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Qudit low-density parity-check codes

Daniel J. Spencer, Andrew Tanggara, Tobias Haug, Derek Khu, Kishor Bharti

TL;DR

The paper develops a comprehensive framework for qudit quantum LDPC codes by promoting the underlying algebra to finite fields $\bbF_q$ and reformulating code constructions via chain/cochain complexes. It applies this quditization to multiple LDPC families—qudit bivariate bicycle codes, qudit hypergraph product codes (including La-cross codes), SHYPS simplex-derived subsystem codes, high-dimensional expander codes, and fiber bundle codes—deriving parameters and, in several cases, numerical decoding results. A key result is that qudit bivariate bicycle codes satisfy $d_X=d_Z=d$, with code parameters $\llbracket n, k, d\rrbracket_q$ and explicit formulas for $n,k,d$ in terms of kernel dimensions and gcds; the La-cross and SHYPS constructions extend similarly with appropriate seed codes. The work demonstrates that qudit LDPC codes can achieve hardware-compatible error correction with competitive scaling, and it provides both the theoretical framework and numerical evidence to motivate further development of decoding algorithms and fault-tolerant gate implementations in the qudit setting.

Abstract

Qudits offer significant advantages over qubit-based architectures, including more efficient gate compilation, reduced resource requirements, improved error-correction primitives, and enhanced capabilities for quantum communication and cryptography. Yet, one of the most promising family of quantum error correction codes, namely quantum low-density parity-check (LDPC) codes, have been so far mostly restricted to qubits. Here, we generalize recent advancements in LDPC codes from qubits to qudits. We introduce a general framework for finding qudit LDPC codes and apply our formalism to several promising types of LDPC codes. We generalize bivariate bicycle codes, including their coprime variant; hypergraph product codes, including the recently proposed La-cross codes; subsystem hypergraph product (SHYPS) codes; high-dimensional expander codes, which make use of Ramanujan complexes; and fiber bundle codes. Using the qudit generalization formalism, we then numerically search for and decode several novel qudit codes compatible with near-term hardware. Our results highlight the potential of qudit LDPC codes as a versatile and hardware-compatible pathway toward scalable quantum error correction.

Qudit low-density parity-check codes

TL;DR

The paper develops a comprehensive framework for qudit quantum LDPC codes by promoting the underlying algebra to finite fields and reformulating code constructions via chain/cochain complexes. It applies this quditization to multiple LDPC families—qudit bivariate bicycle codes, qudit hypergraph product codes (including La-cross codes), SHYPS simplex-derived subsystem codes, high-dimensional expander codes, and fiber bundle codes—deriving parameters and, in several cases, numerical decoding results. A key result is that qudit bivariate bicycle codes satisfy , with code parameters and explicit formulas for in terms of kernel dimensions and gcds; the La-cross and SHYPS constructions extend similarly with appropriate seed codes. The work demonstrates that qudit LDPC codes can achieve hardware-compatible error correction with competitive scaling, and it provides both the theoretical framework and numerical evidence to motivate further development of decoding algorithms and fault-tolerant gate implementations in the qudit setting.

Abstract

Qudits offer significant advantages over qubit-based architectures, including more efficient gate compilation, reduced resource requirements, improved error-correction primitives, and enhanced capabilities for quantum communication and cryptography. Yet, one of the most promising family of quantum error correction codes, namely quantum low-density parity-check (LDPC) codes, have been so far mostly restricted to qubits. Here, we generalize recent advancements in LDPC codes from qubits to qudits. We introduce a general framework for finding qudit LDPC codes and apply our formalism to several promising types of LDPC codes. We generalize bivariate bicycle codes, including their coprime variant; hypergraph product codes, including the recently proposed La-cross codes; subsystem hypergraph product (SHYPS) codes; high-dimensional expander codes, which make use of Ramanujan complexes; and fiber bundle codes. Using the qudit generalization formalism, we then numerically search for and decode several novel qudit codes compatible with near-term hardware. Our results highlight the potential of qudit LDPC codes as a versatile and hardware-compatible pathway toward scalable quantum error correction.

Paper Structure

This paper contains 26 sections, 15 theorems, 184 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fields and rings
  4. Homological algebra and chain complexes
  5. Tensor product of chain complexes
  6. Tensor product of co-chain complexes
  7. Coding theory
  8. Quditization
  9. Qudit bivariate bicycle codes
  10. Coprime qudit bivariate bicycle codes
  11. Numerical code search
  12. Qudit hypergraph product codes
  13. Algebraic framework
  14. Homology framework
  15. Qudit La-cross codes
  16. ...and 11 more sections

Key Result

Proposition 1

The matrices $A, B \in \mathbb{F}\xspace_q^{\ell m \times \ell m}$, defined above, define a qudit bivariate bicycle code with parameters $\llbracket n, k, d \rrbracket\xspace_q$, where where $d_X$ and $d_Z$ are the code distances for $X$-type and $Z$-type errors defined as where $\operatorname{rs}\left( \cdot \right)\xspace$ denotes row space and $\abs{v}$ is the Hamming weight of a vector $v \i

Figures (3)

  • Figure 1: Code capacity for the best qudit bivariate bicycle codes we have found. We plot logical error $p_\text{L}$ against physical error $p$. We show (a) regular, $q = 3$ with $d_\text{fit}=\{3.3, 5.1, 7.0\}$; (b) regular, $q = 5$ with $d_\text{fit}=\{5.8, 5.1, 4.4, 4.6, 7.1\}$; (c) coprime, $q = 3,5,7$ with $d_\text{fit}=\{5.0,4.8,4.5\}$. The subscript on each code in the legend corresponds to the local qudit dimension $q$ and the superscript denotes the weight of the parity check operators of the code.
  • Figure 2: Code capacity of the best found qudit La-cross codes. We plot logical error $p_\text{L}$ against physical error $p_{\text{err}}$. The subscript on each code in the legend corresponds to the local qudit dimension $q$. We fit with \ref{['eq:heuristic_fit']} where we find $d_\text{fit}=\{4.7,4.8,3.1\}$.
  • Figure 3: Minimum and maximum weights of classical $q$-ary simplex code parity-check matrix as a function of $r$.

Theorems & Definitions (50)

  • Definition 1: Finite field
  • Definition 2: Ideal of a ring
  • Definition 3: Linear code
  • Proposition 1: Qudit multivariate bicycle code parameters
  • proof
  • Lemma 1: Qudit bivariate bicycle codes via extension of scalars
  • proof
  • Proposition 2: Qudit coprime bivariate bicycle code dimension
  • proof
  • Proposition 3: Qudit hypergraph product code parameters
  • ...and 40 more