The near-capacity performance of classical low-density parity check (LDPC) codes and their efficient iterative decoding makes quantum LDPC (QLPDC) codes a promising candidate for quantum error correction. In this paper, we present a comprehensive survey of QLDPC codes from the perspective of code design as well as in terms of their decoding algorithms. We also conceive a modified non-binary decoding algorithm for homogeneous Calderbank–Shor–Steane-type QLDPC codes, which is capable of alleviating the problems imposed by the unavoidable length-four cycles. Our modified decoder outperforms the state-of-the-art decoders in terms of their word error rate performance, despite imposing a reduced decoding complexity. Finally, we intricately amalgamate our modified decoder with the classic uniformly reweighted belief propagation for the sake of achieving an improved performance.

Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies

Powerful quantum error correction codes (QECCs) are required for stabilizing and protecting fragile qubits against the undesirable effects of quantum decoherence. Similar to classical codes, hashing bound approaching QECCs may be designed by exploiting a concatenated code structure, which invokes iterative decoding. Therefore, in this paper, we provide an extensive step-by-step tutorial for designing extrinsic information transfer (EXIT) chart-aided concatenated quantum codes based on the underlying quantum-to-classical isomorphism. These design lessons are then exemplified in the context of our proposed quantum irregular convolutional code (QIRCC), which constitutes the outer component of a concatenated quantum code. The proposed QIRCC can be dynamically adapted to match any given inner code using EXIT charts, hence achieving a performance close to the hashing bound. It is demonstrated that our QIRCC-based optimized design is capable of operating within 0.4 dB of the noise limit.

pdf/the-road-from-classical-to-quantum-codes-a-hashing-bound-4iozepc2if.pdf

The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

We address the problem of encoding entanglement-assisted (EA) quantum error-correcting codes (QECCs) and of the corresponding complexity. We present an iterative algorithm from which a quantum circuit composed of CNOT, H, and S gates can be derived directly with complexity O(n2) to encode the qubits being sent. Moreover, we derive the number of each gate consumed in our algorithm according to which we can design EA QECCs with low encoding complexity. Another advantage brought by our algorithm is the easiness and efficiency of programming on classical computers.

http://iopscience.iop.org/article/10.1088/1674-1056/21/2/020304/pdf

Encoding entanglement-assisted quantum stabilizer codes

In this paper, we generalize the feedback sum-product algorithm to the decoding of sparse quantum codes over random balanced X-Z Channels, thereby significantly improving the performance of the concerned sparse quantum codes. We point out that, due to the randomness of the channel, degeneracy errors would be degreased greatly and the improvement of the performance of the sparse quantum codes come from the new decoding algorithm, where more reliable and useful information is feedback to the decoder by using the feedback strategy. As the end of this paper, simulations are given to verify our statements.

Sum-product decoding for sparse quantum codes over random balanced X-Z channel

Decoding sparse quantum codes can be accomplished by syndrome-based decoding through using the sum-product algorithm (SPA). We significantly improve this decoding scheme by developing a new feedback adjustment strategy for the standard SPA. In our feedback strategy, we exploit not only the syndrome but also the values of the frustrated checks on individual qubits of the code and the channel model. Consequently, our decoding algorithm, on the one hand, can break the symmetric degeneracy, and on the other hand, can feed back more useful information to the SPA decoder to help the decoder determine a valid output, thereby significantly improving the decoding ability of the decoder. Moreover, our algorithm does not increase the measurement complexity compared with the previous method, but takes full advantage of the measured information.

Feedback iterative decoding of sparse quantum codes

In this paper, a feedback sum-product decoding algorithm of sparse quantum codes for X-Z Pauli channels is developed. Compared with the previous decoding algorithm, our feedback strategy exploits not just the syndrome but also the values of the frustrated checks on individual qubits of the code and the character of the channel model with the portion of each error to adjust the probability distribution of information nodes. Due to the smart adjustment, our decoding algorithm, on one hand, can break the symmetric degeneracy, and on the other hand, can feed back more useful information to the SPA decoder to help the decoder determine a valid output, thereby significantly improving the decoding ability of the decoder. Moreover, our algorithm, which is based on GF(4), overcomes the limitation caused by decoding in GF(2). Finally, we want to point out that, our method does not increase the measurement overhead in comparison wioth the previous methods, as the extra information comes for free from the requisite stabilizer measurement.

Feedback sum-product decoding of sparse quantum codes for X-Z Pauli channels

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Feedback iterative decoding of sparse quantum codes

Feedback sum-product decoding of sparse quantum codes for X-Z Pauli channels