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Emergent Unitary Designs for Encoded Qubits from Coherent Errors and Syndrome Measurements

Zihan Cheng1,*, Eric Huang2, Vedika Khemani3, Michael J. Gullans2, and Matteo Ippoliti1

  • *Contact author: zihan_cheng@utexas.edu
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PRX Quantum 6, 030333 – Published 22 August, 2025

DOI: https://doi.org/10.1103/bnld-2chd

Abstract

Unitary k-designs are distributions of unitary gates that match the Haar distribution up to its kth statistical moment. They are a crucial resource for randomized quantum protocols. However, their implementation on encoded logical qubits is nontrivial due to the need for magic gates, which can require a large resource overhead. In this work, we propose an efficient approach to generate unitary designs for encoded qubits in surface codes by applying local unitary rotations (“coherent errors”) on the physical qubits followed by syndrome measurement and error correction. We prove that, under some conditions on the coherent errors (notably including all single-qubit unitaries) and on the error-correcting code, this process induces a unitary transformation of the logical subspace. We numerically show that the ensemble of logical unitaries (indexed by the random syndrome outcomes) converges to a unitary design in the thermodynamic limit, provided that the density or strength of coherent errors is above a finite threshold. This “unitary design” phase transition coincides with the code’s coherent error threshold under optimal decoding. Furthermore, we propose a classical algorithm to simulate the protocol based on a “staircase” implementation of the surface code encoder and decoder circuits. This enables a mapping to a (1+1)-dimensional monitored circuit, where we observe an entanglement phase transition (and thus a classical complexity phase transition of the decoding algorithm) coinciding with the aforementioned unitary design phase transition. Our results provide a practical way to realize unitary designs on encoded qubits, with applications including quantum state tomography and benchmarking in error-correcting codes.

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Popular Summary

Quantum randomness is an important resource in many quantum information processing tasks, such as benchmarking, tomography, and demonstrations of quantum advantage. However, on error-corrected quantum computers, the implementation of highly random unitary operations can be challenging because of the resource overhead of non-Clifford, or “magic,” gates. Our work presents a scalable approach to avoid this overhead and generate highly random distributions of unitary gates, known as unitary designs, directly on encoded logical qubits.

Our protocol is based on the intentional application of “coherent errors” to the physical qubits, followed by syndrome measurement and error correction. For wide classes of coherent errors and codes, we show that, despite the presence of measurements, the overall evolution of the logical information is unitary. We find that, in the surface code, this ensemble of unitary operations (one per possible syndrome) becomes highly random in the infinite-system limit when the coherent errors exceed a critical threshold. Remarkably, this threshold also coincides with the code’s optimal error-correction threshold and with an entanglement phase transition in an associated one-dimensional model. The latter phenomenon enables efficient classical simulation based on matrix product states below the threshold.

Our work paves the way for practical realizations of unitary designs on logical qubits, with potential applications including quantum state tomography, benchmarking, and random circuit sampling with logical qubits.

Article Text

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PRX Quantum 6, 030333– Published 22 August, 2025
  • Received 26 April 2025
  • Revised 10 July 2025
  • Accepted 24 July 2025
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