Laplacian versus adjacency matrix in quantum walk search

Abstract

A quantum particle evolving by Schrödinger’s equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace’s operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuous-time quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for non-regular graphs and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences.

Appendices

Appendix 1: Eigensystem for Laplacian walk

The search Hamiltonian, when walking with the Laplacian, is given in (2). Finding the exact eigenvectors and eigenvalues of H is intractable, but they can be approximated for large N using degenerate perturbation theory, which also gives a way to find the critical $\gamma$’s [12]. To do this, we separate the Hamiltonian (2) into its leading- and higher-order terms, i.e., $H = H^{(0)} + H^{(1)} + \dots$, where

$$\begin{aligned} H^{(0)} = -\gamma \begin{pmatrix} \frac{1}{\gamma } - N_2 &{} 0 &{} 0 &{} 0 \\ 0 &{} \frac{1}{\gamma } - N_1 &{} 0 &{} 0 \\ 0 &{} 0 &{} -N_2 &{} \sqrt{N_1 N_2} \\ 0 &{} 0 &{} \sqrt{N_1 N_2} &{} -N_1 \\ \end{pmatrix}, \\ H^{(1)} = -\gamma \begin{pmatrix} 0 &{} 0 &{} 0 &{} \sqrt{k_1N_2} \\ 0 &{} 0 &{} \sqrt{k_2N_1} &{} 0 \\ 0 &{} \sqrt{k_2N_1} &{} 0 &{} 0 \\ \sqrt{k_1N_2} &{} 0 &{} 0 &{} 0 \\ \end{pmatrix}. \end{aligned}$$

The idea is to first find the eigenvectors and eigenvalues of the leading-order Hamiltonian $H^{(0)}$, which is a much simpler matrix. Then we add the next-order corrections $H^{(1)}$ and see how this perturbation modifies them. This gives an approximation for the eigenvectors and eigenvalues of H.

To begin, the eigenvectors and eigenvalues of $H^{(0)}$ are

$$\begin{aligned}&{\left| a \right\rangle }, \quad \gamma N_2 - 1 \\&{\left| b \right\rangle }, \quad \gamma N_1 - 1 \\&{\left| r \right\rangle } = \frac{1}{\sqrt{N}} \left( \sqrt{N_1} {\left| c \right\rangle } + \sqrt{N_2} {\left| d \right\rangle } \right) , \quad 0 \\&\frac{1}{\sqrt{N}} \left( -\sqrt{N_2} {\left| c \right\rangle } + \sqrt{N_1} {\left| d \right\rangle } \right) , \quad \gamma (N_1 + N_2). \end{aligned}$$

Note that the third eigenvector, which we call ${\left| r \right\rangle }$, is approximately ${\left| s \right\rangle }$ for large N since the $k_1$ and $k_2$ terms in ${\left| s \right\rangle }$ are dominated by $N_1$ and $N_2$ for large N.

Now we include the perturbation $H^{(1)}$ to see how these leading-order eigenvectors and eigenvalues change. If they are non-degenerate, then $H^{(1)}$ can only contribute higher-order terms to the eigenvectors and eigenvalues, so the starting state ${\left| s \right\rangle } \approx {\left| r \right\rangle }$ is still an approximate eigenvector of the perturbed system for large N [2]. If the leading-order eigenvectors are degenerate, however, the behavior is vastly different. For example, say

$$\begin{aligned} \gamma _a = \frac{1}{N_2} \end{aligned}$$

so that ${\left| a \right\rangle }$ and ${\left| r \right\rangle }$ are degenerate to leading-order. Then the addition of $H^{(1)}$ causes two linear combinations of them

$$\begin{aligned} \alpha _a {\left| a \right\rangle } + \alpha _r {\left| r \right\rangle } \end{aligned}$$

become eigenstates of the perturbed system [2]. To find the coefficients, we solve the eigenvalue problem

$$\begin{aligned} \begin{pmatrix} H_{aa} &{}&{} H_{ar} \\ H_{ra} &{}&{} H_{rr} \\ \end{pmatrix} \begin{pmatrix} \alpha _a \\ \alpha _r \\ \end{pmatrix} = E \begin{pmatrix} \alpha _a \\ \alpha _r \\ \end{pmatrix}, \end{aligned}$$

where $H_{ar} = \langle a | H^{(0)} + H^{(1)} | r \rangle$, etc. Evaluating the matrix elements,

$$\begin{aligned} \begin{pmatrix} 0 &{}&{} -\sqrt{\frac{k_1}{N}} \\ -\sqrt{\frac{k_1}{N}} &{}&{} 0 \\ \end{pmatrix} \begin{pmatrix} \alpha _a \\ \alpha _r \\ \end{pmatrix} = E \begin{pmatrix} \alpha _a \\ \alpha _r \\ \end{pmatrix}. \end{aligned}$$

