Two-particle scattering on general graphs

TL;DR

This work starts to develop a full theory of multi-particle scattering on graphs and gives initial applications to build multi-particle gadgets with different properties.

Abstract

Quantum walks in general graphs, or more specifically scattering on graphs, encompass enough complexity to perform universal quantum computation. Any given quantum circuit can be broken down into single- and two-qubit gates, which can then be translated into subgraphs -- gadgets -- that implement such unitaries on the logical qubits, simulated by particles traveling along a sparse graph. In this work, we start to develop a full theory of multi-particle scattering on graphs and give initial applications to build multi-particle gadgets with different properties.

Figures (17)

• Figure 1: On the left we represent the labels for the vertices of the finite graph G. On the right we label the vertices on each rail. Note that the vertices connecting to rails have two labels, i.e., $n$ is equivalent to $(1,n)$ for $n\in\{1,2,\cdots,N\}$.
• Figure 2: From left to right: The graph AC(4), the single particle transmission coefficient $|t|^2$ and the single particle reflection coefficient $|r|^2$ as a function of the energy of the incident particle.
• Figure 3: Distinguishable particle elastic reflection and transmission probabilities, $|r_{\chi\rightarrow\chi}^D|^2$ and $|t_{\chi\rightarrow\chi}^D|^2$ and boson elastic reflection and transmission probabilities, $|r_{\chi\rightarrow\chi}^B|^2$ and $|t_{\chi\rightarrow\chi}^B|^2$ in the presence of an evanescent bound state, for the graph AC(4). Dashed line: Corresponding single-particle reflection/transmission probability.
• Figure 4: Elastic transmission probabilities for distinguishable particles and bosons in the presence of a confined bound state for the graph AC(4). Dashed line: Single-particle transmission probability.
• Figure 5: a) Sum of absolute value squared of transmission and reflection coefficients in the presence of a evanescent bound state for the graph AC(4). b) Sum of ejection probabilities of the evanescent bound state, when the scattered particle incomes in rail $1$ and scatters to rail $1$ or $2$ for the graph AC(4).

Theorems & Definitions (5)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Lemma 1
• proof