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Quantum Hitting Time according to a given distribution

P. Boito, G. M. Del Corso

TL;DR

This paper analyzes quantum hitting time for discrete-time Szegedy walks, establishing a quadratic speedup for time-reversible Markov chains and extending the notion to general start distributions beyond the stationary one. It presents a self-contained proof that $QH_{\pi,M} = O(\sqrt{H_{\pi,M}})$ under reversibility and develops a generalized framework for arbitrary starting distributions with corresponding bounds, supported by extensive numerical experiments. The results illuminate how quantum diffusion can outperform classical hitting times in network search tasks and offer insights into distribution-sensitive quantum diffusion as a tool for graph analysis and centrality. Overall, the work broadens the applicability of quantum hitting time and provides practical guidance for leveraging quantum walks in search and network tasks.

Abstract

In this work we focus on the notion of quantum hitting time for discrete-time Szegedy quantum walks, compared to its classical counterpart. Under suitable hypotheses, quantum hitting time is known to be of the order of the square root of classical hitting time: this quadratic speedup is a remarkable example of the computational advantages associated with quantum approaches. Our purpose here is twofold. On one hand, we provide a detailed proof of quadratic speedup for time-reversible walks within the Szegedy framework, in a language that should be familiar to the linear algebra community. Moreover, we explore the use of a general distribution in place of the stationary distribution in the definition of quantum hitting time, through theoretical considerations and numerical experiments.

Quantum Hitting Time according to a given distribution

TL;DR

This paper analyzes quantum hitting time for discrete-time Szegedy walks, establishing a quadratic speedup for time-reversible Markov chains and extending the notion to general start distributions beyond the stationary one. It presents a self-contained proof that under reversibility and develops a generalized framework for arbitrary starting distributions with corresponding bounds, supported by extensive numerical experiments. The results illuminate how quantum diffusion can outperform classical hitting times in network search tasks and offer insights into distribution-sensitive quantum diffusion as a tool for graph analysis and centrality. Overall, the work broadens the applicability of quantum hitting time and provides practical guidance for leveraging quantum walks in search and network tasks.

Abstract

In this work we focus on the notion of quantum hitting time for discrete-time Szegedy quantum walks, compared to its classical counterpart. Under suitable hypotheses, quantum hitting time is known to be of the order of the square root of classical hitting time: this quadratic speedup is a remarkable example of the computational advantages associated with quantum approaches. Our purpose here is twofold. On one hand, we provide a detailed proof of quadratic speedup for time-reversible walks within the Szegedy framework, in a language that should be familiar to the linear algebra community. Moreover, we explore the use of a general distribution in place of the stationary distribution in the definition of quantum hitting time, through theoretical considerations and numerical experiments.
Paper Structure (12 sections, 5 theorems, 92 equations, 4 figures, 4 tables)

This paper contains 12 sections, 5 theorems, 92 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Classical random walks
  4. Quantum walks
  5. Quantum hitting time
  6. Quantum hitting time for symmetric walks
  7. Quantum hitting time for time-reversible walks
  8. Modified walk operator
  9. Quadratic speedup for reversible walks
  10. Quantum hitting time for a general distribution
  11. Numerical experiments
  12. Conclusions

Key Result

Theorem 2.3

With the notation introduced above, let $\theta_1,\ldots,\theta_{\ell}\in(0,\frac{\pi}{2})$ be (possibly repeated) angles such that the singular values of $D$ belonging to the open interval $(0,1)$ can be written as $\cos(\theta_k)$, $k=1,\ldots,\ell$. Let $u_k,v_k$, for $k=1,\ldots,\ell$, be the as and the (non-normalized) associated eigenvectors are Moreover:

Figures (4)

  • Figure 1: Logarithmic plot comparing the dependence on the graph size $n$ of mean quantum hitting time, the $MSH$ and mean hitting-time for Barbell graphs of increasing size.
  • Figure 2: Plot of quantum and square root of classical hitting times for a barbell graph with $90$ nodes and stationary distribution. It is interesting to note how quantum hitting time distinguishes more clearly among nodes in the "bar" (i.e., nodes indexed from $30$ to $60$).
  • Figure 3: Plot of quantum and classical hitting times for a barbell graph with $180$ nodes and Dirac-like distribution centered at the first node. Here quantum and square root of classical hitting times display a similar behavior.
  • Figure 4: Plot of quantum hitting time and square root of classical hitting time for a random-8-regular graph with $100$ nodes and Dirac-like distribution centered at the first node (labeled as node $0$). Note that the 8 nodes adjacent to node $0$ have have a remarkably lower quantum hitting time, whereas the difference is much less marked for the square root of the classical hitting time.

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2.3: Spectral Theorem
  • Remark 4
  • Definition 3.1
  • Definition 3.2
  • Lemma 4.1
  • ...and 6 more