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Simplified Quantum Weight Reduction with Optimal Bounds

Min-Hsiu Hsieh, Xingjian Li, Ting-Chun Lin

TL;DR

This work introduces a new procedure for quantum weight reduction that combines geometric insights with coning techniques, which simplifies Hastings' previous approach while achieving better parameters.

Abstract

Quantum weight reduction is the task of transforming a quantum code with large check weight into one with small check weight. Low-weight codes are essential for implementing quantum error correction on physical hardware, since high-weight measurements cannot be executed reliably. Weight reduction also serves as a critical theoretical tool, which may be relevant to the quantum PCP conjecture. We introduce a new procedure for quantum weight reduction that combines geometric insights with coning techniques, which simplifies Hastings' previous approach while achieving better parameters. Given an arbitrary $[[n,k,d]]$ quantum code with weight $w$, our method produces a code with parameters $[[O(n w^2 \log w), k, Ω(d w)]]$ with check weight $5$ and qubit weight $6$. When applied to random dense CSS codes, our procedure yields explicit quantum codes that surpass the square-root distance barrier, achieving parameters $[[n, \tilde O(n^{1/3}), \tilde Ω(n^{2/3})]]$. Furthermore, these codes admit a three-dimensional embedding that saturates the Bravyi-Poulin-Terhal (BPT) bound. As a further application, our weight reduction technique improves fault-tolerant logical operator measurements by reducing the number of ancilla qubits.

Simplified Quantum Weight Reduction with Optimal Bounds

TL;DR

This work introduces a new procedure for quantum weight reduction that combines geometric insights with coning techniques, which simplifies Hastings' previous approach while achieving better parameters.

Abstract

Quantum weight reduction is the task of transforming a quantum code with large check weight into one with small check weight. Low-weight codes are essential for implementing quantum error correction on physical hardware, since high-weight measurements cannot be executed reliably. Weight reduction also serves as a critical theoretical tool, which may be relevant to the quantum PCP conjecture. We introduce a new procedure for quantum weight reduction that combines geometric insights with coning techniques, which simplifies Hastings' previous approach while achieving better parameters. Given an arbitrary quantum code with weight , our method produces a code with parameters with check weight and qubit weight . When applied to random dense CSS codes, our procedure yields explicit quantum codes that surpass the square-root distance barrier, achieving parameters . Furthermore, these codes admit a three-dimensional embedding that saturates the Bravyi-Poulin-Terhal (BPT) bound. As a further application, our weight reduction technique improves fault-tolerant logical operator measurements by reducing the number of ancilla qubits.

Paper Structure

This paper contains 35 sections, 14 theorems, 25 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. History of weight reduction
  4. Construction of weight reduced code
  5. Analysis of distance
  6. Preliminaries
  7. Notation and useful lemmas
  8. Chain complex and CSS codes
  9. Square complex from quantum code
  10. Framework of quantum code embedding
  11. Weight reduction of a simplicial complex
  12. Weight reduction of a dense simplicial complex
  13. Weight reduction of a sparse simplicial complex
  14. Weight reduction of a quantum code
  15. Weight reduction of a general quantum code
  16. ...and 20 more sections

Key Result

Theorem 1.1

Given a $[[n,k,d]]$ quantum code with weight $\leq w$, there exists a weight reduction procedure that generates a $[[O(nw^2\log w),k,\Omega(dw)]]$ quantum code with check weight $\le 5$ and qubit weight $\le 6$.

Figures (16)

  • Figure 1: An example of our map that takes a 2-simplicial simplex and its boundaries in $X$ to planes in $\mathcal{X}$. The corresponding objects on both sides use the same color.
  • Figure 2: Figure showing the gluing procedure between $P_{v_1}$ and $P_{v_2}$. The deep violent region is $[0,1]\times P_{e}$.
  • Figure 3: The leaf swap operation on trees.
  • Figure 4: The local cone structure in the construction of berdnikov2022parsimonious.
  • Figure 5: The tree rotation gadget can optimize the constants in our construction.
  • ...and 11 more figures

Theorems & Definitions (24)

  • Theorem 1.1: Main theorem
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1: Hoeffding's inequality
  • Definition 2.2: Chain complex
  • Theorem 3.1: berdnikov2022parsimonious
  • Theorem 3.2
  • proof
  • Lemma 3.3: berdnikov2022parsimonious
  • Lemma 3.4
  • ...and 14 more