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Good quantum codes with addressable and parallelizable transversal non-Clifford gates

Virgile Guémard

TL;DR

The paper addresses fault-tolerant quantum computation with transversal non-Clifford gates by constructing asymptotically good qudit CSS codes derived from Stichtenoth algebraic-geometry codes. It develops a framework for addressable and parallelizable inter-block C^{m-1}Z gates, proving depth overhead scales as O(k^{m-1}) for m-block interdependencies and enabling parallel gate execution under symmetry constraints. The work combines function-field theory, AG-code constructions, and puncturing techniques to produce quantum codes with a defined m-multiplication property, extendable to qubit implementations, and provides circuit-level insights showing significant depth reductions for dense gate sets and potential optimizations for sparse-connectivity regimes. This advances practical fault-tolerance by offering scalable, structure-exploiting codes that support large sets of transversal non-Clifford operations with favorable depth characteristics.

Abstract

In this work, we prove that for any $m>1$, there exists a family of good qudit quantum codes supporting transversal logical $\mathsf{C}^{m-1}\mathsf{Z}$ gates that can address specified logical qudits and be largely executed in parallel. Building on the family of good quantum error-correcting codes presented in He et al. (2025), which support addressable and transversal logical $\mathsf{CCZ}$ gates, we extend their framework and show how to perform large sets of gates in parallel. The construction relies on the classical algebraic geometry codes of Stichtenoth (IEEE Trans. Inf. Theory, 2006). Our results lead to a substantial reduction in the depth overhead of multi-control-$Z$ circuits. In particular, we show that the minimal depth of any logical $\mathsf{C}^{m-1}\mathsf{Z}$ circuit involving qudits from $m$ distinct code blocks is upper bounded by $O(k^{m-1})$, where $k$ is the code dimension. While this overhead is optimal for dense $\mathsf{C}^{m-1}\mathsf{Z}$ circuits, for sparse circuits we discuss how the depth overhead can be significantly reduced by exploiting the structure of the quantum code.

Good quantum codes with addressable and parallelizable transversal non-Clifford gates

TL;DR

The paper addresses fault-tolerant quantum computation with transversal non-Clifford gates by constructing asymptotically good qudit CSS codes derived from Stichtenoth algebraic-geometry codes. It develops a framework for addressable and parallelizable inter-block C^{m-1}Z gates, proving depth overhead scales as O(k^{m-1}) for m-block interdependencies and enabling parallel gate execution under symmetry constraints. The work combines function-field theory, AG-code constructions, and puncturing techniques to produce quantum codes with a defined m-multiplication property, extendable to qubit implementations, and provides circuit-level insights showing significant depth reductions for dense gate sets and potential optimizations for sparse-connectivity regimes. This advances practical fault-tolerance by offering scalable, structure-exploiting codes that support large sets of transversal non-Clifford operations with favorable depth characteristics.

Abstract

In this work, we prove that for any , there exists a family of good qudit quantum codes supporting transversal logical gates that can address specified logical qudits and be largely executed in parallel. Building on the family of good quantum error-correcting codes presented in He et al. (2025), which support addressable and transversal logical gates, we extend their framework and show how to perform large sets of gates in parallel. The construction relies on the classical algebraic geometry codes of Stichtenoth (IEEE Trans. Inf. Theory, 2006). Our results lead to a substantial reduction in the depth overhead of multi-control- circuits. In particular, we show that the minimal depth of any logical circuit involving qudits from distinct code blocks is upper bounded by , where is the code dimension. While this overhead is optimal for dense circuits, for sparse circuits we discuss how the depth overhead can be significantly reduced by exploiting the structure of the quantum code.
Paper Structure (24 sections, 18 theorems, 71 equations)

This paper contains 24 sections, 18 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear codes with transitivity and multiplication properties
  4. Transitive codes.
  5. Codes with multiplication properties.
  6. Function fields and algebraic geometry codes
  7. Function fields.
  8. Divisors and Riemann-Roch spaces.
  9. Towers of function fields.
  10. Automorphisms of function fields.
  11. Algebraic geometry codes.
  12. Asymptotically good family of transitive AG codes with multiplication property
  13. Stichtenoth's AG code construction.
  14. Transitive group action.
  15. Multiplication property.
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\ell$ be a prime power, and set $q = \ell^2$. Then, for any integer $m\geq 2$ satisfying $\ell \geq 2(m+1)$, there exists an asymptotically good family of CSS codes $\mathcal{Q}$ with parameters $[[n,k,d]]_q$ over $q$-dimensional qudits such that, for every integer $\widetilde{m}$ with $1 < \wi

Theorems & Definitions (33)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Proposition 2.5
  • Theorem 2.6: Stichtenoth2006
  • Definition 2.7
  • Lemma 2.8
  • ...and 23 more