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.

量子计算涉及跨多个量子比特准备纠缠状态。这需要对噪声和栅极缺陷保持稳定的高效制备方案。在这里，我们演示了在存在相干和非相干误差的情况下，使用 54 个系统量子比特使用基于测量的协议生成最简单的长程阶数 — Ising 阶数。我们实现了一个恒定深度准备协议，该协议使用测量的经典解码来识别长程顺序，否则这些顺序将被量子测量的随机性所隐藏。通过实验调整误码率，我们证明了这种解码的长程顺序在两个空间维度上的稳定性，直到属于不寻常的 Nishimori 普遍性类的临界相变。尽管在经典系统中，西森物理学需要微调多个参数，但在这里它是测量概率的 Born 规则的直接结果。我们的研究展示了可以在 100 个量子比特以上的量子处理器上探索的新兴现象。