Fast quantum search of multiple vertices based on electric circuits

Abstract

The realization of fast quantum search of multiple vertices is very important when considering the realistic requirement of searching multiple data in the database simultaneously. However, the experimental demonstration of searching multiple vertices is difficult because it requires the preparation of many target vertices simultaneously. Here, we report that the fast quantum search of multiple vertices is experimentally realized in electric circuits. Through the dedicated design in the electric circuit, we can exhibit the quantum dynamics on the graphs, which are regarded as quantum walks. Our theoretical and experimental results demonstrate that we can reach the largest searching probability at time $O\left( {\sqrt N } \right)$. Such probability strongly depends on the locations of target vertices in the graphs. Our results based on electric circuits provide one stable and easily integrated platform to implement fast quantum search.

Appendices

Appendix A: The theoretical derivation of searching for $k$ vertices on a joined complete graph with $N$ vertices

As shown in Fig. 1a, the graph has $N$ vertices and the target vertices are $k$ vertices which are all located on the left side of the joined complete graph. We mark the target vertices in red and group the other vertices in the graph according to the different connectivity. The system will evolve in a five-dimensional subspace. The five orthogonal basis vectors in the subspace can be expressed as $\left| a \right\rangle = \frac{1}{\sqrt k }\sum\nolimits_{i \in red} {\left| i \right\rangle }$, $\left| b \right\rangle = \frac{1}{{\sqrt {\frac{N}{2} - k - 1} }}\sum\nolimits_{i \in blue} {\left| i \right\rangle }$, $\left| c \right\rangle = \left| {{\text{orange}}} \right\rangle$, $\left| d \right\rangle = \left| {{\text{green}}} \right\rangle$, and $\left| e \right\rangle = \frac{1}{{\sqrt {\frac{N}{2} - 1} }}\sum\nolimits_{i \in white} {\left| i \right\rangle }$. Substituting these orthogonal basis vector into $H = - \gamma A - \sum\nolimits_{i \in marked}^{i = 1,...,k} {\left| i \right\rangle } \left\langle i \right|$, we rewrite the Hamiltonian in this five-dimensional space. If $k$ is far smaller than $N$, the Hamiltonian $H$ can be simplified and decomposed into three terms according to $N$, $\sqrt N$ and 1 as

$$ H^{(0)} = - \gamma \left( {\begin{array}{*{20}c} {\frac{1}{\gamma }} & 0 & 0 & 0 & 0 \\ 0 & \frac{N}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{N}{2} \\ \end{array} } \right),H^{(1)} = - \gamma \left( {\begin{array}{*{20}c} 0 & {\sqrt{\frac{kN}{2}} } & 0 & 0 & 0 \\ {\sqrt{\frac{kN}{2}} } & 0 & {\sqrt{\frac{N}{2}} } & 0 & 0 \\ 0 & {\sqrt{\frac{N}{2}} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\sqrt{\frac{N}{2}} } \\ 0 & 0 & 0 & {\sqrt{\frac{N}{2}} } & 0 \\ \end{array} } \right),H^{(2)} = - \gamma \left( {\begin{array}{*{20}c} 0 & 0 & {\sqrt k } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ {\sqrt k } & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right) $$

(A1)

