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Quantum State Designs from Minimally Random Quantum Circuits

Jonathon Riddell, Katja Klobas, and Bruno Bertini

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PRX Quantum 7, 033047 – Published 8 September, 2026

DOI: https://doi.org/10.1103/kfzr-2k5r

Abstract

Random many-body states and matrices are both a useful tool to model certain physical systems and an important asset for quantum computation. Realizing them, however, generally requires an exponential (in system size) amount of resources. Recent research has presented a way out by showing that one can generate random states and matrices, or more precisely a controlled approximation of them, by applying a quantum circuit built in terms of few-body unitary gates. Most of this research has been focussed on the case of quantum circuits composed by fully random unitary gates. Here we consider what happens for circuits that, instead, involve a minimal degree of randomness. Specifically, we concentrate on two different settings: (a) brickwork quantum circuits with a single one-qudit random matrix at a boundary; (b) brickwork quantum circuits with fixed interactions but random one-qudit gates everywhere. We show that, for any given initial state, (a) and (b) produce a distribution of states approaching the Haar distribution in the limit of large circuit depth. More precisely, we show that the moments of the distribution produced by our circuits can approximate the ones of the Haar distribution in a depth proportional to the system size. Interestingly we find that in both Cases (a) and (b) the relaxation to the Haar distribution occurs in two steps—this is in contrast with what happens in fully random circuits. Moreover, we show that choosing appropriately the fixed interactions, for example taking the local gate to be a dual-unitary gate with high enough entangling power, minimally random circuits produce a Haar random distribution more rapidly than fully random circuits. In particular, dual-unitary circuits with maximal entangling power—i.e. perfect tensors—appear to provide the optimal quantum state design preparation for any design number. Our results also suggest that the minimally random settings (a) and (b) give an optimal framework to prepare unitary designs, i.e., random unitary matrices.

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Random numbers are a useful resource for many different aspects of today’s life: their use spans from mundane tasks, such as shuffling music playlists, to applications in science and cryptography. Analogously, also the quantum version of random numbers—random unitary matrices and states—has a wealth of different applications ranging from understanding the physics of black holes to specific algorithms of quantum computation. The problem is that, as for their classical analogue, generating large random unitary matrices and states is very expensive in terms of resources. Recent research has found a way around this problem by showing that one can “approximate” random unitary matrices by applying a sequence of few-body random operations—a random quantum circuit—which is much more manageable in terms of resources. Here we show that if the quantum circuit generates complex enough dynamics, one does not need all the quantum operations to be random: a single one-site operation is enough. In other words, we show that there is a trade-off between randomness and dynamical chaoticity. This observation has important implications on our understanding of quantum many-body dynamics and can lead to a more efficient implementation of random matrices and states.

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PRX Quantum 7, 033047– Published 8 September, 2026
  • Received 24 March 2025
  • Revised 27 September 2025
  • Accepted 8 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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