Quantum experiments and graphs II: Quantum interference, computation, and state generation

Abstract

We present a conceptually new approach to describe state-of-the-art photonic quantum experiments using Graph Theory. There, the quantum states are given by the coherent superpositions of perfect matchings. The crucial observation is that introducing complex weights in graphs naturally leads to quantum interference. The new viewpoint immediately leads to many interesting results, some of which we present here. Firstly, we identify a new and experimentally completely unexplored multiphoton interference phenomenon. Secondly, we find that computing the results of such experiments is #P-hard, which means it is a classically intractable problem dealing with the computation of a matrix function Permanent and its generalization Hafnian. Thirdly, we explain how a recent no-go result applies generally to linear optical quantum experiments, thus revealing important insights to quantum state generation with current photonic technology. Fourthly, we show how to describe quantum protocols such as entanglement swapping in a graphical way. The uncovered bridge between quantum experiments and Graph Theory offers a novel perspective on a widely used technology, and immediately raises many follow-up questions.

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

“…All linear optical elements such as mode shifter, are feasibly describable in hypergraphs with an internal vertex set. For further details about the graph description of linear optics, see [16]. Here we only show the corresponding initial and final hypergraphs in Fig.…”

“…Recently, an entirely different method of connecting graphs in quantum physics has been introduced, by showing that Graphs can capture essential elements of quantum optical experiments [15][16][17]. The graph-experiment connection exploits the fact that the most common sources of photonic entanglement are spontaneous parametric down-conversion (SPDC) [18], which is a nonlinear process that probabilistic creates photon pairs.…”

“…Those photon pairs are then interpreted as two vertices connected by an edge. This simple idea has been exploited in [15][16][17] to understand better the generalization of quantum states, quantum information protocols and for gaining new insights towards quantum computation.…”

“…Hypergraphs are generalizations of graphs [19,20], which have caused a significant interest in applications arXiv:2003.01910v2 [quant-ph] 16 Mar 2020 vertex edge hypergraph -H graph -G Figure 1. An example of graphs and hypergraphs.…”

Progress on Integrated Quantum Photonic Sources with Silicon

“…These authors used four paths to prepare two entangled photon‐pair sources and subsequently arbitrarily regulated these sources to construct various four‐photon graph states. By further reducing the loss and increasing the rates, we can display more complex graph state construction with more photon numbers directly on the silicon chip …”