Noisy quantum parameter estimation with indefinite causal order

Optimal probe states and measurements are always required to achieve the best estimation precision in quantum metrology. In the actual physical environment, ubiquitous noise hinders estimation precision and amplifies the challenges of optimizing the probe states and measurements. In this study we extend a theoretical proposal, presented in Phys. Rev. A 103, 032615 (2021), to encompass noisy general Pauli channels for SU(2) phase estimation. By utilizing a superposition of different causal orders of two channels, we establish theoretically a probe-state-independent criterion for the estimation precision. We show that the probe-state-independent property results from the noncommutativity of the Kraus operators of the parameter-encoding channels. Based on this criterion, one can find some parameter-encoding channels without the requirement of precisely preparing the probe state. Moreover, our scheme requires only deterministic projection measurement on the control qubit. In this way, one can simultaneously avoid optimizing both the probe states and measurements. In addition, we also show that the estimation precision and probe-state-independent property of the indefinite causal order scheme are independent of representations of the Kraus operators. This alleviates the experimental challenges in realizing quantum channels. We demonstrate experimentally the advantages of the indefinite causal order for phase estimation of an SU(2) unitary transformation in the three kinds of quantum noise channels. Our result shows that the dynamic evolution of indefinite causal order can outperform the conventional cascaded estimation scenario at high noise levels, exhibiting high robustness and feasibility in practical estimation tasks.