Quantum walker as a probe for its coin parameter

Abstract

In discrete-time quantum walk (DTQW) the walker's coin space entangles with the position space after the very first step of the evolution. This phenomenon may be exploited to obtain the value of the coin parameter θ by performing measurements on the sole position space of the walker. In this paper, we evaluate the ultimate quantum limits to precision for this class of estimation protocols, and use this result to assess measurement schemes having limited access to the position space of the walker in one dimension. We find that the quantum Fisher information (QFI) of the walker's position space Hw(θ) increases with θ and with time which, in turn, may be seen as a metrological resource. We also find a difference in the QFI of bounded and unbounded DTQWs, and provide an interpretation of the different behaviors in terms of interference in the position space. Finally, we compare Hw(θ) to the full QFI H f (θ), i.e., the QFI of the walkers position plus coin state, and find that their ratio is dependent on θ, but saturates to a constant value, meaning that the walker may probe its coin parameter quite faithfully.

Multi-parameter quantum metrology with discrete-time quantum walks

“…The minimum value of the F θ is compatible with the results in Ref. [43], where numerical evaluation for each initial state of the coin was required. 47) for θ = π/4 as a function of time when a maximizing (ry = 0, solid red line) or a minimizing (eigenstates of σy, dashed blue line) initial state is selected.…”

“…This time, however, the initial state of the coin is parametrized by γ. The elements of the QFIm of the full space can be derived by means of (41)(42)(43) and are…”

“…In view of their use for in the realization of quantum protocols, the characterization of quantum walks at the quantum level is a necessary step. Indeed, quantum metrological schemes have been proposed and precision benchmarks have been obtained for either the discrete [43]-or continuous-time quantum walks [44].…”

Estimation of one-dimensional discrete-time quantum walk parameters by using machine learning algorithms

“…It has also been shown that it is indeed possible to estimate the coin parameter of a DTQW with a singleparameter coin. However, the approaches that rely on QFI to predict the parameters in a DTQW share a similar constraint, viz, even though H w (θ) is a continuous, single-valued function over θ, the variations in the plot are small, and thus contribute to the error in determining θ [57]. In this work, we train various machine learning models and demonstrate that such models can indeed estimate the quantum walk parameters to a good accuracy.…”

“…It is known that the Quantum Fisher Information (QFI) [52][53][54][55] H w (θ) of a quantum walker's position quantifies a bound to the amount of information that can be extracted from the probability distribution [56,57]. It has also been shown that it is indeed possible to estimate the coin parameter of a DTQW with a singleparameter coin.…”