# Vertices cannot be hidden from quantum spatial search for almost all random graphs

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## Abstract

In this paper, we show that all nodes can be found optimally for almost all random Erdős–Rényi \(\mathcal G(n,p)\) graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices, though under different conditions. The first one requires \(p=\omega (\log ^8(n)/n)\), while the second requires \(p\ge (1+\varepsilon )\log (n)/n\), where \(\varepsilon >0\). The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the \(\Vert \cdot \Vert _\infty \) norm. At the same time for \(p<(1-\varepsilon )\log (n)/n\), the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight.

## Keywords

Quantum spatial search Random graphs Continuous-time quantum walk Erdős–Rényi graphs## 1 Introduction

Quantum walk is a topic of great interest in quantum information theory [1, 2, 3]. Numerous possible applications were already discovered, including quantum spatial search [1, 4], Google algorithm [5, 6, 7] or quantum transport [8, 9]. Throughout this article, we consider quantum spatial search procedure, which is an example of an algorithm yielding a result up to quadratically faster than its classical counterpart. Since the very first paper describing it was published [1], plenty of new results have appeared in the literature. This includes the noise resistance [10], efficiency analysis [1, 4, 11, 12, 13], imperfect implementation [14] and difference in implementation [15].

Unfortunately, most of the results concern very specific graph classes like complete graphs [1, 10] or their simplex [14], and binary trees [13]. Due to some kind of ‘symmetry,’ it was not necessary to make analysis for all vertices separately (as, for example, in complete graphs or hypercubes), or at least it could be easily fixed (for example by the level in binary trees). The first big step toward the generalization into a large collection of graphs is the work of Chakraborty et al. [4], where Erdős–Rényi random graph model \(\mathcal G(n,p)\) was analyzed (with *n*, *p* standing for the number of vertices and probability of an edge being present, respectively). The authors have proven that for almost all graphs almost all vertices can be found optimally. Since there are already known examples of graphs for which some vertices are searched in \(\Theta (n^{\frac{1}{2}+a})\) time for \(a>0\) [1, 13] (throughout this paper \(O,o,\Omega ,\omega ,\Theta ,\sim \) denote asymptotic relations, see [16]), the result cannot be strengthened into ‘all graphs.’

*i*,

*j*. Otherwise, creating Bell states or quantum transport will be at least very difficult.

What is more, due to the laws of quantum mechanics, the measurement time needs to be known since the beginning. This includes not only differences in the complexity, but a constant as well. For example, if for two different nodes \(v,v'\) we have \(\langle {v}|{\lambda _1}\rangle =\frac{1}{\sqrt{n}}\) and \(\langle {v'}|{\lambda _1}\rangle =\frac{2}{\sqrt{n}}\), then different measurement times should be chosen for each.

Both effects mentioned above can be described as hiding nodes in the graphs. Finally, we propose a following research problem: *can we actually ‘hide’ a vertex in a random Erdős–Rényi graph*? We have managed to show that in the case of adjacency matrix, \(p=\omega (\log ^3(n)/(n\log ^2(\log (n)))\) is a sufficient requirement for all-vertices optimal search. Under further constraint \(p=\omega (\log ^8(n)/n)\), we have common time measurement. Moreover, we went a step further than the authors of [4] and studied also Laplacian matrix, which led us to tighter results. In the case of Laplacian matrix, \(p>(1+\varepsilon )\log (n)/n\), for constant \(\varepsilon >0\), is sufficient for common time measurement; however in the \(p=\Theta (\log (n)/n)\) case, it may not be true that almost surely the probability 1 of a successful measurement is achieved. If \(p<(1-\varepsilon )\log (n)/n\), then a random graph contains almost surely isolated nodes [17]; hence, it is possible to hide a vertex for both adjacency and Laplacian matrix.

## 2 Element-wise optimality for adjacency matrix

*A*or Laplacian \(L=D-A\), where

*D*is the degree matrix. In [4], authors have proven that for a random Erdős–Rényi graph in case of an adjacency matrix almost all vertices from almost all graphs can be found optimally. We say some property holds almost surely for all graphs, when the probability of choosing random graph having such is \(1-o(1)\). The result was based on the following simplified lemma.

