Experimental investigation of equivalent Laplacian and adjacency quantum walks on irregular graphs

The continuous-time quantum walk denotes the evolution of a particle on a given graph governed by Schrödinger's equation. For different applications, the corresponding Hamiltonian is proportional to either the Laplacian or the adjacency matrix of the graph. The two quantum walks are equivalent on regular graphs, since each vertex has the same degree. However, for irregular graphs, the evolutions of the two quantum walks are generally different. In this paper, we report an experimental investigation of the two quantum walks on irregular graphs with single photons and interferometric networks. We demonstrate that it is possible to obtain equivalent probability distributions with the two quantum walks on specific irregular graphs, where the particle is initially localized at a vertex or uniformly distributed at multiple vertices. Our results not only deepen the understanding of the equivalence between the two quantum walks, but also extend the application of the continuous-time quantum walk.

Figure 1

(a) Irregular graphs with (a) five and (b) six vertices. The probability distributions of the two QWs are equivalent, if the particle is initially localized at one of the green vertices.

Figure 2

(a) Experimental setup. The heralded single photons are generated and are injected into the optical network to simulate the CTQW on irregular graphs. The first polarizing beam splitter (PBS), half-wave plates (HWPs), and two beam displacers (BDs) are used to prepare the initial states. The Laplacian and adjacency QWs can be simulated by an interferometric network, consisting of WPs and BDs. The probability distributions are obtained by projecting the final state into the basis states via a PBS. The photons are detected by APDs. (b) The $5\times 5$ unitary matrix is decomposed into a product of ten two-level unitary matrices. The green nodes represent the two-level unitary operators. The connection between two matrices indicates that they can be realized at the same time.

Figure 3

Experimental results of the probability distributions of two QWs at time $t=\{10,{10}^2,{10}^3,{10}^4\}$ for the irregular graph with five vertices. The blue and green hatched bars represent the experimental results of the Laplacian and adjacency QWs, respectively. The open bars correspond to the theoretical predictions of the probability distributions of the two QWs. (a)–(d) indicate the results for QW starting at vertex 1, vertex 2, vertices 2 and 3, and vertices 1, 4, and 5, respectively. Error bars indicate the statistical uncertainty, which are obtained based on assuming Poissonian statistics.

Figure 4

Experimental results of the probability distributions of two QWs at time $t=\{10,{10}^2,{10}^3,{10}^4\}$ for the irregular graph with six vertices. The red and green hatched bars represent the experimental results of the Laplacian and adjacency QWs, respectively. The open bars correspond to the theoretical predictions of the probability distributions of the two QWs. (a)–(d) indicate the results of the two QWs when starting at vertex 1, vertex 2, vertices 2 and 4, and vertices 3 and 5, respectively. Error bars indicate the statistical uncertainty, which is obtained based on assuming Poissonian statistics.