This has solutions

$$\begin{aligned} \frac{1}{\sqrt{2}} \left( \pm {\left| a \right\rangle } + {\left| r \right\rangle } \right) , \quad E = \mp \sqrt{\frac{k_1}{N}}. \end{aligned}$$

Since ${\left| r \right\rangle } \approx {\left| s \right\rangle }$, for large N, the (unnormalized) eigenvectors and eigenvalues of H when $\gamma = \gamma _a$ are

$$\begin{aligned} {\left| s \right\rangle } \pm {\left| a \right\rangle }, \quad E = \mp \sqrt{\frac{k_1}{N}}, \end{aligned}$$

as stated in the main text of the paper.

Note that ${\left| s \right\rangle } + {\left| a \right\rangle }$ and ${\left| s \right\rangle } - {\left| a \right\rangle }$ are non-degenerate eigenstates of the perturbed system, so the first-order correction $H^{(1)}$ has lifted the degeneracy. Then any higher-order corrections ($H^{(2)}$, $H^{(3)}$, etc.) will not significantly change these eigenstates and eigenvalues, meaning any contributions from them will go to zero more quickly than the terms we have derived [2], and hence we can safely ignore them.

Using the method of Sect. IV of [16], we can find how close $\gamma$ must be to its critical value. Say $\gamma$ is within $\epsilon$ of its critical value, i.e., $\gamma = \gamma _a + \epsilon = 1/N_2 + \epsilon$. If $\epsilon$ is small such that ${\left| a \right\rangle }$ and ${\left| r \right\rangle }$ are near-degenerate to leading-order, then the perturbation still causes $\alpha _a {\left| a \right\rangle } + \alpha _r {\left| r \right\rangle }$ to be eigenvectors of the perturbed system. To find the coefficients, one solves an eigenvalue problem, which has a leading-order term in $\epsilon$ scaling as $\epsilon N_2$ due to the component $H_{aa} = \langle a | H^{(0)} + H^{(1)} | a \rangle$. Thus for the system to retain the error-free energy gap of $\Theta (1/\sqrt{N})$, we need $\epsilon N_2 = o(1/\sqrt{N})$. That is, $\epsilon = o(1/N^{3/2})$, which is the precision stated in the main text of the paper.

Similarly, if

$$\begin{aligned} \gamma _b = \frac{1}{N_1} \end{aligned}$$

so that ${\left| b \right\rangle }$ and ${\left| r \right\rangle }$ are degenerate to leading-order, then the perturbation $H^{(1)}$ causes two linear combinations $\alpha _b {\left| b \right\rangle } + \alpha _r {\left| r \right\rangle }$ to be eigenvectors of the perturbed system. Solving a similar eigenvalue problem for the coefficients $\alpha _b$ and $\alpha _r$, we find that the perturbed eigenstates are

$$\begin{aligned} \frac{1}{\sqrt{2}} \left( \pm {\left| b \right\rangle } + {\left| r \right\rangle } \right) , \quad E = \mp \sqrt{\frac{k_2}{N}}. \end{aligned}$$

Since ${\left| r \right\rangle } \approx {\left| s \right\rangle }$, for large N, the (unnormalized) eigenvectors and eigenvalues of H when $\gamma = \gamma _b$ are

$$\begin{aligned} {\left| s \right\rangle } \pm {\left| b \right\rangle }, \quad E = \mp \sqrt{\frac{k_2}{N}}, \end{aligned}$$

as stated in the main text of the paper.

For the precision with which $\gamma$ must be chosen to its critical value, we can again use the argument of [16] to show that $\gamma$ must be within $o(1/N^{3/2})$ of $\gamma _b$, as stated in the main text of the paper.