We can neglect $H^{(2)}$ in Eq. (A1) when $N$ is large. $\left| a \right\rangle$, $\left| b \right\rangle$, $\left| e \right\rangle$ are the eigenstates of $H^{(0)}$. $\frac{1}{\gamma }$,$\frac{N}{2}$ and $\frac{N}{2}$ are the degenerate eigenvalues corresponding to $\left| a \right\rangle$, $\left| b \right\rangle$ and $\left| e \right\rangle$, respectively. When $\frac{1}{\gamma } = \frac{N}{2}$, the critical value $\gamma_{c} = \frac{2}{N}$ is obtained. The eigenvectors $\left| \varphi \right\rangle$ of the perturbation system $H^{\left( 0 \right)} + H^{\left( 1 \right)}$ are $\left| {\varphi_{1} } \right\rangle = \frac{1}{\sqrt 2 }\left( {\left| a \right\rangle + \left| b \right\rangle } \right)$, $\left| {\varphi_{2} } \right\rangle = \left| e \right\rangle$ and $\left| {\varphi_{3} } \right\rangle = \frac{1}{\sqrt 2 }\left( { - \left| a \right\rangle + \left| b \right\rangle } \right)$. The initial state for large $N$ is re-expressed as $\left| {\varphi \left( {0} \right)} \right\rangle { = }\left| s \right\rangle = \frac{1}{\sqrt N }\sum\nolimits_{i}^{i = 1,2...N} {\left| i \right\rangle } \approx \frac{1}{2}\left( {\left| {\varphi_{{1}} } \right\rangle + \left| {\varphi_{{3}} } \right\rangle } \right) + \frac{1}{\sqrt 2 }\left| {\varphi_{{2}} } \right\rangle$. The success amplitude of finding target state $\left| a \right\rangle$ is $\left\langle {a\left| {\varphi (t)} \right\rangle } \right. = \frac{1}{2}(\left\langle a \right|e^{{ - i(H^{\left( 0 \right)} + H^{\left( 1 \right)} )t}} \left| {\varphi_{1} } \right\rangle + \left\langle a \right|e^{{ - i(H^{\left( 0 \right)} + H^{\left( 1 \right)} )t}} \left| {\varphi_{3} } \right\rangle ) + \frac{1}{\sqrt 2 }\left\langle a \right|e^{{ - i(H^{\left( 0 \right)} + H^{\left( 1 \right)} )t}} \left| {\varphi_{2} } \right\rangle$, and the success probability of finding $\left| a \right\rangle$ at time t is $p = \left| {\left\langle a \right|\left. {\varphi (t)} \right\rangle } \right|^{2} = \frac{1}{2}\sin^{2} \sqrt{\frac{2k}{N}} t$. So the maximum search success probability of finding the target vertices is $\frac{1}{2}$, and the time to find the maximum probability is $t = \frac{\pi }{2}\sqrt{\frac{N}{2k}}$.

In the main text, we have theoretically verified that the dynamics of voltages at different points in the circuit are consistent with those of vertices on the graph described by quantum theory. According to the results above, the $k$ vertices in the joined complete graph with $N$ vertices can be found at time $t = \frac{\pi }{2}\sqrt{\frac{N}{2k}}$, and the corresponding search probability is $\frac{1}{2}$. In the circuit, the initial state is set as ${|}\phi_{{H_{c} }} (0)) = \left( {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right) \otimes |\varphi (0))$, where $|\varphi (0)) = \frac{1}{\sqrt N }\sum\nolimits_{{i{ = 1}}}^{N} {|i)}$. The dynamics of each point in the electric circuit are governed by $i\frac{{\partial \phi_{{H_{c} }} (t)}}{\partial t} = H_{c} \phi_{{H_{c} }} (t)$. At time T, the evolution result of each point is $|\phi_{{H_{c} }} (T)) = e^{{ - iH_{c} T}} |\phi_{{H_{c} }} (0)) = = \left( {\begin{array}{*{20}c} {\cos (H^{\prime}T)|\varphi (0))} \\ {\sin (H^{\prime}T)|\varphi (0))} \\ \end{array} } \right)$. According to Eq. (6) in the main text, when left multiplying $(\begin{array}{*{20}c} 1 & { - i} \\ \end{array} ) \otimes I_{N}$ by $|\phi_{{H_{c} }} (T))$, we can obtain the same results as those from the dynamics on graph ${|}\varphi (T)) = ((\begin{array}{*{20}c} 1 & { - i} \\ \end{array} ) \otimes I_{N} )|\phi_{{H_{c} }} (T)) = e^{{ - iH^{\prime}T}} |\varphi (0))$. Therefore, in the electric circuit, we can also find $k$ target points at time $t = \frac{\pi }{2}\sqrt{\frac{N}{2k}}$.

Moreover, in the circuit experiments, we also study the situations that the values of $\gamma$ are not the optimal one. From Fig. 4a and b, we study the search on the joined complete graph with $N = 12,k = 2$. Here, the values are not optimal one, $\gamma = \frac{1}{3}$ and $\gamma = \frac{1}{12}$, respectively. We can find that the success probabilities in Fig. 4a and b are both smaller than that in the optimal case (Fig. 2d).

