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

Published

August 22, 2025

Author(s)

Zihan Cheng, Eric Huang, Vedika Khemani, Michael Gullans, Matteo Ippoliti

Abstract

Unitary k-designs are distributions of unitary gates that match the Haar distribution up to its k-th statistical moments. They serve as a crucial resource for randomized quantum protocols. However, their implementation on encoded logical qubits is nontrivial due to the need for magic gates, which often 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 on the physical qubits ("coherent errors") followed by rounds of error correction. We prove that for specific classes of physical rotations (including all single-qubit unitaries) and under certain conditions 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 syndrome outcomes) converges to a unitary design in the thermodynamic limit, provided the density or strength of coherent errors is larger than a critical value. This "unitary design" phase transition coincides with the code's coherent error threshold. 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+1D 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 correction codes.

Citation

PRX Quantum

Volume

6

Issue

3

Pub Type

Journals

Download Paper

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

Local Download

Keywords

Quantum algorithms, quantum error correction, unitary designs

Citation

Cheng, Z. , Huang, E. , Khemani, V. , Gullans, M. and Ippoliti, M. (2025), Emergent Unitary Designs for Encoded Qubits from Coherent Errors and Syndrome Measurements, PRX Quantum, [online], https://doi.org/10.1103/bnld-2chd, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=959193 (Accessed September 3, 2025)