Fig. 7

Apart from a factor of $-\gamma$, the a Hamiltonian for search on the complete bipartite graph, where the walk is effected by the Laplacian, and b the leading-order terms. Note $N_{ki} = N_i - k_i$

This calculation using degenerate perturbation theory can also be understood diagrammatically [24]. The search Hamiltonian (2) can be depicted as shown in Fig. 7a. Keeping only the leading-order terms, $H^{(0)}$ is shown in Fig. 7b, and the diagram reveals its four eigenvectors: ${\left| a \right\rangle }$, ${\left| b \right\rangle }$, and two linear combinations of ${\left| c \right\rangle }$ and ${\left| d \right\rangle }$ (one of which we called ${\left| r \right\rangle }$). By choosing ${\left| a \right\rangle }$ to be degenerate with ${\left| r \right\rangle }$, the perturbation, which restores the missing edges, causes ${\left| a \right\rangle }$ and ${\left| r \right\rangle }$ to mix. Similarly, if ${\left| b \right\rangle }$ is degenerate with ${\left| r \right\rangle }$, the perturbation causes them to mix.

Appendix 2: Eigensystem for adjacency walk

Similar to the last section, we approximate the eigenvectors and eigenvalues of the search Hamiltonian (3) using degenerate perturbation theory [12]. Breaking H into its leading- and higher-order terms,

$$\begin{aligned} H^{(0)}&= -\gamma \begin{pmatrix} \frac{1}{\gamma } &{} 0 &{} 0 &{} 0 \\ 0 &{} \frac{1}{\gamma } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \sqrt{N_1 N_2} \\ 0 &{} 0 &{} \sqrt{N_1 N_2} &{} 0 \\ \end{pmatrix}, \\ H^{(1)}&= -\gamma \begin{pmatrix} 0 &{} 0 &{} 0 &{} \sqrt{k_1N_2} \\ 0 &{} 0 &{} \sqrt{k_2N_1} &{} 0 \\ 0 &{} \sqrt{k_2N_1} &{} 0 &{} 0 \\ \sqrt{k_1N_2} &{} 0 &{} 0 &{} 0 \\ \end{pmatrix}. \end{aligned}$$

$H^{(0)}$ has eigenvectors and eigenvalues

$$\begin{aligned}&{\left| a \right\rangle }, \quad -1 \\&{\left| b \right\rangle }, \quad -1 \\&{\left| u \right\rangle } = \frac{1}{\sqrt{2}} \left( {\left| c \right\rangle } + {\left| d \right\rangle } \right) , \quad -\gamma \sqrt{N_1 N_2} \\&{\left| v \right\rangle } = \frac{1}{\sqrt{2}} \left( -{\left| c \right\rangle } + {\left| d \right\rangle } \right) , \quad \gamma \sqrt{N_1 N_2}. \end{aligned}$$

Note that the third eigenvector, which we call ${\left| u \right\rangle }$, is approximately ${\left| \sigma \right\rangle }$ for large N. Also, the fourth eigenvector, which we call ${\left| v \right\rangle }$, is approximately ${\left| \delta \right\rangle }$ for large N.

Now we include the perturbation $H^{(1)}$. If ${\left| u \right\rangle }$ is non-degenerate to leading-order, then the starting state ${\left| \sigma \right\rangle } \approx {\left| u \right\rangle }$ approximately remains an eigenstate of the perturbed system. On the other hand, if

$$\begin{aligned} \gamma _* = \frac{1}{\sqrt{N_1 N_2}}, \end{aligned}$$

then ${\left| a \right\rangle }$, ${\left| b \right\rangle }$, and ${\left| u \right\rangle }$ are degenerate to leading-order, and the three linear combinations of them

$$\begin{aligned} \alpha _a {\left| a \right\rangle } + \alpha _b {\left| b \right\rangle } + \alpha _u {\left| u \right\rangle } \end{aligned}$$

will be eigenvectors of the perturbed system. To find the coefficients, we solve the eigenvalue problem

$$\begin{aligned} \begin{pmatrix} H_{aa} &{} H_{ab} &{} H_{au} \\ H_{ba} &{} H_{bb} &{} H_{bu} \\ H_{ua} &{} H_{ub} &{} H_{uu} \\ \end{pmatrix} \begin{pmatrix} \alpha _a \\ \alpha _b \\ \alpha _u \\ \end{pmatrix} = E \begin{pmatrix} \alpha _a \\ \alpha _b \\ \alpha _u \\ \end{pmatrix}, \end{aligned}$$