Fig. 4

a and b, the same structure as Fig. 2c, the value of $\gamma { = }\frac{1}{{3}}$ and $\gamma { = }\frac{1}{{{12}}}$. c and d, the same structure as Fig. 3a, the value of $\gamma { = }\frac{1}{{4}}$ and $\gamma { = }\frac{1}{{{12}}}$. e and f, the same structure as Fig. 3d, the value of $\gamma { = }\frac{1}{{3}}$ and $\gamma { = }\frac{1}{{{12}}}$

In Fig. 4c and d, we study the search on the joined complete graph with $N = 12,k = 3,k_{1} = 2$. Here, the values of $\gamma$ are not optimal one, $\gamma = \frac{1}{4}$ and $\gamma = \frac{1}{12}$, respectively. We can find that the success probabilities in Fig. 4c and d are both smaller than that in the optimal case (Fig. 3c). In Fig. 4e and f, we study the search on the complete graph with $N = 6,k = 2$. Here, the values of $\gamma$ are not optimal one, $\gamma = \frac{1}{3}$ and $\gamma = \frac{1}{12}$, respectively. We can find that the maximum probabilities in Fig. 4e and f are both smaller than that in the optimal case (Fig. 3f).

Appendix B: experimental details for the joined complete graph ($N = 12,k = 2$)

Here we provide the electric design for the structure shown in Fig. 2. The experimental circuit board shown in Fig. 2a corresponds to the graph structure in Fig. 2c. The circuit board is controlled by a touch switch to connect to the power supply. When the switch is turned off, each point of the circuit board evolves according to Kirchhoff's current law. The evolution equation of each point is

$$ \begin{aligned} & C_{1} \frac{d}{dt}V_{1} = \frac{{V_{13} }}{{R_{1,13} }} + \frac{{V_{14} }}{{R_{1,14} }} + \frac{{V_{15} }}{{R_{1,15} }} + \frac{{V_{16} }}{{R_{1,16} }} + \frac{{V_{17} }}{{R_{1,17} }} + \frac{{V_{18} }}{{R_{1,18} }} \\ & - \underbrace {{\left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{1,13} }} + \frac{1}{{R_{1,14} }} + \frac{1}{{R_{1,15} }} + \frac{1}{{R_{1,16} }} + \frac{1}{{R_{1,17} }} + \frac{1}{{R_{1,18} }}} \right)}}_{{Z_{1} }}V_{1} \\ & \ldots \\ & C_{13} \frac{d}{dt}V_{13} = \frac{{V_{1} }}{{R_{13,1} }} + \frac{{V_{2} }}{{R_{13,2} }} + \frac{{V_{3} }}{{R_{13,3} }} + \frac{{V_{4} }}{{R_{13,4} }} + \frac{{V_{5} }}{{R_{13,5} }} + \frac{{V_{6} }}{{R_{13,6} }} \\ & \; - \underbrace {{\left( {\frac{1}{{R_{13} }} + \frac{1}{{R_{13,1} }} + \frac{1}{{R_{13,2} }} + \frac{1}{{R_{13,3} }} + \frac{1}{{R_{13,4} }} + \frac{1}{{R_{13,5} }} + \frac{1}{{R_{13,6} }}} \right)}}_{{Z_{13} }}V_{13} \\ & \ldots \\ \end{aligned} $$

(B1)

Observing the above formula, we can see that the terms of resistance and capacitance on the right side of the evolution equation at each point can be integrated into a matrix $H_{c} = \left( {\begin{array}{*{20}c} {Z_{m} } & {H^{\prime}} \\ { - H^{\prime^{T}} } & {Z_{n} } \\ \end{array} } \right)$ with

