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Fiber Bundle Codes: Breaking the $N^{1/2} \operatorname{polylog}(N)$ Barrier for Quantum LDPC Codes

Matthew B. Hastings, Jeongwan Haah, Ryan O'Donnell

TL;DR

The paper introduces fiber bundle codes, a twisted generalization of the homological product, to construct quantum LDPC codes with improved distance. By pairing a simple fiber (a cycle) with a random base LDPC code and carefully designed twists, the authors derive a code on $N$ qubits with distances $d_X=\Omega(N^{1/2}/\operatorname{polylog}(N))$ and $d_Z=\Omega(N^{3/4}/\operatorname{polylog}(N))$ while keeping stabilizer weights and qubit participation polylogarithmic. Through a weight-reduction procedure and distance balancing, these codes are converted into LDPC codes with $d=\Omega(N^{3/5}/\operatorname{polylog}(N))$ and with $\Theta(N^{3/5}/\operatorname{polylog}(N))$ logical qubits, achieving the long-standing barrier for quantum LDPC codes. The results are complemented by partial decoding algorithms for cohomology, homology, and erasure scenarios, leveraging randomized base codes, twist graphs, and Lipschitz-homeomorphism ideas to relate distances across equivalent constructions. This work advances the practical design of LDPC quantum codes by fusing topological insights with probabilistic coding theory, offering (i) a concrete distance-achieving construction and (ii) groundwork toward efficient decoders. The framework connects homological algebra, fiber bundle topology, and quantum error correction to push toward scalable, low-weight stabilizers with nontrivial distances.

Abstract

We present a quantum LDPC code family that has distance $Ω(N^{3/5}/\operatorname{polylog}(N))$ and $\tildeΘ(N^{3/5})$ logical qubits. This is the first quantum LDPC code construction which achieves distance greater than $N^{1/2} \operatorname{polylog}(N)$. The construction is based on generalizing the homological product of codes to a fiber bundle.

Fiber Bundle Codes: Breaking the $N^{1/2} \operatorname{polylog}(N)$ Barrier for Quantum LDPC Codes

TL;DR

The paper introduces fiber bundle codes, a twisted generalization of the homological product, to construct quantum LDPC codes with improved distance. By pairing a simple fiber (a cycle) with a random base LDPC code and carefully designed twists, the authors derive a code on qubits with distances and while keeping stabilizer weights and qubit participation polylogarithmic. Through a weight-reduction procedure and distance balancing, these codes are converted into LDPC codes with and with logical qubits, achieving the long-standing barrier for quantum LDPC codes. The results are complemented by partial decoding algorithms for cohomology, homology, and erasure scenarios, leveraging randomized base codes, twist graphs, and Lipschitz-homeomorphism ideas to relate distances across equivalent constructions. This work advances the practical design of LDPC quantum codes by fusing topological insights with probabilistic coding theory, offering (i) a concrete distance-achieving construction and (ii) groundwork toward efficient decoders. The framework connects homological algebra, fiber bundle topology, and quantum error correction to push toward scalable, low-weight stabilizers with nontrivial distances.

Abstract

We present a quantum LDPC code family that has distance and logical qubits. This is the first quantum LDPC code construction which achieves distance greater than . The construction is based on generalizing the homological product of codes to a fiber bundle.

Paper Structure

This paper contains 48 sections, 31 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results, motivation, and outline
  3. Quantum codes, chain complexes, and (co)homology
  4. Fiber Bundle Codes
  5. Products of chain complexes
  6. Trivial bundles
  7. Twisted bundles
  8. Isomorphisms on (co)homology groups
  9. Circle bundle over classical codes
  10. Remark.
  11. The random base code, with twists
  12. A random base code
  13. Cohomology representative weight
  14. Twists
  15. Homology representative weight
  16. ...and 33 more sections

Key Result

Theorem 1.1

There exists a family of quantum codes on $N$ qubits with $d_X=\Omega(N^{1/2}/\operatorname{polylog}(N))$ and $d_Z=\Omega(N^{3/4}/\operatorname{polylog}(N))$ where all stabilizer generators have weight at most $\operatorname{polylog}(N)$ and all qubits participate in at most $\operatorname{polylog}(

Figures (2)

  • Figure 1: Fiber bundles over base $1$-complexes. (1) depicts a trivial bundle built from two $1$-complexes, each representing a long line or a circle. Any bundle $1$-cell that is a lift of a base $1$-cell is referred to as a horizontal$1$-cell, indicated by $h$ in the figure. Any bundle $1$-cell that vanishes upon projection onto the base is referred to as a vertical$1$-cell, indicated by $v$ in the figure. The projection is defined in \ref{['def:Pi']}. (2) depicts some twisting. Since a fiber is acted on by an automorphism, the entire fiber over a base cell is shifted. Note that the shown twist can be removed using gauge redundancy. To draw a nonremovable twisting, we must have had a cycle in the base. (3) is the same as (2) but base $1$-cells are positioned on the right-hand side and base $0$-cells on the left-hand side. (4) introduces a "$1$"-cell (red) of the base that has three boundary $0$-cells.
  • Figure 2: Illustration of cell combining. The image shows a sequence of three steps. We begin with two "$1$-cells", shown at the top left. One cell (which corresponds to $e_1$ in the definition) is colored white, while the other cell (which corresponds to $e_2$) is colored red. The straight lines represent "$0$-cells" in the boundary of these "$1$-cells." The straight line between the two cells corresponds to $v$. We show pictorially a sequence as $v$ is deformed, until eventually $e_2$ shrinks to nothing and the line which corresponds to $v$ has been mapped to the other four "$0$-cells" in the boundary of $e_2$.

Theorems & Definitions (68)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 58 more