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Quantum search by continuous-time quantum walk on t-designs

Pedro H. G. Lugão, Renato Portugal

TL;DR

This paper addresses the problem of quantum search via continuous-time quantum walks on combinatorial $t$-designs with multiple marked elements. It leverages a detailed spectral analysis of symmetric $t$-designs and their incidence bipartite graphs to derive closed-form eigenstructures, enabling an analytic evaluation of the CTQW-based search efficiency. The main contributions show that for certain symmetric $t$-designs the search runs in $O(\sqrt{n})$ time regardless of the number of marked elements, with the success probability generally $o(1)$ but approaching 1 in specific configurations; it also provides a comprehensive treatment of single and multi-mark scenarios, including explicit expressions for optimal time and success probability and generalizations to $m$ marked vertices in a single part. These results advance understanding of quantum-walk-based search on structured bipartite graphs and offer a framework for extending analytic time-complexity results to broader combinatorial designs, potentially informing practical quantum search algorithms on symmetric incidence structures.

Abstract

This work examines the time complexity of quantum search algorithms on combinatorial $t$-designs with multiple marked elements using the continuous-time quantum walk. Through a detailed exploration of $t$-designs and their incidence matrices, we identify a subset of bipartite graphs that are conducive to success compared to random-walk-based search algorithms. These graphs have adjacency matrices with eigenvalues and eigenvectors that can be determined algebraically and are also suitable for analysis in the multiple-marked vertex scenario. We show that the continuous-time quantum walk on certain symmetric $t$-designs achieves an optimal running time of $O(\sqrt{n})$, where $n$ is the number of points and blocks, even when accounting for an arbitrary number of marked elements. Upon examining two primary configurations of marked elements distributions, we observe that the success probability is consistently $o(1)$, but it approaches 1 asymptotically in certain scenarios.

Quantum search by continuous-time quantum walk on t-designs

TL;DR

This paper addresses the problem of quantum search via continuous-time quantum walks on combinatorial -designs with multiple marked elements. It leverages a detailed spectral analysis of symmetric -designs and their incidence bipartite graphs to derive closed-form eigenstructures, enabling an analytic evaluation of the CTQW-based search efficiency. The main contributions show that for certain symmetric -designs the search runs in time regardless of the number of marked elements, with the success probability generally but approaching 1 in specific configurations; it also provides a comprehensive treatment of single and multi-mark scenarios, including explicit expressions for optimal time and success probability and generalizations to marked vertices in a single part. These results advance understanding of quantum-walk-based search on structured bipartite graphs and offer a framework for extending analytic time-complexity results to broader combinatorial designs, potentially informing practical quantum search algorithms on symmetric incidence structures.

Abstract

This work examines the time complexity of quantum search algorithms on combinatorial -designs with multiple marked elements using the continuous-time quantum walk. Through a detailed exploration of -designs and their incidence matrices, we identify a subset of bipartite graphs that are conducive to success compared to random-walk-based search algorithms. These graphs have adjacency matrices with eigenvalues and eigenvectors that can be determined algebraically and are also suitable for analysis in the multiple-marked vertex scenario. We show that the continuous-time quantum walk on certain symmetric -designs achieves an optimal running time of , where is the number of points and blocks, even when accounting for an arbitrary number of marked elements. Upon examining two primary configurations of marked elements distributions, we observe that the success probability is consistently , but it approaches 1 asymptotically in certain scenarios.
Paper Structure (6 sections, 1 theorem, 50 equations)

This paper contains 6 sections, 1 theorem, 50 equations.

Table of Contents

  1. Introduction
  2. Spectral decomposition of symmetric t-designs
  3. Quantum search
  4. Single marked vertex
  5. Multiple marked vertices
  6. Final remarks

Key Result

Theorem 1

Let $\mathcal{D}$ be a symmetric $t$-design (i.e., $v=b$ and $r=k$). Then, the eigenvalues of the adjacency matrix of $\mathcal{D}$ are $\{\pm k,\pm \sqrt{r-\lambda_2}\}$ and the spectral idempotents are where $M = \frac{1}{v\sqrt{r-\lambda_2}} \left(-kJ + vN\right)$ and $J$ is a matrix of ones.

Theorems & Definitions (3)

  • Definition 1: $t$-design
  • Theorem 1
  • proof