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Unifying quantum spatial search, state transfer and uniform sampling on graphs: simple and exact

Qingwen Wang, Ying Jiang, Lvzhou Li

TL;DR

The paper develops a universal, succinct framework of alternating quantum walks that unifies quantum spatial search, perfect state transfer, and exact uniform sampling on graphs whose Laplacian eigenvalues are all integers. By organizing the spectral data into a depth-based hierarchy and iteratively transferring amplitude through a chain of subspaces, it proves that the target states can be reached with total time $O\left(2^{d_L}\sqrt{N}\right)$, where $d_L$ is the depth of the eigenvalue set. For vertex-transitive graphs with integer eigenvalues, the framework yields deterministic quantum spatial search, exact uniform sampling, and PST in $O(\sqrt{N})$ time, with a specialized treatment completing the case of complete bipartite graphs. The results provide a universal method that improves and unifies prior work, reduces reliance on case-by-case spectral analysis, and has potential implications for derandomization and broader graph problems.

Abstract

This article presents a novel and succinct algorithmic framework via alternating quantum walks, unifying quantum spatial search, state transfer and uniform sampling on a large class of graphs. Using the framework, we can achieve exact uniform sampling over all vertices and perfect state transfer between any two vertices, provided that eigenvalues of Laplacian matrix of the graph are all integers. Furthermore, if the graph is vertex-transitive as well, then we can achieve deterministic quantum spatial search that finds a marked vertex with certainty. In contrast, existing quantum search algorithms generally has a certain probability of failure. Even if the graph is not vertex-transitive, such as the complete bipartite graph, we can still adjust the algorithmic framework to obtain deterministic spatial search, which thus shows the flexibility of it. Besides unifying and improving plenty of previous results, our work provides new results on more graphs. The approach is easy to use since it has a succinct formalism that depends only on the depth of the Laplacian eigenvalue set of the graph, and may shed light on the solution of more problems related to graphs.

Unifying quantum spatial search, state transfer and uniform sampling on graphs: simple and exact

TL;DR

The paper develops a universal, succinct framework of alternating quantum walks that unifies quantum spatial search, perfect state transfer, and exact uniform sampling on graphs whose Laplacian eigenvalues are all integers. By organizing the spectral data into a depth-based hierarchy and iteratively transferring amplitude through a chain of subspaces, it proves that the target states can be reached with total time , where is the depth of the eigenvalue set. For vertex-transitive graphs with integer eigenvalues, the framework yields deterministic quantum spatial search, exact uniform sampling, and PST in time, with a specialized treatment completing the case of complete bipartite graphs. The results provide a universal method that improves and unifies prior work, reduces reliance on case-by-case spectral analysis, and has potential implications for derandomization and broader graph problems.

Abstract

This article presents a novel and succinct algorithmic framework via alternating quantum walks, unifying quantum spatial search, state transfer and uniform sampling on a large class of graphs. Using the framework, we can achieve exact uniform sampling over all vertices and perfect state transfer between any two vertices, provided that eigenvalues of Laplacian matrix of the graph are all integers. Furthermore, if the graph is vertex-transitive as well, then we can achieve deterministic quantum spatial search that finds a marked vertex with certainty. In contrast, existing quantum search algorithms generally has a certain probability of failure. Even if the graph is not vertex-transitive, such as the complete bipartite graph, we can still adjust the algorithmic framework to obtain deterministic spatial search, which thus shows the flexibility of it. Besides unifying and improving plenty of previous results, our work provides new results on more graphs. The approach is easy to use since it has a succinct formalism that depends only on the depth of the Laplacian eigenvalue set of the graph, and may shed light on the solution of more problems related to graphs.
Paper Structure (18 sections, 13 theorems, 71 equations, 2 figures, 2 tables)

This paper contains 18 sections, 13 theorems, 71 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Quantum spatial search
  3. Derandomization
  4. State transfer
  5. Uniform sampling
  6. Alternating quantum walks
  7. Graph
  8. Our contribution
  9. Technical overview
  10. Related work
  11. Paper organization
  12. Preliminaries
  13. Algorithmic framework
  14. Applications to specific graphs
  15. Conclusion and Discussion
  16. ...and 3 more sections

Key Result

Theorem 1

Let $G=(V,E)$ be a graph with $N$ vertices whose Laplacian matrix $L$ has only integer eigenvalues and $|s\rangle=\frac{1}{\sqrt{N}}\sum_{v\in V} |v \rangle$. Given any $m\in V$, there is an integer $p \in O(2^{{d_L}}\sqrt{N})$ and real numbers $\gamma, \theta_j,t_j\in [0,2\pi)$$(j\in \{1,2,\dots,p where ${d_L}$ is the depth of the eigenvalue set of $L$ and will be illustrated in Definition de1.

Figures (2)

  • Figure 1: The process of computing $d_M$, $\Lambda_0$, $|w_0\rangle$ and $\Lambda_k$, $\overline{\Lambda}_{k}$, $|w_k\rangle$, $|\overline{w}_{k}\rangle$ for $k\in \{1,\dots,d_M\}$.
  • Figure 2: The quantum circuit of $A$ in Theorem \ref{['app_th2']}

Theorems & Definitions (25)

  • Theorem 1
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2: RN7
  • ...and 15 more