Depth dependence of ZX-calculus reduction fraction

Determine whether the fraction of gates removed by automated ZX-calculus rewriting from single-qubit Solovay–Kitaev circuits remains constant as the target approximation accuracy is tightened beyond the three recursion levels studied.

Background

The paper numerically evaluates automated ZX-calculus post-processing of single-qubit circuits synthesized by the Solovay–Kitaev algorithm over the Clifford+T gate set. Across recursion depths 2, 3, and 4, the rewriting removes approximately 26.6–30.1% of total gates and 18.5–22.2% of T gates, with the fractional reduction appearing nearly constant despite substantial circuit growth.

The authors explicitly identify the limited depth range as preventing a conclusion about whether this approximately constant fractional saving persists at higher recursion depths, corresponding to more stringent approximation targets. The observed reduction in target-to-target variability with increasing depth provides suggestive evidence but does not resolve the question.

References

The second is depth: the flat behaviour reported here covers three recursion levels, and whether the fraction remains constant as the accuracy target is tightened further is a question the present data cannot settle, though the collapse of the spread with depth suggests it will.

— Numerical Evaluation of ZX Calculus Optimization for Solovay Kitaev Quantum Circuit Synthesis  (2608.22810 - Silva et al., 24 Aug 2026) in Section 6, Conclusion, subsection “Future work”