Continuous-time quantum walks on strongly regular graphs with loops and its application to spatial search for multiple marked vertices

Abstract

The continuous-time quantum walk (CTQW) on the strongly regular graph is studied in this paper, and the exact transition probability distribution between any two vertices of the graph is provided by using the method of counting the walks between these two vertices. The CTQW is also considered on the perturbed strongly regular graph (SRG) by adding multiple loops on given vertices. Spatial search using CTQW can be regarded as a special case of CTQW on the perturbed SRG by adding loops. Combined with the approach of walk counting, the proper parameter settings and search time of spatial search for both a single marked vertex and two marked vertices are provided. The results show that both kinds of spatial search can be undertaken in $O(\sqrt{N})$ time in a degree k SRG with N vertices, by setting jump rate of CTQW to be 1 / k.

Appendix

The number of walks $L_s$ from one vertex to a non-adjacent vertex in SRG, we obtain it by using Mathematica, is

$$\begin{aligned} L_s=\frac{Q(s)}{k N (c-k)^2 \left( c-k+\lambda _1\right) \left( c-k+\lambda _2\right) }, \end{aligned}$$

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where

$$\begin{aligned} Q(s)= & {} -\,k^5 (k-c)^s+2 c k^4 (k-c)^s \nonumber \\&-\,c^2 k^3 (k-c)^s-c^2 m_1 \lambda _1^3 (k-c)^s \nonumber \\&+\,c k m_1 \lambda _1^3 (k-c)^s-c^2 m_2 \lambda _2^3 (k-c)^s \nonumber \\&+\,c k m_2 \lambda _2^3 (k-c)^s-c m_2 \lambda _1 \lambda _2^3 (k-c)^s \nonumber \\&+\,c k^3 N (k-c)^s-2 c^2 k^2 N (k-c)^s \nonumber \\&+\,c^3 k N (k-c)^s+k^4 \lambda _1 (k-c)^s \nonumber \\&-\,c k^3 \lambda _1 (k-c)^s-c k^2 M \lambda _1 (k-c)^s \nonumber \\&+\,c^2 kN \lambda _1 (k-c)^s+k^4 \lambda _2 (k-c)^s \nonumber \\&-\,c k^3 \lambda _2 (k-c)^s-c m_1 \lambda _1^3 \lambda _2 (k-c)^s \nonumber \\&-\,c k^2 N \lambda _2 (k-c)^s+c^2 k N \lambda _2 (k-c)^s \nonumber \\&-\,k^3 \lambda _1 \lambda _2 (k-c)^s+c k N \lambda _1 \lambda _2 (k-c)^s \nonumber \\&+\,c^4 k^{s+1}+c^4 m_1 \lambda _1^{s+1}-c k^3 m_1 \lambda _1^{s+1} \nonumber \\&+\,3 c^2 k^2 m_1 \lambda _1^{s+1}-3 c^3 k m_1 \lambda _1^{s+1}+c^4 m_2 \lambda _2^{s+1} \nonumber \\&-\,c k^3 m_2 \lambda _2^{s+1}+3 c^2 k^2 m_2 \lambda _2^{s+1} \nonumber \\&-\,3 c^3 k m_2 \lambda _2^{s+1}+c^3 m_2 \lambda _1 \lambda _2^{s+1} \nonumber \\&+\,c k^2 m_2 \lambda _1 \lambda _2^{s+1}-2 c^2 k m_2 \lambda _1 \lambda _2^{s+1} \nonumber \\&-\,4 c^3 k^{s+2}+6 c^2 k^{s+3}-4 c k^{s+4} \nonumber \\&+\,k^{s+5}+c^3 k^{s+1} \lambda _1-3 c^2 k^{s+2} \lambda _1 \nonumber \\&+\,3 c k^{s+3} \lambda _1-k^{s+4} \lambda _1+c^3 k^{s+1} \lambda _2 \nonumber \\&+\,c^3 m_1 \lambda _1^{s+1} \lambda _2+c k^2 m_1 \lambda _1^{s+1} \lambda _2 \nonumber \\&-\,2 c^2 k m_1 \lambda _1^{s+1} \lambda _2-3 c^2 k^{s+2} \lambda _2 \nonumber \\&+\,3 c k^{s+3} \lambda _2-k^{s+4} \lambda _2+c^2 k^{s+1} \lambda _1 \lambda _2 \nonumber \\&-\,2 c k^{s+2} \lambda _1 \lambda _2+k^{s+3} \lambda _1 \lambda _2. \end{aligned}$$

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