iSWAP maximises the second-moment spectral gap in random quantum circuits

• URL: http://arxiv.org/abs/2607.29551v1
• Date: Fri, 31 Jul 2026 15:42:31 GMT
• Title: iSWAP maximises the second-moment spectral gap in random quantum circuits
• Abstract summary: We prove that the $mathrmiSWAP$ gate maximises the spectral gap of the Hermitian second-moment operator on every connected graph.<n>We also prove that the polyhedral cone defined by asymmetric four-point inequalities is invariant under the transpose of the $mathrmiSWAP$ semigroup.
• Score: 3.1020753556218543
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove that the $\mathrm{iSWAP}$ gate maximises the spectral gap of the Hermitian second-moment operator on every connected graph with at least three vertices, among all two-local unitary circuit ensembles. We further prove that the polyhedral cone defined by asymmetric four-point inequalities is invariant under the transpose of the $\mathrm{iSWAP}$ semigroup, yielding a componentwise comparison certificate for its positive Perron--Frobenius eigenvector. These results resolve a conjecture of Kong, Li, and Liu.

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