Hardness of phase-polynomial synthesis with identity linear transformation

Determine whether exact phase-polynomial circuit synthesis remains computationally intractable when the final linear transformation is fixed to the identity matrix, so that any hardness arises from the intermediate parity checkpoints associated with a nontrivial phase polynomial.

Background

The paper proves that minimum-CNOT exact synthesis of phase-polynomial circuits is NP-hard even when the phase polynomial is trivial. This reduction derives all hardness from the prescribed final linear transformation and does not use the intermediate parity checkpoints induced by nonzero terms of a phase polynomial.

The unresolved issue is whether those checkpoint constraints alone can cause computational intractability. The extremal case fixes the final transformation to the identity matrix, thereby isolating the contribution of the phase polynomial and its required intermediate parities. The paper explicitly states that its result does not resolve this case.

References

This reduction, however, deliberately places all of the hardness in the target linear transformation $A$ and does not exploit the intermediate parity requirements introduced by a nontrivial phase polynomial. It naturally leaves a question: do the checkpoint constraints themselves intrinsically contribute to the complexity of exact phase polynomial synthesis? In the extremal case where we fix the target linear transform $A$ as $I_n$, can we still prove that exact phase polynomial synthesis is still intractable in this setting? This question essentially asks whether the hardness of exact phase polynomial synthesis arises from the phase polynomial $f$, the target transformation $A$, or both. Our result does not answer this question, which we leave as an interesting direction for future work.

— Vanilla Exact Synthesis of CNOT Circuits is NP-hard  (2609.04160 - Li et al., 3 Sep 2026) in Section 4, subsection “Exact Phase Polynomial Synthesis” (Section 4.4)