$$ H^{\prime} = \frac{1}{C}\left( {\begin{array}{*{20}c} {\frac{1}{{R_{1,13} }}} & {\frac{1}{{R_{1,14} }}} & {\frac{1}{{R_{1,15} }}} & {\frac{1}{{R_{1,16} }}} & {\frac{1}{{R_{1,17} }}} & {\frac{1}{{R_{1,18} }}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {\frac{1}{{R_{2,13} }}} & {\frac{1}{{R_{2,14} }}} & {\frac{1}{{R_{2,15} }}} & {\frac{1}{{R_{2,16} }}} & {\frac{1}{{R_{2,17} }}} & {\frac{1}{{R_{2,18} }}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {\frac{1}{{R_{3,13} }}} & {\frac{1}{{R_{3,14} }}} & 0 & {\frac{1}{{R_{3,16} }}} & {\frac{1}{{R_{3,17} }}} & {\frac{1}{{R_{3,18} }}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {\frac{1}{{R_{4,13} }}} & {\frac{1}{{R_{4,14} }}} & {\frac{1}{{R_{4,15} }}} & 0 & {\frac{1}{{R_{4,17} }}} & {\frac{1}{{R_{4,18} }}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {\frac{1}{{R_{5,13} }}} & {\frac{1}{{R_{5,14} }}} & {\frac{1}{{R_{5,15} }}} & {\frac{1}{{R_{5,16} }}} & 0 & {\frac{1}{{R_{5,18} }}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {\frac{1}{{R_{6,13} }}} & {\frac{1}{{R_{6,14} }}} & {\frac{1}{{R_{6,15} }}} & {\frac{1}{{R_{6,16} }}} & {\frac{1}{{R_{6,17} }}} & 0 & {\frac{1}{{R_{6,19} }}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{1}{{R_{7,18} }}} & 0 & {\frac{1}{{R_{7,20} }}} & {\frac{1}{{R_{7,21} }}} & {\frac{1}{{R_{7,22} }}} & {\frac{1}{{R_{7,23} }}} & {\frac{1}{{R_{7,24} }}} \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{R_{8,19} }}} & 0 & {\frac{1}{{R_{8,21} }}} & {\frac{1}{{R_{8,22} }}} & {\frac{1}{{R_{8,23} }}} & {\frac{1}{{R_{8,24} }}} \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{R_{9,19} }}} & {\frac{1}{{R_{9,20} }}} & 0 & {\frac{1}{{R_{9,22} }}} & {\frac{1}{{R_{9,23} }}} & {\frac{1}{{R_{9,24} }}} \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{R_{10,19} }}} & {\frac{1}{{R_{10,20} }}} & {\frac{1}{{R_{10,21} }}} & 0 & {\frac{1}{{R_{10,23} }}} & {\frac{1}{{R_{10,24} }}} \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{R_{11,19} }}} & {\frac{1}{{R_{11,20} }}} & {\frac{1}{{R_{11,21} }}} & {\frac{1}{{R_{11,22} }}} & 0 & {\frac{1}{{R_{11,24} }}} \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{R_{12,19} }}} & {\frac{1}{{R_{12,20} }}} & {\frac{1}{{R_{12,21} }}} & {\frac{1}{{R_{12,22} }}} & {\frac{1}{{R_{12,23} }}} & 0 \\ \end{array} } \right) $$

(B2)

$Z_{m}$ and $Z_{n}$ represent two matrices with the same dimension as $H^{\prime}$. The diagonal terms in the matrix $Z_{m}$ ($Z_{n}$) are $Z_{{1}}$ to $Z_{{{12}}}$ ($Z_{{{13}}}$ to $Z_{{{24}}}$) in the above equation, and the remaining elements in the matrix are 0. In the discussion of the main text, the values of $R_{m,n}$ and $R_{n,m}$ satisfy $R_{m,n} = - R_{n,m}$, which corresponds to $R_{13,1} = - R_{1,13}$,$R_{13,2} = - R_{2,13}$,$R_{13,3} = - R_{3,13}$,…,$R_{24,11} = - R_{11,24}$ in the circuit structure of Fig. 2a. Then, the Hamiltonian of the circuit structure can be integrated as $H_{c} = \left( {\begin{array}{*{20}c} {Z_{m} } & {H^{\prime}} \\ { - H^{\prime^{T}} } & {Z_{n} } \\ \end{array} } \right)$. Since the search matrix of the regular graph is symmetric, there is a relationship $H^{\prime} = H^{\prime^{T}}$, and the Hamiltonian of the circuit can be substituted into $H_{c} = \left( {\begin{array}{*{20}c} {Z_{m} } & {H^{\prime}} \\ { - H^{\prime}} & {Z_{n} } \\ \end{array} } \right)$.

