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A quantum walk inspired model for distributed computing on arbitrary graphs

Mathieu Roget, Giuseppe Di Molfetta

TL;DR

The paper presents a quantum-walk–inspired framework for distributed quantum computation on arbitrary graphs, mapping walk amplitudes to edge qubits and employing a two-state coin with edge polarity. It shows two practical distributed protocols for reproducing the walk dynamics under all-to-all and cycle node-connectivity regimes, with respective communication costs and circuit depths. The approach enables edge- and node-searching applications, achieving quadratic speedups on several graph families (grid, hypercube, complete, and scale-free), demonstrated through rigorous analysis and extensive numerical experiments. This work bridges quantum walk research and distributed quantum computation, offering scalable, locality-preserving methods for quantum search and highlighting the practical trade-offs between connectivity and communication overhead.

Abstract

A discrete time quantum walk is known to be the single-particle sector of a quantum cellular automaton. For a long time, these models have interested the community for their nice properties such as locality or translation invariance. This work introduces a model of distributed computation for arbitrary graphs inspired by quantum cellular automata. As a by-product, we show how this model can reproduce the dynamic of a quantum walk on graphs. In this context, we investigate the communication cost for two interaction schemes. Finally, we explain how this particular quantum walk can be applied to solve the search problem and present numerical results on different types of topologies.

A quantum walk inspired model for distributed computing on arbitrary graphs

TL;DR

The paper presents a quantum-walk–inspired framework for distributed quantum computation on arbitrary graphs, mapping walk amplitudes to edge qubits and employing a two-state coin with edge polarity. It shows two practical distributed protocols for reproducing the walk dynamics under all-to-all and cycle node-connectivity regimes, with respective communication costs and circuit depths. The approach enables edge- and node-searching applications, achieving quadratic speedups on several graph families (grid, hypercube, complete, and scale-free), demonstrated through rigorous analysis and extensive numerical experiments. This work bridges quantum walk research and distributed quantum computation, offering scalable, locality-preserving methods for quantum search and highlighting the practical trade-offs between connectivity and communication overhead.

Abstract

A discrete time quantum walk is known to be the single-particle sector of a quantum cellular automaton. For a long time, these models have interested the community for their nice properties such as locality or translation invariance. This work introduces a model of distributed computation for arbitrary graphs inspired by quantum cellular automata. As a by-product, we show how this model can reproduce the dynamic of a quantum walk on graphs. In this context, we investigate the communication cost for two interaction schemes. Finally, we explain how this particular quantum walk can be applied to solve the search problem and present numerical results on different types of topologies.

Paper Structure

This paper contains 37 sections, 7 theorems, 47 equations, 11 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Contribution
  2. Model of Quantum Walk on graphs
  3. The model
  4. Polarity : Why and How
  5. Relation with other models
  6. Flip-flop quantum walk
  7. Szegedy quantum walk
  8. What about searching ?
  9. Conclusion
  10. Distributed implementation
  11. Model of distributed quantum computation
  12. Edge register
  13. Node register
  14. Reproduction of the quantum walk's dynamic
  15. Edges Register
  16. ...and 22 more sections

Key Result

Theorem 1

Let $U$ be a $2\times 2$ unitary operator (a one qubit gate). It holds that Furthermore, $\Lambda(U)$ is realized by the following circuit: @C=1em @R=.7em 1 U 1 -1 In this circuit, the CNOT applications are forcing the controlled application of $U$ to be applied only on the basis states $\ket{01}$ and $\ket{10}$.

Figures (11)

  • Figure 1: The edge $(u,v)$ and how are placed the amplitudes for the two possible polarities.
  • Figure 2: Example of a walk on a path of size 3.
  • Figure 3: A graphical representation of the model of distributed computation. Show how the qubits are positioned and connected for a given graph. Two node connectivity are considered: all-to-all and cycle.
  • Figure 4: Application of a MCT (multi-controlled X gate) on an edge controlled by the whole node register via a CNOT gate and an anscillary qubit. $Q_0,q_1$ are the qubits of the node register, $q_2$ is the anscillary qubit and $a_i$ is the targeted edge qubit. Both circuits are equivalent.
  • Figure 5: Circuit of one step of the quantum walk for the path graph $u-v-w$. The circuit applies successively the oracle on $(u,v)$, the coin, $T$, $D$, $T^{-1}$.
  • ...and 6 more figures

Theorems & Definitions (17)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 2
  • Theorem 3: Batched application
  • proof
  • Theorem 4
  • proof
  • ...and 7 more