Joint error-budget optimization with phase squashing

Determine how synthesis error and phase-squashing error should be jointly allocated, and determine whether applying phase squashing to a synthesized circuit can recover part of the approximation gap that exact semantics-preserving optimization cannot cross.

Background

The exactness barrier applies to optimizers that preserve the synthesized unitary exactly, whereas phase squashing deliberately permits controlled approximation. Because a synthesized circuit already has approximation error relative to its target, the paper identifies an unresolved composition problem: how to divide an overall error budget between initial synthesis and subsequent approximate rewriting, and whether phase squashing can exploit the available approximation slack to improve the circuit beyond the exactness floor.

References

A synthesized circuit already carries an approximation error from synthesis, so composing an approximation-aware optimizer with a synthesis pass raises a genuine question our exact analysis does not answer: how the two error budgets should be jointly allocated, and whether phase squashing applied to a synthesized circuit can recover part of the gap that the exactness barrier forbids exact methods from touching.

— An Exactness Barrier for ZX-Calculus Optimization of Synthesized Clifford+T Circuits  (2608.22801 - Kam et al., 24 Aug 2026) in Section 6, Discussion and Open Problems, subsection “Relation to phase squashing”