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.

翻译：容错量子计算需要最小化非克利福德门，其通过魔幻态蒸馏实现的方式主导了资源成本。虽然 $T$ 门数量最小化已得到充分研究，但专用的 $CCZ$ 工厂将自然目标转向直接托佛利门最小化。我们针对此问题发展了代数方法，该方法基于托佛利门数量与 $\mathbb{F}_2$ 上张量分解之间的关联。在标准基准测试中，这些方法匹配或改进了所有已报道的托佛利门和 $T$ 门数量结果，且大多数电路在单CPU上可在1分钟内完成，而先前工作需使用数千个TPU。