Quantum Codes with Transversal $CCZ$ Gates and Sublinear $Z$-Stabilizers

2026-06-21

AI Summary

This work addresses the construction of quantum CSS codes that simultaneously support transversal CCZ gates, high dimension, large distance, and a sublinear-weight generating set for Z-type stabilizers. Building upon an algebraic puncturing framework, the authors integrate algebraic expander codes with a projective-multiplicity variant of multiplication-friendly codes and establish a refined puncturing theorem. This theorem guarantees the desired code properties under purely local puncturing conditions, substantially relaxing the requirement on global dual distance. As a result, the paper provides explicit constructions: over growing alphabets, it achieves CSS codes with parameters $[[N, \Theta(N), \Omega(N^{1/m})]]$ for any integer $m \geq 3$; when restricted to a fixed prime field, the construction still yields near-linear dimension, distance $\Omega(N^{1/m}/\mathrm{polylog}\,N)$, and sublinear locality for Z-stabilizers.

Abstract

We construct quantum CSS codes with transversal \(CCZ\) gates whose \(Z\)-stabilizers admit sublinear-weight generating sets. We build on the algebraic puncturing framework of Guruswami and Golowich \cite{GG24}, which turns classical codes with the required Schur-product and distance conditions into CSS codes with transversal \(CCZ\). However, applying the framework directly to the algebraic expander codes of \cite{KT26} runs into their small dual distance, and therefore produces only sublinear quantum dimension. Our main technical step is a refined puncturing theorem in which the global dual-distance assumption is replaced by a condition only on the selected puncturing set. Applying this theorem to algebraic expander codes gives explicit growing-alphabet CSS codes with parameters \([[N,Θ(N),Ω(N^{1/m})]]\), for every fixed \(m\geq 3\), and with transversal \(CCZ\) gates. Moreover, the \(Z\)-stabilizer space has an explicit generating set of weight \(O(N^{1/m})\). We also reduce the alphabet to a fixed prime field using a projective-multiplicity version of multiplication-friendly codes. The resulting fixed-prime-field CSS code triples, of length \(n\), still have transversal \(CCZ\) gates. Their dimension is near-linear, their distance is \(n^{1/m}\) up to polylogarithmic factors, and the \(Z\)-stabilizer locality remains sublinear, again up to polylogarithmic losses.

Problem

• quantum CSS codes
• transversal CCZ gates
• sublinear Z-stabilizers
• quantum error correction
• stabilizer locality

Innovation

• transversal CCZ gate
• sublinear stabilizers
• algebraic puncturing
• CSS codes
• multiplication-friendly codes