Nishimori transition across the error threshold for constant-depth quantum circuits

Quantum computing involves the preparation of entangled states across many qubits. This requires efficient preparation protocols that are stable to noise and gate imperfections. Here we demonstrate the generation of the simplest long-range order—Ising order—using a measurement-based protocol on 54 system qubits in the presence of coherent and incoherent errors. We implement a constant-depth preparation protocol that uses classical decoding of measurements to identify long-range order that is otherwise hidden by the randomness of quantum measurements. By experimentally tuning the error rates, we demonstrate the stability of this decoded long-range order in two spatial dimensions, up to a critical phase transition belonging to the unusual Nishimori universality class. Although in classical systems Nishimori physics requires fine-tuning multiple parameters, here it arises as a direct result of the Born rule for measurement probabilities. Our study demonstrates the emergent phenomena that can be explored on quantum processors beyond a hundred qubits.