Computation with quantum walks

Salvador Elías Venegas-Andraca

Norio Konno, Takuya Machida

Recently Mc Gettrick introduced and studied a discrete-time 2-state quantum walk (QW) with a memory in one dimension. He gave an expression for the amplitude of the QW by path counting method. Moreover he showed that the return probability of the walk is more than 1/2 for any even time. In this paper, we compute the stationary distribution by considering the walk as a 4-state QW without memory. Our result is consistent with his claim. In addition, we obtain the weak limit theorem of the rescaled QW. This behavior is strikingly different from the corresponding classical random walk and the usual 2-state QW without memory as his numerical simulations suggested.

Salvador Elías Venegas-Andraca

Asif Shakeel

Quantum walks (QWs) are of interest as examples of uniquely quantum behavior and are applicable in a variety of quantum search and simulation models. Implementing QWs on quantum devices is useful from both points of view. We describe a prototype one-dimensional QW algorithm that economizes resources required in its implementation. Our algorithm needs only a single shift (increment) operation. It also allows complete flexibility in choosing the shift circuit, a resource intensive part of QW implementations. This is desirable for Noisy Intermediate-Scale Quantum (NISQ) devices, in which fewer computations implies faster execution and reduced effects of noise and decoherence. We implement versions of the algorithm, with two different shift circuit structures, on publicly accessible IBM quantum computers.

Salvador Elías Venegas-Andraca

Bereneice Sephton, Angela L. Dudley, Andrew Forbes

In the avid search for means to increase computational power in comparison to that which is currently available, quantum walks (QWs) have become a promising option with derived quantum algorithms providing an associated speed up compared to what is currently used for implementation in classical computers. It has additionally been shown that the physical implementation of QWs will provide a successful computational basis for a quantum computer. It follows that considerable drive for finding such means has been occurring over the 20+ years since its introduction with phenomena such as electrons and photons being employed. Principal problems encountered with such quantum systems involve the vulnerability to environmental influence as well as scalability of the systems. Here we outline how to perform the QW due to interference characteristics inherent in the phenomenon, to mitigate these challenges. We utilize the properties of vector beams to physically implement such a walk in orbital angular momentum space by manipulating polarization and exploiting the non-separability of such beams.