Quantum experiments and hypergraphs: Multiphoton sources for quantum interference, quantum computation, and quantum entanglement

Abstract

We introduce the concept of hypergraphs to describe quantum optical experiments with probabilistic multi-photon sources. Every hyperedge represents a correlated photon source, and every vertex stands for an optical output path. Such general graph description provides new insights for producing complex high-dimensional multi-photon quantum entangled states, which go beyond limitations imposed by pair creation via spontaneous parametric down-conversion. Furthermore, properties of hypergraphs can be investigated experimentally. For example, the NP-Complete problem of deciding whether a hypergraph has a perfect matchin can be answered by experimentally detecting multi-photon events in quantum experiments. By introducing complex weights in hypergraphs, we show a general many-particle quantum interference and manipulating entanglement in a pictorial way. Our work paves the path for the development of multi-photon high-dimensional state generation and might inspire new applications of quantum computations using hypergraph mappings.

Graph Picture of Linear Quantum Networks and Entanglement

“…For example, vertices and (hyper-) edges correspond to output modes and photon sources in Ref. [35][36][37][38], which is fundamentally distinct from our correspondence relation in Table 1. The reason why the final state is computed from perfect matchings in both works is that both postselect states without bunching at the output.…”

“…At this point, it is worth emphasizing the distinctness of our graph picture for understanding entanglement in LQN from that in Refs. [35][36][37][38]. While both approaches exploit the graph theory for implementing experimental schemes to generate multipartite genuinely entangled states, the physical setups to be mapped to graphs are disparate.…”

“…The graph analogies used in Refs. [35][36][37][38] describe multipartite entanglement generations with a mixture of probabilistic photon pair sources (spontaneous parametric down-conversion crystals, SPDC) and single-photon sources. Since most of the processes strongly depend on photon pair sources, we can consider that the analogies are based on optical systems.…”

“…The advantage of associating graph theory with quantum information processing has been validated in many works: contextuality in the graph and hypergraph frameworks [26,27], quantum graph states [28][29][30][31], Gaussian boson sampling exploited to solve the computational complexity problems in graph theory [32][33][34], and the linkage of undirected graphs to optical setups involv-ing multiple crystals [35][36][37][38] Here, we show that the graph picture of LQNs is also beneficial for computing the entanglement of no-bunching states in LQNs. Moreover, by showing that the graph structure is closely related to the entanglement property of the corresponding LQNs, we can reversely propose a protocol for designing optimal LQNs that generate specific N -partite entangled states.…”

Quantum Hypergraph States in Noisy Quantum Channels

“…To go beyond graph states, while maintaining the important connection to graphs, this concept is generalized to quantum hypergraph states [10,11]. The hypergraph states have been widely applied to different problems of quantum information and computation, such as, error correction [12][13][14] and quantum blockchain [15], measurement based quantum computation [16][17][18][19], study of quantum entanglement [20][21][22][23][24][25][26], continuous variable quanutm information [27], quantum optics [28,29], and neural network [30]. The size of the Hilbert space for a graph or a hypergraph state scales exponentially with the number of qubits.…”