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On Quantum Weight Reduction

M. B. Hastings

TL;DR

This work develops a comprehensive weight-reduction framework for quantum LDPC codes, introducing copy/gauging, thickening, and the novel coning operation to reduce stabilizer weights while tracking impacts on qubit count and distance. By combining these tools, any LDPC code can be transformed into an LDPC code with stabilizer weights bounded by a small constant (e.g., at most 5) and controlled overhead, enabling scalable distance properties. A central achievement is constructing LDPC codes with distance $\tilde{\Omega}(N^{2/3})$ and $\tilde{\Omega}(N^{2/3})$ logical qubits, starting from simple codes and using thickening and coning, with further improvements via distance balancing and soundness enhancements. The paper also presents a linear-distance example with high-weight $Z$-stabilizers that, after thickening/coning, yields LDPC codes with strong distance properties, and outlines potential for derandomization and fault-tolerant Clifford operations on LDPC codes through the weighting techniques.

Abstract

We give a general procedure for weight reducing quantum codes. This corrects a previous work\cite{owr}, and introduces a new technique that we call "coning" to effectively induce high weight stabilizers in an LDPC code. As one application, any LDPC code (with arbitrary $O(1)$ stabilizer weights) may be turned into a code where all stabilizers have weight at most $5$ at the cost of at most a constant factor increase in number of physical qubits and constant factor reduction in distance. Also, by applying this technique to a quantum code whose $X$-stabilizers are derived from a classical log-weight random code and whose $Z$-stabilizers have linear weight, we construct an LDPC quantum code with distance $\tilde Ω(N^{2/3})$ and $\tildeΩ(N^{2/3})$ logical qubits.

On Quantum Weight Reduction

TL;DR

This work develops a comprehensive weight-reduction framework for quantum LDPC codes, introducing copy/gauging, thickening, and the novel coning operation to reduce stabilizer weights while tracking impacts on qubit count and distance. By combining these tools, any LDPC code can be transformed into an LDPC code with stabilizer weights bounded by a small constant (e.g., at most 5) and controlled overhead, enabling scalable distance properties. A central achievement is constructing LDPC codes with distance and logical qubits, starting from simple codes and using thickening and coning, with further improvements via distance balancing and soundness enhancements. The paper also presents a linear-distance example with high-weight -stabilizers that, after thickening/coning, yields LDPC codes with strong distance properties, and outlines potential for derandomization and fault-tolerant Clifford operations on LDPC codes through the weighting techniques.

Abstract

We give a general procedure for weight reducing quantum codes. This corrects a previous work\cite{owr}, and introduces a new technique that we call "coning" to effectively induce high weight stabilizers in an LDPC code. As one application, any LDPC code (with arbitrary stabilizer weights) may be turned into a code where all stabilizers have weight at most at the cost of at most a constant factor increase in number of physical qubits and constant factor reduction in distance. Also, by applying this technique to a quantum code whose -stabilizers are derived from a classical log-weight random code and whose -stabilizers have linear weight, we construct an LDPC quantum code with distance and logical qubits.

Paper Structure

This paper contains 18 sections, 16 theorems, 15 equations, 1 figure.

Table of Contents

  1. Notation and Definitions
  2. Outline
  3. Copying and Gauging
  4. New $X$-Stabilizers
  5. New $Z$-Stabilizers
  6. $X$-Reduced Code Properties
  7. Thickening
  8. Thickening Construction
  9. Choosing Heights
  10. Coning
  11. Comparison of Reduced Cone Code and Induced Code
  12. Examples With Toric Code and Geometric Interpretation
  13. Unreasonable Codes
  14. Putting It Together
  15. Coding Applications: Weight Reducing a Linear Distance Code
  16. ...and 3 more sections

Key Result

Lemma 1

The $X$-reduced code has the following parameters:

Figures (1)

  • Figure 1: A: part of a cellulation of a toric code. We show a polygon representing a high weight $Z$-stabilizer. B: The result with "rough" boundary conditions if we use ${\cal B}_1$ rather than $\overline {\cal B}_1$ in the cone construction. Solid lines represent qubits, vertices represent $X$-stabilizer. There are $Z$-stabilizers on triples of lines inside the polygon, corresponding to two neighboring edges entering the polygon and one edge in the polygon between those edges. The dashed line shows the graph $G_1$. The edges of this graph correspond to $0$-cells of ${\cal B}_1$ and the vertices (shown as solid circles) correspond to $1$-cells. C: Cone code with one additional high weight $X$-stabilizer. Note the relation with a dual cellulation of B: the dashed line of B bounds the attached $2$-cell in ${\cal S}_1$, which dually becomes a $0$-cell in C. D: Reduced cone code. Dashed lines show the cellulation of complex ${\cal S}_1$. The resulting stabilizers in the reduced cone code are on the dual cellulation.

Theorems & Definitions (33)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • proof
  • Definition 1
  • ...and 23 more