Optimal synthesis complexity with arbitrary Clifford primitives

Extend the NP-hardness result for optimal Clifford circuit synthesis to the model in which an arbitrary one- or two-qubit Clifford operation is treated as a primitive gate.

Background

The paper proves NP-hardness for a specific elementary gate set containing H, S, S†, CNOT, CZ, and Pauli gates. Its reduction relies on the finer structure of these individual elementary operations, particularly the ability to eliminate Hadamard and CNOT gates from short circuits implementing graph-associated CZ unitaries.

The authors identify a natural broader model in which any one- or two-qubit Clifford operation may be used as a primitive. They state that their proof does not immediately apply to this model and leave extending the complexity result to future work.

References

A natural extension of the current result is to count an arbitrary one- or two-qubit Clifford as a primitive gate. The present proof uses the finer elementary structure of $H,S,CNOT$, and CZ and therefore does not immediately extend to that model. We leave this to future work.

— (A Variant of) Clifford Circuit Synthesis is NP-Complete  (2610.02029 - Keller et al., 1 Oct 2026) in Synopsis