Unitary Designs of Symmetric Local Random Circuits

Yosuke Mitsuhashi; Ryotaro Suzuki; Tomohiro Soejima; Nobuyuki Yoshioka

doi:10.1103/PhysRevLett.134.180404

Unitary Designs of Symmetric Local Random Circuits

Yosuke Mitsuhashi ^1,*, Ryotaro Suzuki ^2,†, Tomohiro Soejima ^3,‡, and Nobuyuki Yoshioka ^4,5,6,§

¹Department of Basic Science, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
²Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin 14195, Germany
³Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
⁴Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
⁵Quantum Computing Center, RIKEN Cluster for Pioneering Research (CPR), Wako-shi, Saitama 351-0198, Japan
⁶JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan

^*Contact author: mitsuhashi@noneq.t.u-tokyo.ac.jp
^†Contact author: ryotaro.suzuki@fu-berlin.de
^‡Contact author: tomohiro_soejima@g.harvard.edu
^§Contact author: nyoshioka@ap.t.u-tokyo.ac.jp

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Phys. Rev. Lett. 134, 180404 – Published 7 May, 2025

DOI: https://doi.org/10.1103/PhysRevLett.134.180404

Abstract

We have established the method of characterizing the unitary design generated by a symmetric local random circuit. Concretely, we have shown that the necessary and sufficient condition for the circuit asymptotically forming a $t$ -design is given by simple integer optimization for general symmetry and locality. By using the result, we explicitly give the maximal order of unitary design under the $Z_{2}$ , U(1), and SU(2) symmetries for general locality. This work reveals the relation between the fundamental notions of symmetry and locality in terms of randomness.

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Unitary Designs of Symmetric Local Random Circuits

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Characterization of randomness in quantum circuits of continuous gate sets

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Unitary Designs of Symmetric Local Random Circuits

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Characterization of randomness in quantum circuits of continuous gate sets

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