Discrete-time Quantum Walks in Qudit Systems

TL;DR

This paper introduces a novel framework for implementing discrete-time quantum walks in higher-dimensional qudit systems, offering more efficient quantum circuits and scalability for quantum algorithms and simulations.

Contribution

It presents the first one-dimensional quantum walk in d-dimensional systems with circuit realizations, extending to multi-dimensional walks and scalable circuit designs in qudit systems.

Findings

Efficient quantum circuits for odd-dimensional systems.

Extension of quantum walks to multi-dimensional lattices.

Scalable circuit designs for various search spaces.

Abstract

Quantum walks contribute significantly to developing quantum algorithms and quantum simulations. Here, we introduce a first of its kind one-dimensional quantum walk in the $d$-dimensional quantum domain, where $d>2$, and show its equivalence for circuit realization in an arbitrary finite-dimensional quantum logic for utilizing the advantage of larger state space, which helps to reduce the run-time of the quantum walks as compared to the conventional binary quantum systems. We provide efficient quantum circuits for the implementation of discrete-time quantum walks (DTQW) in one-dimensional position space in any finite-dimensional quantum system when the dimension is odd using an appropriate logical mapping of the position space on which a walker evolves onto the multi-qudit states. With example circuits for various qudit state spaces, we also explore scalability in terms of $n$-qudit $d$-ary quantum systems. Further, the extension of one-dimensional DTQW to $d$-dimensional DTQW using $2d$-dimensional coin space on $d$-dimensional lattice has been studied, where $d>=2$. Thereafter, the circuit design for the implementation of scalable $d$-dimensional DTQW in $d$-ary quantum systems has been portrayed. Lastly, we exhibit the circuit design for the implementation of DTQW using different coins on various search spaces.