{"work":{"id":"8c7cc112-2d30-4ebb-94dc-25f74f198934","openalex_id":null,"doi":null,"arxiv_id":"1905.12053","raw_key":null,"title":"Unitary designs from statistical mechanics in random quantum circuits","authors":null,"authors_text":"N","year":2019,"venue":"quant-ph","abstract":"Random quantum circuits are proficient information scramblers and efficient generators of randomness, rapidly approximating moments of the unitary group. We study the convergence of local random quantum circuits to unitary $k$-designs. Employing a statistical mechanical mapping, we give an exact expression of the distance to forming an approximate design as a lattice partition function. In the statistical mechanics model, the approach to randomness has a simple interpretation in terms of domain walls extending through the circuit. We analytically compute the second moment, showing that random circuits acting on $n$ qudits form approximate 2-designs in $O(n)$ depth, as is known. Furthermore, we argue that random circuits form approximate unitary $k$-designs in $O(nk)$ depth and are thus essentially optimal in both $n$ and $k$. We can show this in the limit of large local dimension, but more generally rely on a conjecture about the dominance of certain domain wall configurations.","external_url":"https://arxiv.org/abs/1905.12053","cited_by_count":null,"metadata_source":"pith","metadata_fetched_at":"2026-07-09T07:26:03.283131+00:00","pith_arxiv_id":"1905.12053","created_at":"2026-05-09T05:55:31.114361+00:00","updated_at":"2026-07-09T07:26:03.283131+00:00","title_quality_ok":true,"display_title":"Unitary designs from statistical mechanics in random quantum circuits","render_title":"Unitary designs from statistical mechanics in random quantum circuits"},"hub":{"state":{"work_id":"8c7cc112-2d30-4ebb-94dc-25f74f198934","tier":"hub","tier_reason":"10+ Pith inbound or 1,000+ external citations","pith_inbound_count":10,"external_cited_by_count":null,"distinct_field_count":2,"first_pith_cited_at":"2024-08-02T19:18:30+00:00","last_pith_cited_at":"2026-05-04T18:00:24+00:00","author_build_status":"not_needed","summary_status":"needed","contexts_status":"needed","graph_status":"needed","ask_index_status":"not_needed","reader_status":"not_needed","recognition_status":"not_needed","updated_at":"2026-06-26T08:14:46.804050+00:00","tier_text":"hub"},"tier":"hub","role_counts":[{"context_role":"background","n":4}],"polarity_counts":[{"context_polarity":"background","n":4}],"runs":{},"summary":{},"graph":{},"authors":[]}}