Published October 4, 2026
| Version v1
Preprint
Open
Ancilla-Free Parity Network Synthesis is NP-complete
Authors/Creators
- 1. University of Engineering and Technology, Vietnam National University, Hanoi
-
2.
Phenikaa University
Description
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This is a preprint; it has not been peer reviewed.