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Multi-target quantum walk search on Johnson graph

Pulak Ranjan Giri

TL;DR

This work analyzes multi-target spatial search on Johnson graphs $J(n,k)$ via discrete-time coined quantum walks with a lackadaisical extension (self-loop of weight $l$) and compares four coin operators. The central contribution is the modified coin $\mathcal{C}_g$, which uses a Grover diffusion focused on the edge space of target vertices, yielding consistently high success probabilities across a range of $k$ (including $k\ge3$) and target counts $M$. Across complete, triangular, and higher-order Johnson graphs, $\mathcal{C}_g$ outperforms $\mathcal{C}_{grov}$, $\mathcal{C}_l$, and $\mathcal{C}_{skw}$ in multi-target scenarios, while ${\mathcal{C}_{grov}}$ aligns with analytic single-target results but falters for multiple targets as $k$ grows. The findings support using $\mathcal{C}_g$ for robust multi-target quantum search on Johnson graphs and motivate future analytic derivations of time complexity and success probabilities.

Abstract

The discrete-time quantum walk on the Johnson graph $J(n,k)$ is a useful tool for performing target vertex searches with high success probability. This graph is defined by $n$ distinct elements, with vertices being all the \(\binom{n}{k}\) $k$-element subsets and two vertices are connected by an edge if they differ exactly by one element. However, most works in the literature focus solely on the search for a single target vertex on the Johnson graph. In this article, we utilize lackadaisical quantum walk--a form of discrete-time coined quantum walk with a wighted self-loop at each vertex of the graph--along with our recently proposed modified coin operator, $\mathcal{C}_g$, to find multiple target vertices on the Johnson graph $J(n,k)$ for various values of $k$. Additionally, a comparison based on the numerical analysis of the performance of the $\mathcal{C}_g$ coin operator in searching for multiple target vertices on the Johnson graph, against various other frequently used coin operators by the discrete-time quantum walk search algorithms, shows that only $\mathcal{C}_g$ coin can search for multiple target vertices with a very high success probability in all the scenarios discussed in this article, outperforming other widely used coin operators in the literature.

Multi-target quantum walk search on Johnson graph

TL;DR

This work analyzes multi-target spatial search on Johnson graphs via discrete-time coined quantum walks with a lackadaisical extension (self-loop of weight ) and compares four coin operators. The central contribution is the modified coin , which uses a Grover diffusion focused on the edge space of target vertices, yielding consistently high success probabilities across a range of (including ) and target counts . Across complete, triangular, and higher-order Johnson graphs, outperforms , , and in multi-target scenarios, while aligns with analytic single-target results but falters for multiple targets as grows. The findings support using for robust multi-target quantum search on Johnson graphs and motivate future analytic derivations of time complexity and success probabilities.

Abstract

The discrete-time quantum walk on the Johnson graph is a useful tool for performing target vertex searches with high success probability. This graph is defined by distinct elements, with vertices being all the -element subsets and two vertices are connected by an edge if they differ exactly by one element. However, most works in the literature focus solely on the search for a single target vertex on the Johnson graph. In this article, we utilize lackadaisical quantum walk--a form of discrete-time coined quantum walk with a wighted self-loop at each vertex of the graph--along with our recently proposed modified coin operator, , to find multiple target vertices on the Johnson graph for various values of . Additionally, a comparison based on the numerical analysis of the performance of the coin operator in searching for multiple target vertices on the Johnson graph, against various other frequently used coin operators by the discrete-time quantum walk search algorithms, shows that only coin can search for multiple target vertices with a very high success probability in all the scenarios discussed in this article, outperforming other widely used coin operators in the literature.

Paper Structure

This paper contains 6 sections, 13 equations, 5 figures.

Table of Contents

  1. Introduction
  2. QW search on Johnson graph
  3. Complete graph
  4. Triangular graph
  5. Johnson graph $J(n,k\ge 3)$
  6. Conclusions

Figures (5)

  • Figure 1: Johnson graph $J(5,1)$(left) and $J(4,2)$(right).
  • Figure 2: (a) Variation of success probability and (b) running time as a function of the self-loop weight $l$ for a Johnson graph $J(10,3)$
  • Figure 3: Multi-target quantum walk search on a complete graph with $N=300$ vertices and with (a) $\mathcal{C}_{g}$, $l=10$ (b) $\mathcal{C}_{grov}$, (c) $\mathcal{C}_{l}$, $l=1$ and (d) $\mathcal{C}_{skw}$ coins for $M=1$(blue), $3$(red), and $6$(green) targets.
  • Figure 4: Multi-target quantum walk search on a $25$-triangular graph $J(25,2)$ with $N=300$ vertices and with (a) $\mathcal{C}_{g}$, $l=1$ (b) $\mathcal{C}_{grov}$, (c) $\mathcal{C}_{l}$, $l=0.1$ and (d) $\mathcal{C}_{skw}$ coins for $M=1$(blue), $3$(red), and $6$(green) targets.
  • Figure 5: Multi-target quantum walk search on Johnson graph for $M=1$(blue), $3$(red), and $6$(green) targets. Top to bottom rows correspond to the Johnson graphs $J(13,3)$, $J(13,4)$, $J(13,5)$ and $J(13,6)$ respectively. Left to right columns correspond to $\mathcal{C}_{g}$, $l=1.0$, $\mathcal{C}_{grov}$, $\mathcal{C}_{l}$, $l=0.1$ and $\mathcal{C}_{skw}$ coins respectively.