Moreover, we adjust the potentiometer at the m point on the circuit board so that the diagonal element of $H_{c}$ is 0. For example, the diagonal term of the Hamiltonian at point 1 is $H_{c(1,1)} = Z_{1} = \frac{1}{{R_{1} }} + \frac{1}{{R_{1,13} }} + \frac{1}{{R_{1,14} }} + \frac{1}{{R_{1,15} }} + \frac{1}{{R_{1,16} }} + \frac{1}{{R_{1,17} }} + \frac{1}{{R_{1,18} }} = 0$. The values $R_{1,13}$,$R_{1,14}$,…,$R_{1,18}$ are chosen as positive values, so we adjust the INICs to meet $\frac{1}{{R_{1} }} = - (\frac{1}{{R_{1,13} }} + \frac{1}{{R_{1,14} }} + \frac{1}{{R_{1,15} }} + \frac{1}{{R_{1,16} }} + \frac{1}{{R_{1,17} }} + \frac{1}{{R_{1,18} }}) < 0$, which is required in our design. To make $H_{c(13,13)} = Z_{13} = 0$, we need the effective resistance at the 13th point to be $\frac{1}{{R_{13} }} = - (\frac{1}{{R_{13,1} }} + \frac{1}{{R_{13,2} }} + \frac{1}{{R_{13,3} }} + \frac{1}{{R_{13,4} }} + \frac{1}{{R_{13,5} }} + \frac{1}{{R_{13,6} }})$, which is exactly the opposite value of that at the 1st point. The resistance values of $R_{13,1}$,$R_{13,2}$,…,$R_{13,6}$ are negative and $R_{13}$ is positive. After choosing the right values for the resistances, the circuit Hamiltonian is expressed as $H_{c} = \left( {\begin{array}{*{20}c} 0 & { - iH^{\prime}} \\ {iH^{\prime}} & 0 \\ \end{array} } \right)$. In this case, the evolution equation in the circuit is $i\frac{{\partial \phi_{{H_{c} }} (t)}}{\partial t} = H_{c} \phi_{{H_{c} }} (t)$. When left multiplying $(\begin{array}{*{20}c} 1 & { - i} \\ \end{array} ) \otimes I_{N}$ to $|\phi_{{H_{c} }} (t))$, we can get ${|}\varphi (t)) = ((\begin{array}{*{20}c} 1 & { - i} \\ \end{array} ) \otimes I_{N} )|\phi_{{H_{c} }} (t)) = e^{{ - iH^{\prime}t}} |\varphi (0))$. $\varphi (t)$ is the quantum evolution state governed by the Hamiltonian $H^{\prime}$ at $t$. In the main text, we follow the design above to carry out the experiment.

Appendix C: theoretical derivations of searching different vertices on different joined complete graphs

As shown in Fig. 5, the total number of vertices is $N$ and the target vertices is $k$ on the joined complete graph ($k_{1}$ marked vertices (red) are located on the left side. $k - k_{1}$ vertices (gray) are distributed on the right. The target vertices are not on the connecting line of the graph). We mark the target vertices in red and gray, and group the other vertices on the graph according to the different connectivity. The system will evolve in a six-dimensional subspace. The six orthogonal basis vectors in the subspace can be expressed as, $\left| a \right\rangle = \frac{1}{{\sqrt {k_{1} } }}\sum\nolimits_{i \in red} {\left| i \right\rangle }$, $\left| b \right\rangle = \frac{1}{{\sqrt {\frac{N}{2} - k_{1} - 1} }}\sum\nolimits_{i \in blue} {\left| i \right\rangle }$, $\left| c \right\rangle = \left| {{\text{orange}}} \right\rangle$, $\left| d \right\rangle = \left| {{\text{green}}} \right\rangle$, $\left| e \right\rangle = \frac{1}{{\sqrt {k - k_{1} } }}\sum\nolimits_{i \in gray} {\left| i \right\rangle }$, and $\left| f \right\rangle = \frac{1}{{\sqrt {\frac{N}{2} - k + k_{1} - 1} }}\sum\nolimits_{i \in white} {\left| i \right\rangle }$. Similar as the derivation shown in Appendix A, we can get the success probability of searching for $\left| {a + e} \right\rangle$ as $P = \left| {\frac{1}{2}e^{it} i(\sin \sqrt {\frac{{2(k - k_{1} )}}{N}} t + \sin \sqrt {\frac{{2k_{1} }}{N}} t)} \right|^{2}$.