## Lemma 1

[4] Let *H* be a Hamiltonian with eigenvalues \(\lambda _1\ge \dots \ge \lambda _n\) satisfying \(\lambda _1=1\) and \(|\lambda _i|\le c <1\) for all \(i>1\) with corresponding eigenvectors \(|\lambda _1\rangle =|s\rangle ,|\lambda _2\rangle ,\dots ,|\lambda _n\rangle \) and let *w* denote a marked vertex. For an appropriate choice of \(r\in [-\frac{c}{1+c},\frac{c}{1-c}]\), the starting state \(|s\rangle \) evolves by the Schr\(\ddot{o}\)dinger’s equation with the Hamiltonian \((1+r)H +|w\rangle \langle w|\) for time \(t = \Theta (\sqrt{n})\) into the state \(|f\rangle \) satisfying \(|\langle {w}|{f}\rangle |^2\ge \frac{1-c}{1+c}+o(1)\).

*r*satisfying

*r*and

*t*is needed.

According to the lemma, two properties of \(M_\mathrm{G}\) are useful in proving search optimality. Firstly, the matrix should have a single outlying eigenvalue. Secondly, if \(|\lambda _1\rangle \) is the eigenvector corresponding to the outlying eigenvalue, one should have \( \vert \langle {w}|{\lambda _1}\rangle \vert =\Theta (\frac{1}{\sqrt{n}})\).

Note that in the limit \(n\rightarrow \infty \), norms cease to be equivalent; thus, different concepts of closeness of vectors can be chosen. In [4], authors choose \(1-|\langle {\psi }|{\phi }\rangle |\) for arbitrary vectors \(|\psi \rangle , |\phi \rangle \), which allows to infer that *o*(*n*) of nodes can be found in time \(\omega (\sqrt{n})\), see the example given in Eq. (1). In order to make statements concerning all vertices, we should study the limit behavior of the principal vector in \(L^{\infty }\) norm \(\Vert \cdot \Vert _\infty \), which bounds the maximal deviation of coordinates. More precisely, we are interested whether \(\Vert | \lambda _{1}\rangle -|s\rangle \Vert _\infty = \frac{o(1)}{\sqrt{n}}\), as this would imply that for an arbitrary marked node \( w\) we have \( \langle {w}|{\lambda _{1}}\rangle =(1+o(1))\frac{1}{\sqrt{n}}\). The above will give us the bound \(\Theta (\frac{1}{|\langle {w}|{\lambda _1}\rangle |}) = \Theta (\sqrt{n})\) for the time needed for quantum spatial algorithm to locate vertex *w*.

Indeed, a convergence of infinity norm was shown by Mitra [18] providing \(p\ge \log ^6(n)/n\). We have managed to weaken the assumptions and thereby strengthen the result.

## Proposition 1

*A*is an adjacency matrix of a random Erdős–Rényi graph \(\mathcal G(n,p)\) with \(p=\omega ( \log ^3(n)/(n\log ^2\log n))\). Let \(|\lambda _{1}\rangle \) denote the eigenvector corresponding to the largest eigenvalue of

*A*and let \(|s\rangle =\frac{1}{\sqrt{n}}\sum _{v}|v\rangle \). Then,

The proof, which follows the concept proposed by Mitra [18], can be found in Section A in Supplementary Materials. This implies that all vertices can be found optimally in \(\Theta (\sqrt{n})\) time for almost all graphs.

*w*in time

*t*can be approximated by [4]

*n*. Thus, we have that in time \(t=\frac{\pi }{2}\sqrt{n}\), the probability of measurement is optimal, independently on a chosen marked node. Finally, we can conclude our results concerning adjacency matrix with the following theorem.

## Theorem 2

Suppose we chose a graph according to Erdős–Rényi \(\mathcal G(n,p)\) model with \(p=\omega (\log ^8(n)/n)\). Then by choosing \(M_\mathrm{G}=\frac{1}{np}A\), where *A* is an adjacency matrix in Eq. (2), almost surely all vertices can be found with probability \(1-o(1)\) with common measurement time approximately \(t=\pi \sqrt{n}/2\).

## 3 Element-wise optimality for Laplacian matrix

*L*. This is a positive semi-definite matrix, where the dimensionality of null space corresponds to the number of connected components. Based on the results from [20], one can show that for \(p=\omega (\log (n) /n)\) all of the other eigenvalues of \(\frac{L}{np}\) converge to 1, see Section C in Supplementary Materials. At the same time, the eigenvector corresponding to the null space is

*exactly*the equal superposition \(|s\rangle = |\mu _n\rangle = \frac{1}{\sqrt{n}}\sum _{v\in V} |v\rangle \). Thus, since for \(p>(1+\varepsilon )\log (n)/n\) a graph is almost surely connected, the Laplacian matrix takes the form

The situation changes in the case of \(p=O(\log (n)/n)\). Note that for both adjacency and Laplacian matrices the evolution does not change the probability of measuring isolated vertices. If \(p<(1-\varepsilon )\log (n)/n\), then graphs almost surely contain such vertices, and hence, you actually *can* hide a vertex in such a graph.

