Tensor Decomposition for Non-Clifford Gate Minimization

2026-02-16

AI Summary

This work addresses the high resource overhead of non-Clifford gates—such as the Toffoli gate—in fault-tolerant quantum computing by proposing a novel, direct optimization approach. For the first time, the Toffoli gate minimization problem is reformulated as a tensor decomposition over the finite field ℤ₂, establishing a direct connection to algebraic structure and circumventing conventional indirect strategies that rely on T-count optimization. The resulting method achieves significant reductions in both Toffoli and T gate counts on standard benchmark circuits, matching or surpassing state-of-the-art results. Notably, the algorithm runs in under one minute on a single CPU, offering a dramatic improvement in computational efficiency compared to existing approaches that require thousands of TPUs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Optimization
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Fault-tolerant quantum computation requires minimizing non-Clifford gates, whose implementation via magic state distillation dominates the resource costs. While $T$-count minimization is well-studied, dedicated $CCZ$ factories shift the natural target to direct Toffoli minimization. We develop algebraic methods for this problem, building on a connection between Toffoli count and tensor decomposition over $\mathbb{F}_2$. On standard benchmarks, these methods match or improve all reported results for both Toffoli and $T$-count, with most circuits completing in under a minute on a single CPU instead of thousands of TPUs used by prior work.

Problem

• non-Clifford gate minimization
• Toffoli count
• fault-tolerant quantum computation
• magic state distillation
• tensor decomposition

Innovation

• Tensor Decomposition
• Non-Clifford Gate Minimization
• Toffoli Count
• Fault-Tolerant Quantum Computation
• Algebraic Methods