P T-symmetric Bayesian parameter estimation on a superconducting quantum processor

To realize the computational advantages of quantum mechanics, it is necessary to be able to effectively perform quantum state tomography, and Bayesian methods provide significant opportunities in this regard, being applicable in the uncertainty regimes, where the local approaches based on Cramer-Rao bound become ill-defined. ${\mathcal{P}}{\mathcal{T}}$ -symmetric quantum mechanics provides a unique advantage for Bayesian parameter estimation having an additional degree of freedom, absent in the regular Hermitian case, which significantly enhances the quantum Fisher information and reduces variance of the obtained results. While previous studies have examined ${\mathcal{P}}{\mathcal{T}}$ -symmetric Bayesian parameter estimation in theory, we conduct experimental studies using IBM Quantum Experience and compare them with theoretical calculations. In our approach, the evolution of a qubit controlled by a ${\mathcal{P}}{\mathcal{T}}$-symmetric Hamiltonian is realized by the dilation method using ancilla qubit. We show that our implementation is consistent with the theoretical model and, in particular, we observe a good correspondence between the theoretical and experimental likelihood distributions. Thus we bridge the gap between theory and practical implementations in ${\mathcal{P}}{\mathcal{T}}$-symmetric quantum mechanics and its practical applications in quantum information science.