Implementation of multidimensional quantum walks using linear optics and classical light

Classical optics can be used to efficiently implement certain quantum information processing tasks with a high degree of control, for example, one-dimensional quantum walks through the space of orbital angular momentum of light directed by its polarization. To explore the potential of quantum information processing with classical light, we here suggest a method to realize $d$-dimensional quantum walks with classical optics—an important step towards robust implementation of certain quantum algorithms. In this scheme, different degrees of freedom of light, such as frequency, orbital angular momentum, and time bins, represent different directions for the walker while the coin to decide which direction the walker takes is realized by employing the polarization combined with different light paths.

Figure 1

(a) Diagram for the optical implementation of a four-dimensional unitary operation. Here, $BS$ stands for the elementary beam-splitter array implementing the operator ${\mathbb{U}}^{\left(4\right)}$ in the CS decomposition (7), which consists out of two beam splitters and two half-wave plates and shown in (b).

Figure 2

Any unitary coin toss operator for a quantum walk in three dimensions can be implemented by a combination of three elementary beam-splitter arrays $U^{\left(4\right)}$ combined with nine unitary transforms of the polarization.

Figure 3

Schematic diagram for the setup for a three-dimensional quantum walk. Here, $a, b$, and $c$ denote the three paths for the light beams, $U_c$ is the six-dimensional coin flip operator, and $S_1, S_2$, and $S_3$ are the three shift operators acting on OAM, time, and the position space, respectively.