Correction of Data and Syndrome Errors by Stabilizer Codes

Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of error information--error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose the idea of quantum data-syndrome (DS) codes that are capable of correcting both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes and provide several CSS-type code constructions of quantum DS codes.Comment: 2 figures. This is a short version of our full paper (in preparation

Quantum convolutional data-syndrome codes

We consider performance of a simple quantum convolutional code in a fault-tolerant regime using several syndrome measurement/decoding strategies and three different error models, including the circuit model.Comment: Abstract submitted for The 20th IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC 2019

Correcting phenomenological quantum noise via belief propagation

Quantum stabilizer codes often face the challenge of syndrome errors due to error-prone measurements. To address this issue, multiple rounds of syndrome extraction are typically employed to obtain reliable error syndromes. In this paper, we consider phenomenological decoding problems, where data qubit errors may occur between two syndrome extractions, and each syndrome measurement can be faulty. To handle these diverse error sources, we define a generalized check matrix over mixed quaternary and binary alphabets to characterize their error syndromes. This generalized check matrix leads to the creation of a Tanner graph comprising quaternary and binary variable nodes, which facilitates the development of belief propagation (BP) decoding algorithms to tackle phenomenological errors. Importantly, our BP decoders are applicable to general sparse quantum codes. Through simulations of quantum memory protected by rotated toric codes, we demonstrates an error threshold of 3.3% in the phenomenological noise model. Additionally, we propose a method to construct effective redundant stabilizer checks for single-shot error correction. Simulations show that BP decoding performs exceptionally well, even when the syndrome error rate greatly exceeds the data error rate.Comment: 14 pages, 9 figures, 1 tabl