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(i) Random Pauli Rotation: choose an $n$-qubit Pauli operator $P$ and an angle $\\theta \\in \\mathbb R / 2\\pi \\mathbb Z$, both uniformly at random, and apply $e^{\\mathrm i \\theta P}$. (ii) Brickwork Random Unitary Circuit: choose $n-1$ unitaries $U_{i}$ uniformly at random from $\\mathsf{SU}(4)$ independently, and apply $U_{2j-1}$ on two qubits $2j-1, 2j$ and then $U_{2j}$ on two qubits $2j, 2j+1$. Importantly, the spectral gaps are independent of $n$ and apply for"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.20919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2026-07-23T04:56:09Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"14b7979a49836210b89648ae9a124c42d1b91332de43aff300bb02c8ee4913ca","abstract_canon_sha256":"145493aea1736a927f10ec3291b51636e63d740ae718c30c30008e0a71721f4f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-24T01:23:40.727672Z","signature_b64":"HPoSzj7umaJ9rG17xODyQ6uaMVTxEL3bJbhLzEnckNTGu1wFexAZ2Rk32iutzVNwQLoMk/ri8CP35aU6Fc56Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"102ed1d914a99c51e81261314258893b9a6f3418ff49e3496dd6551846999c18","last_reissued_at":"2026-07-24T01:23:40.726718Z","signature_status":"signed_v1","first_computed_at":"2026-07-24T01:23:40.726718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random unitary circuits with constant spectral gap","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"quant-ph","authors_text":"Jeongwan Haah, Tim Baer","submitted_at":"2026-07-23T04:56:09Z","abstract_excerpt":"We prove constant lower bounds for the spectral gap of the following random walks on unitary groups $\\mathsf{SU}(2^n)$ on $n$ qubits. (i) Random Pauli Rotation: choose an $n$-qubit Pauli operator $P$ and an angle $\\theta \\in \\mathbb R / 2\\pi \\mathbb Z$, both uniformly at random, and apply $e^{\\mathrm i \\theta P}$. (ii) Brickwork Random Unitary Circuit: choose $n-1$ unitaries $U_{i}$ uniformly at random from $\\mathsf{SU}(4)$ independently, and apply $U_{2j-1}$ on two qubits $2j-1, 2j$ and then $U_{2j}$ on two qubits $2j, 2j+1$. 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