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Deterministic Search on Complete Bipartite Graphs by Continuous Time Quantum Walk

Honghong Lin, Yun Shang

TL;DR

A deterministic search algorithm on complete bipartite graphs with optimal runtime scaling with optimal runtime scaling is presented and a quantum counting algorithm is constructed based on the spectrum structure of the search operator.

Abstract

This paper presents a deterministic search algorithm on complete bipartite graphs. Our algorithm adopts the simple form of alternating iterations of an oracle and a continuous-time quantum walk operator, which is a generalization of Grover's search algorithm. We address the most general case of multiple marked states, so there is a problem of estimating the number of marked states. To this end, we construct a quantum counting algorithm based on the spectrum structure of the search operator. To implement the continuous-time quantum walk operator, we perform Hamiltonian simulation in the quantum circuit model. We achieve simulation in constant time, that is, the complexity of the quantum circuit does not scale with the evolution time.

Deterministic Search on Complete Bipartite Graphs by Continuous Time Quantum Walk

TL;DR

A deterministic search algorithm on complete bipartite graphs with optimal runtime scaling with optimal runtime scaling is presented and a quantum counting algorithm is constructed based on the spectrum structure of the search operator.

Abstract

This paper presents a deterministic search algorithm on complete bipartite graphs. Our algorithm adopts the simple form of alternating iterations of an oracle and a continuous-time quantum walk operator, which is a generalization of Grover's search algorithm. We address the most general case of multiple marked states, so there is a problem of estimating the number of marked states. To this end, we construct a quantum counting algorithm based on the spectrum structure of the search operator. To implement the continuous-time quantum walk operator, we perform Hamiltonian simulation in the quantum circuit model. We achieve simulation in constant time, that is, the complexity of the quantum circuit does not scale with the evolution time.
Paper Structure (5 sections, 3 theorems, 47 equations, 4 figures, 1 algorithm)

This paper contains 5 sections, 3 theorems, 47 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Deterministic Spatial Search Algorithm
  3. Quantum Counting for Spatial Search
  4. Circuit Implementation
  5. Conclusion and Discussion

Key Result

Theorem 1

Let $K_{m,n}$ be a complete bipartite graph with $k$ marked vertices on the order-$n$ part and $A$ be its adjacency matrix. For any $l \ge \frac{\pi}{4}\sqrt{\frac{n}{k}}-\frac{1}{2}$ and $t = \frac{2}{\sqrt{mn}}\arcsin[\sqrt{\frac{n}{k}}\sin(\frac{\pi}{2(2l+1)})]$, starting from the state $|s\rangl

Figures (4)

  • Figure 1: Quantum circuit implementation for the oracle in (\ref{['local_oracle']}).
  • Figure 2: The possible phase set returned by Algorithm 1 for $M = 16$. The best estimation for an angle $\theta$ is either $\theta^+$ or $\theta^-$.
  • Figure 3: The eigenphase distribution of $U(t_0)$ on the unit circle.
  • Figure 4: Quantum circuit for implementation of ${\rm e}^{-i\widetilde{A}t}$.

Theorems & Definitions (5)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3: Quantum Counting on the Complete Bipartite Graph
  • proof