where $H_{ab} = \langle a | H^{(0)} + H^{(1)} | b \rangle$, etc. Evaluating the matrix components, we get

$$\begin{aligned} \begin{pmatrix} -1 &{} 0 &{} -\sqrt{\frac{k_1}{2 N_1}} \\ 0 &{} -1 &{} -\sqrt{\frac{k_2}{2 N_2}} \\ -\sqrt{\frac{k_1}{2 N_1}} &{} -\sqrt{\frac{k_2}{2 N_2}} &{} -1 \\ \end{pmatrix} \begin{pmatrix} \alpha _a \\ \alpha _b \\ \alpha _u \\ \end{pmatrix} = E \begin{pmatrix} \alpha _a \\ \alpha _b \\ \alpha _u \\ \end{pmatrix}. \end{aligned}$$

Solving this yields (unnormalized) eigenstates

$$\begin{aligned} \psi _{-1}&= -\sqrt{\frac{k_2 N_1}{k_1 N_2}} {\left| a \right\rangle } + {\left| b \right\rangle } \\ \psi _-&= \sqrt{\frac{k_1 N_2}{k_2 N_1 + k_1 N_2}} {\left| a \right\rangle } + \sqrt{\frac{k_2 N_1}{k_2 N_1 + k_1 N_2}} {\left| b \right\rangle } + {\left| u \right\rangle } \\ \psi _+&= -\sqrt{\frac{k_1 N_2}{k_2 N_1 + k_1 N_2}} {\left| a \right\rangle } - \sqrt{\frac{k_2 N_1}{k_2 N_1 + k_1 N_2}} {\left| b \right\rangle } + {\left| u \right\rangle } \end{aligned}$$

with corresponding eigenvalues

$$\begin{aligned} E_{-1}&= -1 \\ E_-&= -1 - \sqrt{\frac{k_2 N_1 + k_1 N_2}{2 N_1 N_2}} \\ E_+&= -1 + \sqrt{\frac{k_2 N_1 + k_1 N_2}{2 N_1 N_2}}. \end{aligned}$$

Note that $\psi _\mp$ and its corresponding eigenvalues $E_\mp$ were stated in the main text of the paper, with ${\left| u \right\rangle }$ replaced by ${\left| \sigma \right\rangle }$, assuming large N.

Fig. 8

Apart from a factor of $-\gamma$, the a Hamiltonian for search on the complete bipartite graph, where the walk is effected by the adjacency matrix, and b the leading-order terms. Note $N_{ki} = N_i - k_i$

As before, we can use the argument of [16] to find the precision with which $\gamma$ must be chosen to its critical value. If $\gamma$ is within $\epsilon$ of $\gamma _*$, then there is a term scaling as $\epsilon \sqrt{N_1N_2}$ that appears in the perturbative calculation, which is leading-order in $\epsilon$. For this to be small enough to not interfere with the energy gap of $\Theta (1/\sqrt{N})$, we get $\epsilon = o(1/N^{3/2})$, which is the precision stated in the main text of the paper.

The Hamiltonian (3) can be represented diagrammatically [24], as depicted in Fig. 8a. Keeping only the leading-order terms, $H^{(0)}$ is shown in Fig. 8b, and we see that ${\left| a \right\rangle }$ and ${\left| b \right\rangle }$ are always degenerate since they have the same self-loop. Making these degenerate with ${\left| u \right\rangle }$, which is a combination of ${\left| c \right\rangle }$ and ${\left| d \right\rangle }$, the perturbation restores the missing edges and mixes ${\left| a \right\rangle }$, ${\left| b \right\rangle }$, and ${\left| u \right\rangle }$.

Finally, since $\gamma > 0$, the leading-order eigenvector ${\left| v \right\rangle }$ is never degenerate with the others. So it remains an approximate eigenstate of the perturbed system. Since ${\left| v \right\rangle } \approx {\left| \delta \right\rangle }$, we get that ${\left| \delta \right\rangle }$ is approximately an eigenvector of H for large N, as stated in the main text of the paper.