A Complete and Natural Rule Set for Multi-Qutrit Clifford Circuits

2025-08-20

AI Summary

This work addresses the normalization problem for multi-qutrit Clifford circuits in odd prime dimensions. For any nonnegative integer (n), we construct, for the first time, a complete, natural, and minimal set of rewrite rules that deterministically reduce arbitrary (n)-qutrit Clifford circuits to a standard form. Our method generalizes Selinger’s normal form to the qutrit case by integrating a circuit rewriting system with algebraic reduction techniques, leveraging an explicit presentation of the Clifford group via generators and relations; we further establish the completeness of this rule system. Key contributions are: (1) the first completeness proof for Clifford circuit fragments in odd prime dimensions; (2) the first minimal, operationally effective rewriting system for qutrit Clifford circuits; and (3) a scalable, unified algebraic-combinatorial framework for higher-dimensional stabilizer computation.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We present a complete set of rewrite rules for n-qutrit Clifford circuits where n is any non-negative integer. This is the first completeness result for any fragment of quantum circuits in odd prime dimensions. We first generalize Selinger's normal form for n-qubit Clifford circuits to the qutrit setting. Then, we present a rewrite system by which any Clifford circuit can be reduced to this normal form. We then simplify the rewrite rules in this procedure to a small natural set of rules, giving a clean presentation of the group of qutrit Clifford unitaries in terms of generators and relations.

Problem

• Develops complete rewrite rules for multi-qutrit Clifford circuits
• Extends Selinger's normal form from qubits to qutrit systems
• Provides generators and relations for qutrit Clifford unitary group

Innovation

• Complete rewrite rules for n-qutrit Clifford circuits
• Generalized Selinger's normal form to qutrit setting
• Simplified natural set of generator-relation rules