Emergent unitary designs for encoded qubits from coherent errors and syndrome measurements

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 logical qubits in quantum error correction codes is highly nontrivial due to the need for magic gates, which often requires a large resource overhead. In this work, we propose an efficient approach to generate unitary designs for encoded qubits in surface codes by intentionally applying coherent errors (i.e. local unitary rotations on the physical qubits) and performing syndrome measurements. We prove that for a specific class of physical rotations (including all single qubit unitaries), the syndrome measurements induce unitary transformations of the logical subspace. By tuning the density of physical coherent errors, we numerically show the projected ensemble of logical unitaries forms unitary designs above the optimal coherent an error threshold, indicating the existence of a unitary design phase transition coinciding with the error correction transition. Furthermore, we propose a classical algorithm to simulate the projected ensemble 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.