Praxiological Studies of Quantum Communication

Chapter 5

Abstract

mable and controllable.

1. Introduction

59

Photonics Lattices

Quantum Walks in Quasi-Periodic

Dan Trung Nguyen, Daniel A. Nolan and Nicholas F. Borrelli

We present construction rules for a new class of quasiperiodic photonics lattices (QPL) to realize localized quantum walks (LQWs) deterministically. The new quasiperiodic structures are constructed symmetrically with Fibonacci, Thue-Morse, and other quasiperiodic sequences. Our construction rules allow us to build the symmetrical quasiperiodic photonics lattices. As a result, LQWs with symmetrical probability distributions can be realized in these QPLs. Furthermore, the proposed QPLs are composed with different waveguides providing both on- and off-diagonal deterministic disorders. We show LQWs in the proposed QPLs are highly program-

Quantum walks (QWs) have been used to construct exponential speedup quantum algorithms [1–3] and quantum simulations [4, 5], to implement universal gates for quantum computers [6, 7], etc. For the last few decades, scientists have made tremendous progress on research and development of those areas which are the most promising for solving problems that are intractable by classical computers. Among different schemes (or approaches) quantum photonics has the advantage of highly advanced technology that can easily generate and manipulate almost any desired photons—the walkers, even in room-temperature conditions. Discrete-time QWs (DTQW) have been demonstrated using beam splitter arrays [8, 9], and continuous-time quantum walks (CTQWs) have been investigated both theoretically and experimentally in evanescently coupled parallel waveguide arrays [10–17]. Integrated photonics lattices consisting of evanescently coupled waveguides are perfectly suited for investigation of CTQWs, and in fact, laser-written waveguides were the first systems used to demonstrate quantum walks on a line with coherent light [10–13]. In those photonics lattices, the walking process occurs in the region of evanescent-coupled waveguides. As a result, spacing between waveguides in those lattices is close enough, typically on the order of several micrometers to ensure evanescent coupling to occur. It is well established both theoretically and experimentally that in a uniform and/or periodic array of coupled waveguides (photonics lattices), the probability distribution of single-photon QWs spreads across the waveguide lattice by coupling from one waveguide to its neighbors in a pattern

Keywords: quantum walks, localized quantum walks, photonics lattices,

quasiperiodic, Fibonacci sequence, Thue-Morse sequence

characterized by two strong "ballistic" lobes [11, 12].
