Estimation of shift parameters of a quantum state

Daniel Braun

Alexander Semenovich Holevo

We develop an asymptotic theory of estimation of a shift parameter in a pure quantum state to study the relation between entangled and unentangled covariant estimates in the analytically most transparent way. After recollecting basics of estimation of shift parameter in Sec. 2, we study the structure of the optimal covariant estimate in Sec. 3, showing how entanglement comes into play for several independent trials. In Secs. 4,5 we give the asymptotics of the performance of the optimal covariant estimate comparing it with the ``semiclassical'' unentangled covariant estimation in the regular case of finite variance of the generator of the shift group. Sec. 6 is devoted to covariant estimation in the case where the regularity assumption is violated.

Alexander Semenovich Holevo

The asymptotics of the optimal covariant estimate of a shift parameter of a pure quantum state is studied. In the regular case, where the variance of the generator of the shift group is finite, this problem of quantum statistics turns outto be connected with the information-theoretic approach to the classical central limit theorem proposed by Linnik in 1959. We also consider the quantum analogue of estimation of the shift parameter of the uniform distribution in which the regularity assumption is violated. The advantage of using entanglement is demonstrated in the class of covariant estimates.

Samuele Pedrielli, Christopher J. Anders, Lena Funcke +3 more

Parameter shift rules (PSRs) are key techniques for efficient gradient estimation in variational quantum eigensolvers (VQEs). In this paper, we propose its Bayesian variant, where Gaussian processes with appropriate kernels are used to estimate the gradient of the VQE objective. Our Bayesian PSR offers flexible gradient estimation from observations at arbitrary locations with uncertainty information and reduces to the generalized PSR in special cases. In stochastic gradient descent (SGD), the flexibility of Bayesian PSR allows the reuse of observations in previous steps, which accelerates the optimization process. Furthermore, the accessibility to the posterior uncertainty, along with our proposed notion of gradient confident region (GradCoRe), enables us to minimize the observation costs in each SGD step. Our numerical experiments show that the VQE optimization with Bayesian PSR and GradCoRe significantly accelerates SGD and outperforms the state-of-the-art methods, including sequential minimal optimization.

Carl W. Helstrom