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Quantum Search on Bipartite Multigraphs

Gustavo Alves Bezerra, Andris Ambainis, Renato Portugal

TL;DR

This work addresses quantum spatial search on graphs by extending Szegedy's quantum walk to arbitrary bipartite graphs and introducing Adapted Szegedy Walks, enabling efficient search on bipartite multigraphs with a quadratic speedup of $~O(\tilde{\sqrt{HT}})$ queries. It also generalizes the staggered quantum-walk framework to 2-tessellable graphs, enabling quantum search on line graphs of bipartite multigraphs via a corresponding discriminant framework and QDB/CDB equivalence. The combined approach broadens the applicability of quantum search beyond balanced bipartite graphs, preserving the $\tilde{O}(\sqrt{HT})$ query complexity and leveraging AGJK's algorithm as a key subroutine. The results offer a unifying view of quantum-walk-based search across a wider class of graph families, with potential implications for scalable quantum search in complex network structures.

Abstract

Quantum walks provide a powerful framework for achieving algorithmic speedup in quantum computing. This paper presents a quantum search algorithm for 2-tessellable graphs, a generalization of bipartite graphs, achieving a quadratic speedup over classical Markov chain-based search methods. Our approach employs an adapted version of the Szegedy quantum walk model (adapted SzQW), which takes place on bipartite graphs, and an adapted version of Staggered Quantum Walks (Adapted StQW), which takes place on 2-tessellable graphs, with the goal of efficiently finding a marked vertex by querying an oracle. The Ambainis, Gilyén, Jeffery, and Kokainis' algorithm (AGJK), which provides a quadratic speedup on balanced bipartite graphs, is used as a subroutine in our algorithm. Our approach generalizes existing quantum walk techniques and offers a quadratic speedup in the number of queries needed, demonstrating the utility of our adapted quantum walk models in a broader class of graphs.

Quantum Search on Bipartite Multigraphs

TL;DR

This work addresses quantum spatial search on graphs by extending Szegedy's quantum walk to arbitrary bipartite graphs and introducing Adapted Szegedy Walks, enabling efficient search on bipartite multigraphs with a quadratic speedup of queries. It also generalizes the staggered quantum-walk framework to 2-tessellable graphs, enabling quantum search on line graphs of bipartite multigraphs via a corresponding discriminant framework and QDB/CDB equivalence. The combined approach broadens the applicability of quantum search beyond balanced bipartite graphs, preserving the query complexity and leveraging AGJK's algorithm as a key subroutine. The results offer a unifying view of quantum-walk-based search across a wider class of graph families, with potential implications for scalable quantum search in complex network structures.

Abstract

Quantum walks provide a powerful framework for achieving algorithmic speedup in quantum computing. This paper presents a quantum search algorithm for 2-tessellable graphs, a generalization of bipartite graphs, achieving a quadratic speedup over classical Markov chain-based search methods. Our approach employs an adapted version of the Szegedy quantum walk model (adapted SzQW), which takes place on bipartite graphs, and an adapted version of Staggered Quantum Walks (Adapted StQW), which takes place on 2-tessellable graphs, with the goal of efficiently finding a marked vertex by querying an oracle. The Ambainis, Gilyén, Jeffery, and Kokainis' algorithm (AGJK), which provides a quadratic speedup on balanced bipartite graphs, is used as a subroutine in our algorithm. Our approach generalizes existing quantum walk techniques and offers a quadratic speedup in the number of queries needed, demonstrating the utility of our adapted quantum walk models in a broader class of graphs.

Paper Structure

This paper contains 12 sections, 8 theorems, 80 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete-time Markov chain
  4. Szegedy's quantum walk
  5. Interpolated Szegedy quantum walk
  6. Interpolated quantum search
  7. Staggered quantum walk
  8. Adapting Szegedy Quantum Walks
  9. Adapted Szegedy's quantum walk
  10. Interpolated adapted Szegedy's quantum walk
  11. Searching with Staggered Quantum Walks
  12. Final Remarks

Key Result

Theorem 1

If $P$ is a reversible Markov chain and $p_M \leq 1/9$, Alg. alg:ambainis finds a marked vertex with success probability $\Omega(1)$ in steps where $HT$ is the hitting time of $P$, $\mathbb{S}$ is the setup cost of step step:setup, and $\mathbb{W}$ is the cost of invoking $W(r)$ (which includes the cost of update operation and the cost of querying the oracle).

Figures (6)

  • Figure 1: Markov chain and its associated bipartite graph.
  • Figure 2: Markov chain with a sink and its associated bipartite digraph.
  • Figure 3: Line graphs and tessellations of previous bipartite graphs. Cliques $\alpha_u$ are induced by the red solid edges and vertices, and cliques $\beta_v$ are induced by blue dashed edges and vertices.
  • Figure 4: A bipartite multigraph $\mathcal{G}$ and its line graph $L(\mathcal{G})$. We relabelled $uv0 \to uv$ and $uv1 \to uv'$.
  • Figure 5: Interpretations of interpolated adapted SzQW s. Vertex $4 \in V_1$ is marked. In Fig. \ref{['fig:marked-bip-lsz']}, vertex $4'$ was added by the disjoint union $V_2 \sqcup M$. In Fig. \ref{['fig:new-markov-chain']}, it is depicted the new Markov chain $\mathcal{P}(1) = P_2' P_1'$.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 2
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Corollary 6
  • proof
  • ...and 4 more