Log-domain decoding of quantum LDPC codes over binary finite fields

• URL: http://arxiv.org/abs/2104.00304v3
• Date: Tue, 26 Oct 2021 19:24:02 GMT
• Title: Log-domain decoding of quantum LDPC codes over binary finite fields
• Authors: Ching-Yi Lai and Kao-Yueh Kuo
• Abstract summary: We study the decoding of quantum low-density parity-check (LDPC) codes over binary finite fields GF$(q=2l)$ by the sum-product algorithm, also known as belief propagation (BP) We show that scalar messages suffice for BP decoding of nonbinary quantum codes, rather than vector messages necessary for the conventional BP.
• Score: 4.340338299803562
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A quantum stabilizer code over GF$(q)$ corresponds to a classical additive code over GF$(q^2)$ that is self-orthogonal with respect to a symplectic inner product. We study the decoding of quantum low-density parity-check (LDPC) codes over binary finite fields GF$(q=2^l)$ by the sum-product algorithm, also known as belief propagation (BP). Conventionally, a message in a nonbinary BP for quantum codes over GF$(2^l)$ represents a probability vector over GF$(2^{2l})$, inducing high decoding complexity. In this paper, we explore the property of the symplectic inner product and show that scalar messages suffice for BP decoding of nonbinary quantum codes, rather than vector messages necessary for the conventional BP. Consequently, we propose a BP decoding algorithm for quantum codes over GF$(2^l)$ by passing scalar messages so that it has low computation complexity. The algorithm is specified in log domain by using log-likelihood ratios (LLRs) of the channel statistics to have a low implementation cost. Moreover, techniques such as message normalization or offset can be naturally applied in this algorithm to mitigate the effects of short cycles to improve BP performance. This is important for nonbinary quantum codes since they may have more short cycles compared to binary quantum codes. Several computer simulations are provided to demonstrate these advantages. The scalar-based strategy can also be used to improve the BP decoding of classical linear codes over GF$(2^l)$ with many short cycles.

Related papers

• Decoding Quasi-Cyclic Quantum LDPC Codes [23.22566380210149]
  Quantum low-density parity-check (qLDPC) codes are an important component in the quest for fault tolerance. Recent progress on qLDPC codes has led to constructions which are quantumally good, and which admit linear-time decoders to correct errors affecting a constant fraction of codeword qubits. In practice, the surface/toric codes, which are the product of two repetition codes, are still often the qLDPC codes of choice.
  arXiv  Detail & Related papers  (2024-11-07T06:25:27Z)
• Geometric structure and transversal logic of quantum Reed-Muller codes [51.11215560140181]
  In this paper, we aim to characterize the gates of quantum Reed-Muller (RM) codes by exploiting the well-studied properties of their classical counterparts. A set of stabilizer generators for a RM code can be described via $X$ and $Z$ operators acting on subcubes of particular dimensions.
  arXiv  Detail & Related papers  (2024-10-10T04:07:24Z)
• Fast and Parallelizable Logical Computation with Homological Product Codes [3.4338109681532027]
  High-rate quantum low-density-parity-check (qLDPC) codes promise a route to reduce qubit numbers, but performing computation while maintaining low space cost has required serialization of operations and extra time costs. We design fast and parallelizable logical gates for qLDPC codes, demonstrating their utility for key algorithmic subroutines such as the quantum adder.
  arXiv  Detail & Related papers  (2024-07-26T03:49:59Z)
• Homological Quantum Rotor Codes: Logical Qubits from Torsion [51.9157257936691]
  homological quantum rotor codes allow one to encode both logical rotors and logical qudits in the same block of code. We show that the $0$-$pi$-qubit as well as Kitaev's current-mirror qubit are indeed small examples of such codes.
  arXiv  Detail & Related papers  (2023-03-24T00:29:15Z)
• Neural Belief Propagation Decoding of Quantum LDPC Codes Using Overcomplete Check Matrices [60.02503434201552]
  We propose to decode QLDPC codes based on a check matrix with redundant rows, generated from linear combinations of the rows in the original check matrix. This approach yields a significant improvement in decoding performance with the additional advantage of very low decoding latency.
  arXiv  Detail & Related papers  (2022-12-20T13:41:27Z)
• Quantum message-passing algorithm for optimal and efficient decoding [2.3020018305241328]
  We expand the understanding, formalism, and applicability of the BPQM algorithm. We provide the first formal description of the BPQM algorithm in full detail and without any ambiguity. We show some promising numerical results that indicate that BPQM on factor graphs with cycles can significantly outperform the best possible classical decoder.
  arXiv  Detail & Related papers  (2021-09-16T18:01:12Z)
• Finding the disjointness of stabilizer codes is NP-complete [77.34726150561087]
  We show that the problem of calculating the $c-disjointness, or even approximating it to within a constant multiplicative factor, is NP-complete. We provide bounds on the disjointness for various code families, including the CSS codes,$d codes and hypergraph codes. Our results indicate that finding fault-tolerant logical gates for generic quantum error-correcting codes is a computationally challenging task.
  arXiv  Detail & Related papers  (2021-08-10T15:00:20Z)
• Realization of arbitrary doubly-controlled quantum phase gates [62.997667081978825]
  We introduce a high-fidelity gate set inspired by a proposal for near-term quantum advantage in optimization problems. By orchestrating coherent, multi-level control over three transmon qutrits, we synthesize a family of deterministic, continuous-angle quantum phase gates acting in the natural three-qubit computational basis.
  arXiv  Detail & Related papers  (2021-08-03T17:49:09Z)
• Refined Belief-Propagation Decoding of Quantum Codes with Scalar Messages [4.340338299803562]
  Codes based on sparse matrices have good performance and can be efficiently decoded by belief-propagation (BP) BP decoding of stabilizer codes suffers a performance loss from the short cycles in the underlying Tanner graph. We show that running BP with message normalization according to a serial schedule may significantly improve the decoding performance and error-floor in computer simulation.
  arXiv  Detail & Related papers  (2021-02-14T10:29:58Z)
• Quantum Garbled Circuits [9.937090317971313]
  We show how to compute an encoding of a given quantum circuit and quantum input, from which it is possible to derive the output of the computation and nothing else. Our protocol has the so-called $Sigma$ format with a single-bit challenge, and allows the inputs to be delayed to the last round.
  arXiv  Detail & Related papers  (2020-06-01T17:07:01Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.