{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/random-unitaries-in-extremely-low-depth","title":"Random unitaries in extremely low depth","arxiv_id":"2407.07754","date":"2024-07-10","proceeding":null,"authors":["Thomas Schuster","Jonas Haferkamp","Hsin-Yuan Huang"],"abstract":"We prove that random quantum circuits on any geometry, including a 1D line, can form approximate unitary designs over $n$ qubits in $\\log n$ depth. In a similar manner, we construct pseudorandom unitaries (PRUs) in 1D circuits in $\\text{poly}(\\log n)$ depth, and in all-to-all-connected circuits in $\\text{poly}(\\log \\log n)$ depth. In all three cases, the $n$ dependence is optimal and improves exponentially over known results. These shallow quantum circuits have low complexity and create only short-range entanglement, yet are indistinguishable from unitaries with exponential complexity. Our construction glues local random unitaries on $\\log n$-sized or $\\text{poly}(\\log n)$-sized patches of qubits to form a global random unitary on all $n$ qubits. In the case of designs, the local unitaries are drawn from existing constructions of approximate unitary $k$-designs, and hence also inherit an optimal scaling in $k$. In the case of PRUs, the local unitaries are drawn from existing PRU constructions. Applications of our results include proving that classical shadows with 1D log-depth Clifford circuits are as powerful as those with deep circuits, demonstrating superpolynomial quantum advantage in learning low-complexity physical systems, and establishing quantum hardness for recognizing phases of matter with topological order.","url_abs":"https://arxiv.org/abs/2407.07754v2","url_pdf":"https://arxiv.org/pdf/2407.07754v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"random-unitaries-in-extremely-low-depth","repo_url":"https://github.com/haimengzhao/bounded-gate-tomography","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"jax","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}