Fig. 5

The joined complete graph with $N$ vertices. The number of target vertices is $k$ and there are $k_{1}$ (red) vertices on the left side of the graph structure. The remaining $k - k_{1}$ vertices (gray) are distributed on the right

In the above, we provide the fast quantum search in the joined complete graph where $k_{1}$ vertices are located on the left side and $k - k_{1}$ vertices are distributed on the right. When we focus on the search within the graph involving two target vertices in the left and one in the right (that is $k_{1} = 2$ and $k - k_{1} = 1$), the largest probability of searching these vertices can be achieved at time $t$ when $\cos \sqrt{\frac{2}{N}} t + \sqrt 2 \cos \frac{{2}}{\sqrt N }t = 0$. Although this equation about $t$ cannot be solved analytically, we can obtain the solution for different $N$ as.

From this Table 1, it is clearly seen that the largest search probability for the target vertices can be achieved in the time $t \propto O\left( {\sqrt N } \right)$.

Table 1 The optimal time to reach the largest search probability for different $N$

Appendix D: The fast search of multiple vertices for other scenarios on joined complete graphs

Considering the target data and related structures in the search scenarios, the corresponding target vertices may be located at special positions that just belong to the connection line in the graph. In the following, we provide a discussion of the fast search in these different scenarios. Because the electric simulations are similar to those above, we do not provide the electric design details here and only compare the electric simulation results and the corresponding quantum theoretical results.

In Fig. 6a, we provide the graph structures. The “possible” target vertices are labeled “a,” “c,” “d,” and “e,” which are expressed in red, yellow, green, and gray, respectively. Here, the numbers of “a” and “e” are not limited to only one. Three different scenarios are discussed. First, the target vertices are “a” and “c.” The corresponding numbers for “a” and “c” are k-1 and 1. Following the derivations shown in Appendix C, we can obtain that the fast quantum search is realized at time $t = \frac{\pi }{2}\sqrt{\frac{N}{2k}}$, and the corresponding largest search probability is $p = \left[ {\frac{1}{2\sqrt k }(\sqrt {k - 1} + 1)} \right]^{2}$. In Fig. 6b, we provide the theoretical (red line) and simulation (blue line) results for the simplest case where the numbers for “a” and “c” are both one, and the total number of vertices in the graph $N = 12$. From our results, we can find that both the theoretical derivation and electric simulation provide the same results. The fast search can be realized at time $t = \frac{\pi }{2}\sqrt {\frac{{{12}}}{4}} { = }\frac{\pi }{2}\sqrt {3}$. In Fig. 6c, we show that the results for the numbers for “a” and “c” are two and one, respectively. The fast search can be realized at time $t = \frac{\pi }{2}\sqrt {\frac{{{12}}}{6}} { = }\frac{\pi }{2}\sqrt {2}$. Therefore, our electric simulations verify that fast quantum search can be realized in this first scenario.

Fig. 6

a The joined complete graph with $N$ vertices. The target vertices are labeled “a,” “c,” “d,” and “e.” The total number of these target vertices is $k$. b The simulation and theoretical results of the target vertices “a” and “c” in a when $N = 12,k = {2, }\gamma { = }\frac{1}{6}$. c The simulation and theoretical results of the target vertices “a” and “c” in a when $N = 12,k = {3, }\gamma { = }\frac{1}{6}$. d The simulation and theoretical results of the target vertices “a,” “c,” and “d” in a when $N = 12,k = {3, }\gamma { = }\frac{1}{6}$.e The simulation and theoretical results of the target vertices “a,” “c,” “d,” and “e” in a. The corresponding numbers for “a,” “c,” “d,” and “e” are all 1. In panel b, c, d, and e, the red curves are obtained from the quantum theory, and the blue curves are the circuit simulation results

