of repetition code and error avoiding code is presented. In [34], symmetric code is combined with an asymmetric one, and fault-tolerant circuits are needed to switch between the symmetric and asymmetric encodings.…”

Quantum stabilizer codes embedding qubits into qudits

“…Entanglement fidelity is a reliable performance measure of the efficiency of quantum error correcting codes [23]. Suppose a two dimensional code C is such that C ⊂ H with dim C H = N (here H can be either H ⊗n 2 or H d , hence N is either 2 n or d).…”