Steered discrete-time quantum walks for engineering of quantum states

TL;DR

This paper investigates the capabilities and limitations of steered discrete-time quantum walks in generating quantum states, providing algorithms for accessible states and practical implementation insights with linear optics.

Contribution

It introduces an algorithm for generating all walk-accessible states efficiently and offers a method to construct minimal quantum walks between such states, with practical optical implementation details.

Findings

Not all quantum states are accessible via quantum walks.

All normalized states are accessible in linear optics implementations.

Single q-plate and waveplates can implement generalized quantum steps.

Abstract

We analyze the strengths and limitations of steered discrete time quantum walks in generating quantum states of bipartite quantum systems comprising of a qubit coupled to a qudit system. We demonstrate that not all quantum states in the composite space are accessible through quantum walks, even under the most generalized definition of a quantum step, leading to a bifurcation of the composite Hilbert space into "walk-accessible states" and the "walk-inaccessible states". We give an algorithm for generating any walk-accessible state from a simple-to-realize product state, in a minimal number of walk steps, all of unit step size. We further give a prescription towards constructing minimal quantum walks between any pair of such walk-accessible states. Linear optics has been a popular physical system for implementing coin-based quantum walks, where the composite space is built up of spin and orbital angular momenta of light beams. We establish that in such an implementation, all normalized quantum states are "walk-accessible". Furthermore, any generalized quantum step can be implemented upto a global phase using a single q-plate and a pair of homogeneous waveplates. We then give a quantum walk based scheme for realizing arbitrary vector beams, using only q-plates and waveplates.