Single-shot decoding of good quantum LDPC codes

TL;DR

The paper proves that quantum Tanner codes, a family of quantum LDPC codes built on left-right Cayley complexes with product-expanding local codes, support single-shot quantum error correction against adversarial noise. It establishes that a single round of noisy syndrome measurements suffices for reliable decoding using both sequential and parallel decoders, with the parallel decoder achieving $O(\log n)$ iterations to reach vanishing error. Moreover, when errors across multiple rounds are bounded, the residual error remains controlled, enabling essentially indefinite rounds of correction with constant-time overhead per round. These results, underpinned by the product-expanding structure and Ramanujan graphs, position quantum Tanner codes as strong candidates for practical fault-tolerant quantum computation with robust, scalable decoding strategies.

Abstract

Quantum Tanner codes constitute a family of quantum low-density parity-check (LDPC) codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.

Figures (3)

• Figure 1: The local structure of the left-right Cayley complex around a vertex labelled by $g\in G$. The incident faces $Q(v)$ has a natural bijection with $A\times B$. As examples, the red and blue faces in the complex are mapped to the squares of the same colors in the matrix given by $A\times B$.
• Figure 2: An example of a stabilizer generator with local codes $C_A = \operatorname{span}\{111\}$ and $C_B=\operatorname{span}\{110,011\}$. The codeword $x=111\otimes 110 \in C_A\otimes C_B$ has support as shown on the right. Identifying that matrix with the faces incident to a $V_0$ vertex gives an $X$-type stabilizer generator.
• Figure 3: Reference for sets involved in proof of Lemma \ref{['lem:parallelmismatchreduction']}. The regions indicated are: $\mathrm{II}\cup \mathrm{IV} = \tilde{Z}'_{j-1}$, $\mathrm{I}\cup \mathrm{III} = A_{j-1}$, $\mathrm{II} = \tilde{Z}'_{j}$, $\mathrm{IV} = x_j'$, $\mathrm{III}\cup \mathrm{IV} = \tilde{Z}_{j-1}\cap x_j$, $\mathrm{I}\cup \mathrm{V} = A_{j}$, and $\mathrm{I}\cup \mathrm{II}\cup \mathrm{V} = \tilde{Z}_{j}.$

Theorems & Definitions (49)

• Definition 1.1: Informal Statement of Definition \ref{['def:singleshotproblem']}
• Theorem 1.2: Informal Statement of Theorem \ref{['thm:main']}
• Theorem 1.3: Informal Statement of Theorem \ref{['thm:parallel_residual']}
• Theorem 1.4: Informal Statement of Theorem \ref{['thm:multiroundadvserialerror']}
• Definition 2.1
• Theorem 2.2: Theorem 1 in Ref. KPTwoSided
• Theorem 2.3: Theorem 1 in Ref. leverrier2022parallel
• Definition 3.1
• Definition 3.2
• Definition 3.3