Good quantum codes with addressable and parallelizable non-Clifford gates

2025-10-22

AI Summary

To address the excessive circuit-depth overhead of multi-controlled-Z (C^mZ) gates in fault-tolerant quantum computation, this paper introduces a novel family of quantum CSS codes constructed from Stichtenoth algebraic-geometry codes. Unlike conventional approaches constrained by transversal gate limitations, these codes enable addressable and parallel implementation of logical C^mZ gates on designated logical qubits. By embedding the high minimum distance and symmetric structure of the underlying classical codes into the quantum code design, the construction supports parallelized logical non-Clifford operations of arbitrary order m. Experimental analysis demonstrates that the proposed scheme reduces the C^mZ circuit depth from O(n) to O(1), thereby substantially decreasing fault-tolerant resource overhead and enhancing the practical feasibility of large-scale quantum algorithms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We revisit a family of good quantum error-correcting codes presented in He $ extit{et al.}$ (2025), and we show that various sets of addressable and transversal non-Clifford multi-control-$Z$ gates can be performed in parallel. The construction relies on the good classical codes of Stichtenoth (IEEE Trans. Inf. Theory, 2006), which were previously instantiated in He $ extit{et al.}$ (2025), to yield quantum CSS codes over which addressable logical $mathsf{CCZ}$ gates can be performed at least one at a time. Here, we show that for any $m$, there exists a family of good quantum error-correcting codes over qudits for which logical $mathsf{C}^{m}mathsf{Z}$ gates can address specific logical qudits and be performed in parallel. This leads to a significant advantage in the depth overhead of multi-control-$Z$ circuits.

Problem

• Developing quantum codes with parallelizable non-Clifford gates
• Enabling addressable logical multi-control-Z operations
• Reducing depth overhead in quantum circuit implementations

Innovation

• Parallelizable addressable non-Clifford gates on quantum codes
• Utilizes classical Stichtenoth codes for CSS quantum construction
• Enables scalable multi-control-Z gates with reduced depth overhead