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Quantum walks on finite and bounded infinite graphs

Chris Godsil, Steve Kirkland, Sarojini Mohapatra, Hermie Monterde, Hiranmoy Pal

Abstract

A weighted graph $G$ with countable vertex set is bounded if there is an upper bound on the maximum of the sum of absolute values of all edge weights incident to a vertex in $G$. In this paper, we prove a fundamental result on equitable partitions of bounded weighted graphs with twin subgraphs and use this fact to construct finite and bounded infinite graphs with pair and plus state transfer with the adjacency matrix as a Hamiltonian. We show that for each $k \ge 3$, (i) there are infinitely many connected unweighted graphs with maximum degree $k$ admitting pair state transfer at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$, and (ii) there are infinitely many signed graphs with exactly one negative edge weight and whose underlying unweighted graphs have maximum degree $k$ admitting plus state transfer at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$. Parallel results are proven for perfect state transfer between a plus state and a pair state, and for the existence of sedentary pair and plus states. We further prove that almost all connected unweighted finite planar graphs admit pair state transfer at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$, and almost all connected unweighted finite planar graphs can be assigned a single negative edge weight resulting in plus state transfer, or perfect state transfer between a plus state and a pair state, at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$. Analogous results are shown to hold for unweighted finite trees. Using blow-up graphs, Cayley graphs and graphs with tails, we construct new infinite families of (finite and infinite) unweighted graphs and signed graphs admitting pair or plus state transfer.

Quantum walks on finite and bounded infinite graphs

Abstract

A weighted graph with countable vertex set is bounded if there is an upper bound on the maximum of the sum of absolute values of all edge weights incident to a vertex in . In this paper, we prove a fundamental result on equitable partitions of bounded weighted graphs with twin subgraphs and use this fact to construct finite and bounded infinite graphs with pair and plus state transfer with the adjacency matrix as a Hamiltonian. We show that for each , (i) there are infinitely many connected unweighted graphs with maximum degree admitting pair state transfer at , and (ii) there are infinitely many signed graphs with exactly one negative edge weight and whose underlying unweighted graphs have maximum degree admitting plus state transfer at . Parallel results are proven for perfect state transfer between a plus state and a pair state, and for the existence of sedentary pair and plus states. We further prove that almost all connected unweighted finite planar graphs admit pair state transfer at , and almost all connected unweighted finite planar graphs can be assigned a single negative edge weight resulting in plus state transfer, or perfect state transfer between a plus state and a pair state, at . Analogous results are shown to hold for unweighted finite trees. Using blow-up graphs, Cayley graphs and graphs with tails, we construct new infinite families of (finite and infinite) unweighted graphs and signed graphs admitting pair or plus state transfer.

Paper Structure

This paper contains 13 sections, 33 theorems, 14 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Graphs
  3. Quantum walks
  4. Equitable partitions
  5. Twin subgraphs
  6. Signed graphs
  7. Pair and plus state transfer
  8. Infinite families
  9. Blow-ups
  10. Sedentariness
  11. Cayley Graphs
  12. Graphs with tails
  13. Open questions

Key Result

Theorem 1

Let $G$ be a locally finite graph. If $G$ is bounded and $\mathcal{M}$ is the maximum absolute degree in $G$, then $A(G)$ is a bounded linear operator on the Hilbert space $\ell^2(\mathbb{Z}^+)$ of square-summable functions on $V(G)$ and $\|A(G)\|\leq\mathcal{M}$. The converse holds whenever each ed

Figures (9)

  • Figure 1: A graph with $P_2$ as twin subgraphs (left) and an edge-perturbed version (right)
  • Figure 2: Perfect state transfer between $\frac{1}{\sqrt2}({\mathbf e}_1+{\mathbf e}_2)$ and $\frac{1}{\sqrt2}({\mathbf e}_4+{\mathbf e}_5)$ (left), between $\frac{1}{\sqrt2}({\mathbf e}_1-{\mathbf e}_2)$ and $\frac{1}{\sqrt2}({\mathbf e}_4-{\mathbf e}_5)$ (centre), and between $\frac{1}{\sqrt2}({\mathbf e}_1+{\mathbf e}_2)$ and $\frac{1}{\sqrt2}({\mathbf e}_4-{\mathbf e}_5)$ (right)
  • Figure 3: Signed versions of a graph with $P_2$ as twin subgraphs
  • Figure 4: Graphs having $P_3$ as twin subgraphs
  • Figure 5: $\overset{2}{\uplus}~P_3$ with PST between $\frac{1}{\sqrt2}({\mathbf e}_{(0,a)}+{\mathbf e}_{(1,a)})$ and $\frac{1}{\sqrt2}({\mathbf e}_{(0,c)}+{\mathbf e}_{(1,c)})$ (left), signed $\overset{2}{\uplus}~P_3$ with PST between $\frac{1}{\sqrt2}({\mathbf e}_{(0,a)}-{\mathbf e}_{(1,a)})$ and $\frac{1}{\sqrt2}({\mathbf e}_{(0,c)}+{\mathbf e}_{(1,c)})$ (centre), signed $\overset{2}{\uplus}~P_3$ with PST between $\frac{1}{\sqrt2}({\mathbf e}_{(0,a)}-{\mathbf e}_{(1,a)})$ and $\frac{1}{\sqrt2}({\mathbf e}_{(0,c)}-{\mathbf e}_{(1,c)})$ (right)
  • ...and 4 more figures

Theorems & Definitions (68)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 1
  • Remark 2
  • Corollary 1
  • Remark 3
  • ...and 58 more