Long-range entangled states are vital for quantum information processing and quantum metrology. Preparing such states by combining measurements with unitary gates opened new possibilities for efficient protocols with finite-depth quantum circuits. The complexity of these algorithms is crucial for the resource requirements on a large-scale noisy quantum device, while their stability to perturbations decides the fate of their implementation. In this work, we consider stochastic quantum circuits in one and two dimensions comprising randomly applied unitary gates and local measurements. These operations preserve a class of discrete local symmetries, which are broken due to the stochasticity arising from timing and gate imperfections. In the absence of randomness, the protocol generates a symmetry-protected long-range entangled state in a finite-depth circuit. In the general case, by studying the time evolution under this hybrid circuit, we analyze the time to reach the target entangled state. We find two important time scales that we associate with the emergence of certain symmetry generators. The quantum trajectories embody the local symmetry with a time scaling logarithmically with system size, while global symmetries require exponentially long times. We devise error-mitigation protocols that significantly lower both time scales and investigate the stability of the algorithm to perturbations that naturally arise in experiments. We also generalize the protocol to realize toric code and Xu-Moore states in two dimensions, opening avenues for future studies of anyonic excitations. Our results unveil a fundamental relationship between symmetries and dynamics across a range of lattice geometries, which contributes to a broad understanding of the stability of preparation algorithms in terms of phase transitions. Our work paves the way for efficient error correction for quantum state preparation.

中文翻译：

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随机量子电路中对称保护的长程纠缠的稳定性


长程纠缠态对于量子信息处理和量子计量至关重要。通过将测量与酉门相结合来准备这种状态，为有限深度量子电路的高效协议开辟了新的可能性。这些算法的复杂性对于大规模噪声量子设备的资源需求至关重要，而它们对扰动的稳定性决定了其实现的命运。在这项工作中，我们考虑一维和二维的随机量子电路，包括随机应用的单一门和局部测量。这些操作保留了一类离散的局部对称性，但由于时序和门缺陷产生的随机性，这些对称性被破坏。在缺乏随机性的情况下，该协议在有限深度电路中生成对称保护的长程纠缠态。在一般情况下，通过研究这种混合电路下的时间演化，我们分析了达到目标纠缠态的时间。我们发现了两个重要的时间尺度，它们与某些对称生成器的出现相关。量子轨迹体现了局部对称性，其时间与系统大小成对数缩放，而全局对称性则需要指数级的长时间。我们设计了误差减轻协议，显着降低了时间尺度，并研究了算法对实验中自然出现的扰动的稳定性。我们还推广了该协议以实现二维环面码和 Xu-Moore 态，为未来的任意子激发研究开辟了途径。 我们的结果揭示了一系列晶格几何形状的对称性和动力学之间的基本关系，这有助于广泛理解相变制备算法的稳定性。我们的工作为量子态制备的有效纠错铺平了道路。