The \(p\sim p_0\log (n)/n\) for a constant \(p_0>1\) is a smooth transition case between hiding and non-hiding cases mentioned before. In this case based on Exercise III.4 from [21], one can show that \(\mu _1 \sim (1-p_0)(W_{0}(\frac{1-p_0}{\mathrm {e}p_0}))^{-1}\log (n)\) and \(\mu _{n-1} \sim (1-p_0)(W_{-1}(\frac{1-p_0}{\mathrm {e}p_0}))^{-1}\log (n)\), where \(W_{0},W_{-1}\) are Lambert W functions, see Section D in Supplementary Materials. Here we use the notation \(f(n)\sim g(n) \iff f(n)-g(n)=o(g(n))\) . In this case, the \(M_\mathrm{G}=\mathrm {I}-\frac{1}{np}L\) does not imply that both \(\mu _1\) and \(\mu _{n-1}\) converge to 1.

## Theorem 3

Suppose we chose a graph according to Erdős–Rényi \(\mathcal G(n,p)\) model. For \(p=\omega (\log (n)/n)\), by choosing \(M_\mathrm{G}=\frac{1}{np}L\) in Eq. (2), almost surely all vertices can be found with probability \(1-o(1)\) in asymptotic \(\pi \sqrt{n}/2\) time. For \(p\sim p_0\log (n)/n \), by choosing \(M_\mathrm{G} = (1+r)\gamma L\) for some proper *r*, where \(\gamma \) is defined as in Eq. (11), all vertices can be found in \(\Theta (\sqrt{n})\) time with probability bounded from below by the constant in Eq. (12).

*r*and

*t*values as open question.

## Theorem 4

Suppose we chose a graph according to Erdős–Rényi \(\mathcal G(n,p)\) model with \(p\le (1-\varepsilon )\log (n)/n\), where \(\varepsilon >0\). Then for both adjacency and Laplacian matrices, there exist vertices which cannot be found in *o*(*n*) time.

## 4 Conclusion and discussion

In this work, we prove that all vertices can be found optimally with common measurement time \((\pi \sqrt{n})/2\) for almost all Erdős–Rényi graphs for both adjacency and Laplacian matrices under conditions \(p=\omega (\log ^8(n)/n)\) and \(p\ge (1+\varepsilon )\log (n)/n\), respectively. The proof is based on element-wise ergodicity of the eigenvector corresponding to the outlying eigenvalue of adjacency or Laplacian matrix. While under the mentioned constraint adjacency matrix almost surely achieves success probability \(1-o(1)\), the same probability for Laplacian matrix in the \(p\sim p_0\log (n)/n\) case for some \(p_0>1\) can only be bounded from below by some positive constant. At the same time for \(p<(1-\varepsilon )\log (n)/n\), the property does not hold anymore, since almost surely there exist isolated vertices which need \(\Omega (n)\) time to be found.

While our derivation concerning the Laplacian matrix is nearly complete, since only upper bound for success probability is missing in the \(p=\Theta (\log (n)/n)\) case, in our opinion it is possible to weaken the condition on *p* for the adjacency matrix. The first key step would be showing that the largest eigenvalue \(\lambda (\frac{1}{np} A)\) follows \(\mathcal N(1,\frac{1}{n}\sqrt{2(1-p)/p)}\) distribution for \(p\ge (1+\varepsilon )\log (n)/n\). Then, since element-wise convergence of principal vector requires \(p=\omega (\log ^3(n)/(n\log ^2\log n))\), the result would be strengthened to the last mentioned constraint. The second step would be the generalization of the mentioned element-wise convergence theorem.

Further interesting generalization of the result would be the analysis of more general random graph models as well. While this proposition has already been stated [4], our results show that in order to prove security of the quantum spatial search, it would be desirable to analyze the limit behavior of the principal vector in the sense of \(\Vert \cdot \Vert _\infty \) norm.

## Notes

### Acknowledgements

Aleksandra Krawiec, Ryszard Kukulski and Zbigniew Puchała acknowledge the support from the National Science Centre, Poland, under Project Number 2016/22/E/ST6/00062. Adam Glos was supported by the National Science Centre under Project Number DEC-2011/03/D/ST6/00413.

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