Quantum LDPC Codes Achieve Larger Minimum Distance by Breaking Orthogonality Barrier

Quantum error correction is vital for realising the potential of quantum computing, and low-density parity-check (LDPC) codes represent a promising approach to achieving this. Kenta Kasai from the Institute of Science Tokyo, alongside colleagues, demonstrate a significant advance in the design of these quantum codes, addressing a fundamental limitation that has previously hindered their performance. The team have overcome the conventional trade-off between key code characteristics , orthogonality, regularity, girth and minimum distance , by employing a novel construction method utilising permutation matrices with controlled commutativity. This breakthrough allows for the creation of LDPC codes with both large girth and improved minimum distance, potentially leading to more robust and efficient quantum communication and storage. Their work, exemplified by the construction of a girth-8, (3,12)-regular LDPC code achieving a low frame error rate, paves the way for practical implementation of quantum error correction schemes.

Quantum Error Correction with LDPC Codes

This is a very detailed and technical paper describing the construction and performance of a novel quantum error-correcting code.

Quantum error correction is crucial for building fault-tolerant quantum computers, and the authors leverage low-density parity-check (LDPC) codes, originally developed for classical communication, as a key strategy. The work builds upon existing QEC codes like surface codes and generalized bicycle codes, utilising protographs and affine permutation matrices to allow for a structured design and tractable analysis. Performance is evaluated assuming a depolarizing channel, a common model for quantum noise.

In the quantum setting, desirable code characteristics like performance and large minimum distance do not directly apply, as quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. Enforcing both orthogonality and regularity typically reduces the girth and structurally limits the minimum distance. This research overcomes this limitation by utilising permutation matrices with controlled commutativity and restricting orthogonality constraints to only the necessary parts of the construction, preserving regular check-matrix structures and breaking the conventional trade-off between these parameters.

Girth-8 LDPC Codes with Improved Performance

Scientists achieved a breakthrough in low-density parity-check (LDPC) code construction, overcoming a traditional trade-off between orthogonality, regularity, girth, and minimum distance. The research team successfully designed a girth-8, (3,12)-regular LDPC code, demonstrating a significant advancement in code structure.

This was accomplished by employing permutation matrices with controlled commutativity and strategically restricting orthogonality constraints during the construction process, allowing for larger girth and improved minimum distance characteristics. Experiments revealed that, under belief-propagation (BP) decoding combined with a low-complexity post-processing algorithm, the newly constructed code achieves a frame error rate as low as on the depolarizing channel with an error probability of . The team instantiated the general theory and sequential construction for the case of J = 3, L = 12, and P = 768, explicitly defining ∆, Γ, and Ψr, and clarifying the active orthogonality condition for the HX and HZ matrices.

Measurements confirm a specific block-circulant structure for ˆHX and ˆHZ, where each block row is a cyclic shift of the previous row. The active matrices HX and HZ satisfy the condition Ψr = 0 for r ∈ {0, 1, 2, 4, 5}, while allowing Ψ3 to remain non-zero, guaranteeing active orthogonality and preventing the latent part from becoming orthogonal to the active part. The mother-matrix product ˆHX( ˆHZ)T exhibits a defined structure, further validating the code’s unique properties and performance potential.

Detailed analysis identified and mitigated the impact of elementary trapping sets (ETSs), particularly those formed by connecting length-8 cycles. The team constructed an ETS library, enumerating dominant patterns like (6, 2) ETSs with X = 48 and Z = 16, (12, 2) ETSs with X = 23 and Z = 0, and (8, 2) ETSs with X = 48 and Z = 0. The use of affine permutations, defined by parameters fi(x) and gi(x), further contributed to the code’s robust performance and girth of 8.

Girth and Distance in Novel LDPC Codes

This work presents a novel approach to constructing low-density parity-check (LDPC) codes, overcoming limitations previously encountered in quantum code design. Researchers successfully demonstrated a method utilising permutation matrices with controlled commutativity and strategically applied orthogonality constraints, allowing for the creation of codes with both large girth and improved minimum distance characteristics.

This represents a significant advancement as conventional methods often force a trade-off between these desirable properties. Specifically, the authors detail the creation of a girth-8, (3,12)-regular LDPC code which, when paired with a low-complexity post-processing algorithm, achieves a low frame error rate on the depolarizing channel. The authors acknowledge a limitation in that the presented construction is specific to the demonstrated code and further generalisation may require additional investigation.

Future research will focus on extending these techniques to create codes with even larger girth and improved performance across a wider range of parameters, potentially furthering the development of robust quantum error correction schemes.

Quantum error correction is vital for realising the potential of quantum computing, and low-density parity-check (LDPC) codes represent a promising approach to achieving this. This breakthrough allows for the creation of LDPC codes with both large girth and improved minimum distance, potentially leading to more robust and efficient quantum communication and storage. Their work, exemplified by the construction of a girth-8, (3,12)-regular LDPC code achieving a low frame error rate, paves the way for practical implementation of quantum error correction schemes.