A family of unitaries for the quantum period finding algorithm

Igor E. Shparlinski, Arne Winterhof

Xinyu Yang, Long Li, Qi Zhou

This paper introduces and rigorously analyzes a new class of one-dimensional discrete-time quantum walks whose dynamics are governed by a parametrized family of extended CMV matrices. The model generalizes the unitary almost Mathieu operator (UAMO) and exhibits a richer spectral phase diagram, closely resembling the extended Harper's model. It provides the first example of a solvable quasi-periodic quantum walk that exhibits a stable region of purely singular continuous spectrum.

John George Francis, Anil Shaji

We use differentiable programming and gradient descent to find unitary matrices that can be used in the period finding algorithm to extract period information from the state of a quantum computer post application of the oracle. The standard procedure is to use the inverse quantum Fourier transform. Our findings suggest that that this is not the only unitary matrix appropriate for the period finding algorithm, There exist several unitary matrices that can affect out the same transformation and they are significantly different from each other as well. These unitary matrices can be learned by an algorithm. Neural networks can be applied to differentiate such unitary matrices from randomly generated ones indicating that these unitaries do have characteristic features that cannot otherwise be discerned easily.

Florina Piroi, Arne Winterhof

Jonathan M. Robbins

We consider the geometric phase for a family of quantum/classical Hamiltonians in which the effect of changing parameters is simply to induce unitary/canonical transformations. In this case the classical limit of the geometric phase is easily obtained, even when the classical motion is chaotic. The results agree with those previously obtained for general chaotic families, but may be expressed in a simpler form, not involving time integrals of correlation functions. It is also straightforward to establish some results which are problematic in the general case, for example the form of periodic orbit corrections, and the closeness of the classical 2-form. If the parameters are regarded as dynamical variables, evolving slowly so as to maintain adiabaticity, they are subject to geometric magnetism, but not, in contrast to the general case, deterministic friction and Born-Oppenheimer forces. Examples including families of translated and rotated systems are discussed.

Jamie Smith, Michele Mosca