Tight bounds on depth-2 QAC-circuits computing parity
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910906951139328 |
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| author | Fenner, Stephen Grier, Daniel Padé, Daniel Thierauf, Thomas |
| author_facet | Fenner, Stephen Grier, Daniel Padé, Daniel Thierauf, Thomas |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06433 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tight bounds on depth-2 QAC-circuits computing parity Fenner, Stephen Grier, Daniel Padé, Daniel Thierauf, Thomas Quantum Physics Computational Complexity Mathematical Physics F.1.1; F.1.3 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]. |
| title | Tight bounds on depth-2 QAC-circuits computing parity |
| topic | Quantum Physics Computational Complexity Mathematical Physics F.1.1; F.1.3 |
| url | https://arxiv.org/abs/2504.06433 |