Tensor Decomposition for Non-Clifford Gate Minimization

TL;DR

This paper introduces algebraic tensor decomposition methods to minimize non-Clifford gates, specifically Toffoli and T gates, in fault-tolerant quantum circuits, achieving faster and more efficient results.

Contribution

It develops novel algebraic techniques linking Toffoli count to tensor decomposition, improving gate minimization efficiency over existing methods.

Findings

Methods match or improve existing Toffoli and T-count results.

Most circuits are optimized in under a minute on a single CPU.

Significant reduction in computational resources compared to prior work.

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.