Stabilization of symmetry-protected long-range entanglement in stochastic quantum circuits

Long-range entangled states are vital for quantum information processing and quantum metrology. Preparing such entangled states by combining measurements with unitary gates has opened new possibilities for efficient protocols with finite-depth quantum circuits. The complexity of these algorithms is crucial for the resource requirements on a quantum device. The stability of the preparation protocols to perturbations decides the fate of their implementation in large-scale noisy quantum devices. In this work, we consider stochastic quantum circuits in one and two dimensions consisting of randomly applied unitary gates and local measurements. These quantum operations preserve a class of discrete local symmetries, which can be broken due to the stochasticity arising from timing and gate imperfections. In the absence of randomness, the protocol is known to generate a symmetry-protected long-range entangled state in a finite-depth circuit. In the general case, by studying the time evolution under this hybrid quantum circuit, we analyze the time to reach the target entangled state. We find two important time scales which we associate with the emergence of certain symmetry generators. The quantum trajectories embody the local symmetry with a time that scales logarithmically with system size, whereas global symmetries require exponentially long times to appear. We devise error-mitigation protocols that provide significant improvement on both time scales and investigate the stability of the algorithm to perturbations that naturally arise in experiments. We also generalize the protocol to realize the toric code and Xu-Moore states in two dimensions, and open avenues for future studies of anyonic excitations present in those systems. Our work paves the way for efficient error correction for quantum state preparation.