Tight bounds on depth-2 QAC-circuits computing parity
Abstract
We show that the parity of more than three non-target input bits cannot be computed by QAC-circuits of depth-2, not even uncleanly, regardless of the number of ancilla qubits. This result is incomparable with other recent lower bounds on constant-depth QAC-circuits by Rosenthal [ICTS~2021,arXiv:2008.07470] and uses different techniques which may be of independent interest: 1. We show that all members of a certain class of multivariate polynomials are irreducible. The proof applies a technique of Shpilka & Volkovich [STOC 2008]. 2. We give a tight-in-some-sense characterization of when a multiqubit CZ gate creates or removes entanglement from the state it is applied to. The current paper strengthens an earlier version of the paper [arXiv:2005.12169].
Keywords
Cite
@article{arxiv.2504.06433,
title = {Tight bounds on depth-2 QAC-circuits computing parity},
author = {Stephen Fenner and Daniel Grier and Daniel Padé and Thomas Thierauf},
journal= {arXiv preprint arXiv:2504.06433},
year = {2025}
}
Comments
26 pages, 5 figures (including 2 in-line). Some material also appeared in [arXiv:2005.12169]