stabilizing measurement-induced order under coherent errors in quantum circuit

Measurement-induced phases represent a fundamentally nonequilibrium form of quantum order generated by the competition between unitary dynamics and projective measurements. in this work, we numerically investigate measurement-induced ising-like order in hybrid quantum circuits under realistic noise conditions. using qiskit-based simulations of systems with sizes ranging from n=9to n=25qubits, we explicitly incorporate both measurement noise with probability p_s∈[0,0.5]and coherent unitary errors with strength t_a∈[0,0.6]. we introduce an echo-based error-mitigation protocol to suppress coherent errors and systematically quantify its impact on the stability of the ordered phase. by analyzing magnetization, phase diagrams, and susceptibility, we extract effective critical noise thresholds in the range p_c≈0.22 - 0.35, depending on the coherent error strength. our results demonstrate that echo protection significantly extends the parameter regime over which measurement-induced order remains robust. this work provides a quantitative, noise-resilient framework for studying measurement-induced phases using experimentally accessible observables, thereby enhancing their relevance for near-term noisy intermediate-scale quantum devices.