Zihan Cheng

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Unitary k-designs are distributions of unitary gates that match the Haar distribution up to its k-th 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 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 converges to a unitary design in the thermodynamic limit, provided the density or strength of coherent errors is above a finite threshold.

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UT Austin
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