Second, the target vertices are “a,” “c,” and “d.” The corresponding numbers for “a,” “c,” and “d” are k-2, 1, and 1, respectively. Following the similar theoretical derivation in Appendix C, we can obtain that the search probability with time is expressed as $p = \left| {\left\langle {a + c + d\left| {\varphi (t)} \right\rangle } \right.} \right|^{2} = \left| {\frac{1}{\sqrt 6 }e^{it} i\left[ {\frac{1}{{\sqrt {(k - 1)} }}(\sqrt {k - 2} + 1)\sin \sqrt {\frac{2(k - 1)}{N}} t + \sin \sqrt{\frac{2}{N}} t} \right]} \right|^{2}$. When we focus on the simplest case in which the numbers for “a,” “c,” and “d” are all one. The search probability with time $t$ is $p = \left| {\left\langle {a + c + d\left| {\varphi (t)} \right\rangle } \right.} \right|^{2} = \left| {\left( {\frac{1}{\sqrt 3 }\sin \frac{2}{\sqrt N }t + \frac{1}{\sqrt 6 }\sin \sqrt{\frac{2}{N}} t} \right)} \right|^{2}$. Our calculation shows that when the time $t$ satisfies ${2}\cos \frac{{2}}{\sqrt N }t + \cos \sqrt {\frac{{2}}{N}} t = 0$, the largest searching probability of these vertices can be achieved. In the Table 2, we list some optimal times for different numbers of vertices $N$ as.

Table 2 The optimal time to reach the largest search probability for different $N$

From the Table 2, we can find that for different numbers of vertices $N$, the largest search probability can be reached at time $t \propto O\left( {\sqrt N } \right)$. In Fig. 6d, we provide the theoretical results (red line) and electric simulations (blue line) for the simplest case in which the numbers for “a,” “c,” and “d” are all one. The number $N = 12$. The theoretical results and electric simulations show very good coincidence.

Third, the target vertices are “a,” “c,” “d,” and “e.” The corresponding numbers for “a,” “c,” “d,” and “e” are $k_{1} - 1$, 1, 1 and $k - k_{1} - 1$, respectively. Our calculation shows that the search probability with time is $p = \left| {\left\langle {a + c + d + e\left| {\varphi (t)} \right\rangle } \right.} \right|^{2} = \left| {\frac{1}{2\sqrt 2 }e^{it} i\left[ {\frac{1}{{\sqrt {k_{1} } }}(\sqrt {k_{1} - 1} + 1)\sin \sqrt {\frac{{2k_{1} }}{N}} t + (\frac{1}{{\sqrt {k - k_{1} } }} + \sqrt {1 - \frac{1}{{k - k_{1} }}} )\sin \sqrt {\frac{{2(k - k_{1} )}}{N}} t} \right]} \right|^{2}$. When we study the simplest case in which the numbers for “a,” “c,” “d,” and “e” are all one, the largest search probability $p = \sin^{2} \frac{2}{\sqrt N }t$ can be achieved at time $t = \frac{\pi }{2}\frac{\sqrt N }{2}$. In Fig. 6e, our theoretical results (red line) and electric simulations (blue line) show that when the number of total vertices is $N = 12$, the largest probability can be obtained at time $t = \frac{\pi }{2}\frac{{\sqrt {{12}} }}{2}$.

Moreover, we also provide the simulation results for the cases with not optimal $\gamma$. Figure 7a is the simulation result of searching the same vertices as in Fig. 6c, but choosing $\gamma { = }\frac{1}{{{12}}}$. From the comparison of Figs. 7a and 6c, it can be seen that when the value of $\gamma$ deviates from the optimal value $\frac{1}{6}$, the success probability P that can be searched will become smaller, and the time t reaching the maximum probability increases.

Fig. 7

a Simulation results with the same structure as Fig. 6c, $\gamma { = }\frac{1}{{{12}}}$. b Simulation results with the same structure as Fig. 6d, $\gamma { = }\frac{1}{{{12}}}$. c Simulation results with the same structure as Fig. 6e, $\gamma { = }\frac{1}{{{12}}}$

Figure 7bshows the simulation result of searching the same vertices as Fig. 6d, but taking $\gamma { = }\frac{1}{{{12}}}$. From the comparison of Figs. 7b and 6d, it can be seen that when the value of $\gamma$ deviates from the optimal value $\frac{1}{6}$, the success probability P will be smaller, and the time t reaching the maximum probability increases. Figure 7c shows the simulation result of searching the same vertices as Fig. 6e, but selecting $\gamma { = }\frac{1}{{{12}}}$.When comparing Figs. 7c and 6e, it can be seen that when the value of $\gamma$ is not the optimal value, the value of the success probability P decreases, accompanied by the increase of the time reaching the success probability.