Ancilla-Free Parity Network Synthesis is NP-complete

A parity network is a CNOT circuit in which every parity from a prescribed set S appears on some wire; such networks are the CNOT skeletons of phase-polynomial circuits. Amy, Azimzadeh and Mosca (Quantum Science and Technology, 2018) left open the ancilla-free problem with arbitrary output. We prove that deciding whether S admits such a network with at most K CNOTs is NP-complete, even if every CNOT must produce a new parity, all parities share a variable and have weight at most five, and connectivity is a star. Hence CNOT minimization of phase polynomials is NP-hard even with a free linear part. The reduction is from Hamiltonian cycle in grid graphs: networks meeting the trivial lower bound are path covers with at most one path per wire, and chaining many copies of the instance makes a missing Hamiltonian path cost more paths than there are wires. We also show that unrestricted networks can be shorter than fixed-target ones by a factor linear in the number of qubits. This is a preprint; it has not been peer reviewed.