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Quantum State Diffusion on a Graph

John C Vining, Howard A. Blair

TL;DR

Diffusion of quantum states on general graphs is challenging under traditional single-walker or globally informed models. The authors propose a distributed, edge-centric multi-walker framework where each vertex hosts a walker, coin preparation uses $W$-states, and diffusion is implemented via fixed local channels and CCSWAP-based edge interactions within a Watrous-inspired quantum cellular automaton. Major contributions include a formal vertex/edge labeling scheme, explicit $U_W^{v}$ operators for $W$-state coin preparation, an edge-selection distribution $p_{u,v}$ with independence constraints, and demonstrations of both undirected and directed diffusion variants. The approach offers structure-agnostic, localized diffusion on arbitrary graphs at the cost of increased qubit overhead, with future work targeting qubit reduction and integration with cluster-state techniques.

Abstract

Quantum walks have frequently envisioned the behavior of a quantum state traversing a classically defined, generally finite, graph structure. While this approach has already generated significant results, it imposes a strong assumption: all nodes where the walker is not positioned are quiescent. This paper will examine some mathematical structures that underlie state diffusion on arbitrary graphs, that is the circulation of states within a graph. We will seek to frame the multi-walker problem as a finite quantum cellular automaton. Every vertex holds a walker at all times. The walkers will never collide and at each time step their positions update non-deterministically by a quantum swap of walkers at opposite ends of a randomly chosen edge. The update is accomplished by a unitary transformation of the position of a walker to a superposition of all such possible swaps and then performing a quantum measurement on the superposition of possible swaps. This behavior generates strong entanglement between vertex states which provides a path toward developing local actions producing diffusion throughout the graph without depending on the specific structure of the graph through blind computation.

Quantum State Diffusion on a Graph

TL;DR

Diffusion of quantum states on general graphs is challenging under traditional single-walker or globally informed models. The authors propose a distributed, edge-centric multi-walker framework where each vertex hosts a walker, coin preparation uses -states, and diffusion is implemented via fixed local channels and CCSWAP-based edge interactions within a Watrous-inspired quantum cellular automaton. Major contributions include a formal vertex/edge labeling scheme, explicit operators for -state coin preparation, an edge-selection distribution with independence constraints, and demonstrations of both undirected and directed diffusion variants. The approach offers structure-agnostic, localized diffusion on arbitrary graphs at the cost of increased qubit overhead, with future work targeting qubit reduction and integration with cluster-state techniques.

Abstract

Quantum walks have frequently envisioned the behavior of a quantum state traversing a classically defined, generally finite, graph structure. While this approach has already generated significant results, it imposes a strong assumption: all nodes where the walker is not positioned are quiescent. This paper will examine some mathematical structures that underlie state diffusion on arbitrary graphs, that is the circulation of states within a graph. We will seek to frame the multi-walker problem as a finite quantum cellular automaton. Every vertex holds a walker at all times. The walkers will never collide and at each time step their positions update non-deterministically by a quantum swap of walkers at opposite ends of a randomly chosen edge. The update is accomplished by a unitary transformation of the position of a walker to a superposition of all such possible swaps and then performing a quantum measurement on the superposition of possible swaps. This behavior generates strong entanglement between vertex states which provides a path toward developing local actions producing diffusion throughout the graph without depending on the specific structure of the graph through blind computation.
Paper Structure (16 sections, 1 theorem, 36 equations, 2 figures)

This paper contains 16 sections, 1 theorem, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Graph Walkers
  3. Traditional Quantum Graph Walker
  4. Alternative Quantum Graph Walker
  5. State Labeling
  6. State Preparation
  7. Transformations
  8. Leveraging the W-States
  9. Edge selection probability distribution
  10. Advantages
  11. Example
  12. Example functions
  13. Watrous Implementation
  14. Example
  15. Directed Diffusion
  16. ...and 1 more sections

Key Result

Proposition 1

For all edges $\left(u,v \right) \in E$, the state of $\ket{1}_{u,v,1} \ket{1}_{u,v,2} \implies \forall \left(j,k\right) \in E,$ incident to $\left(u,v\right)$, then

Figures (2)

  • Figure 1: Quantum Registry of WQCA block
  • Figure 2: 15 node WQCA with node interacting edges

Theorems & Definitions (6)

  • Definition 1: $n$-W-Statedu_three_2000
  • Example 1
  • Definition 2: Bell Paired
  • Proposition 1: Independent Edges
  • proof
  • Remark 1