Quantum search on Hanoi network

TL;DR

This paper investigates quantum search algorithms on Hanoi networks using regular and lackadaisical quantum walks, demonstrating improved scaling of search times compared to classical methods.

Contribution

It introduces the application of regular and lackadaisical quantum walks to search on Hanoi networks and analyzes their scaling behavior.

Findings

Regular quantum walks scale as O(N^{0.79}√log N) and O(N^{0.65}√log N) for degrees three and four.

Lackadaisical quantum walks scale as O(N^{0.57} log N) and O(N^{0.50} log N) for degrees three and four.

Quantum walks outperform classical exhaustive search on Hanoi networks.

Abstract

Hanoi network has a one-dimensional periodic lattice as its main structure with additional long-range edges, which allow having efficient quantum walk algorithm that can find a target state on the network faster than the exhaustive classical search. In this article, we use regular quantum walks and lackadaisical quantum walks respectively to search for a target state. From the curve fitting of the numerical results for Hanoi network of degree three and four we find that their running time for the regular quantum walks followed by amplitude amplification scales as $\mathcal{O}\left(N^{0.79} \sqrt{\log N}\right)$ and $\mathcal{O}\left(N^{0.65} \sqrt{\log N}\right)$ respectively. And for the search by lackadaisical quantum walks the running time scales as $\mathcal{O}\left(N^{0.57}\log N\right)$ and $\mathcal{O}\left(N^{0.50}\log N\right)$ respectively.

Figures

Equations

$$x_v=2^i\left(2j+1\right),$$

$$|x_v⟩=|i,j⟩,$$

$$|{\psi }_{in}⟩=\frac{1}{\sqrt{d}}\sum _{x_c=0}^{d-1}|x_c⟩\otimes \frac{1}{\sqrt{N}}\sum _{x_v=1}^N|x_v⟩,$$

$$\mathcal{U}=SC,$$

$$\mathcal{C}=C\otimes \left(\mathbb{I}-2|t_v⟩⟨t_v|\right).$$

$$S_{HN3}$$

$$S_{HN4}$$

$$|{\psi }_f⟩={\mathcal{U}}^T|{\psi }_{in}⟩,$$

$$T_{HN3}$$

$$T_{HN4}$$

$$|{\psi }_{in}⟩=|{\psi }_c⟩\otimes \frac{1}{\sqrt{N}}\sum _{x_v=1}^N|x_v⟩,$$

$$|{\psi }_c⟩=\frac{1}{\sqrt{d+l}}\left(\sum _{x_c=0}^{d-1}|x_c⟩+\sqrt{l}|\text{mbox}loop⟩\right).$$

$$C=2|{\psi }_c⟩⟨{\psi }_c|-{\mathbb{I}}_{d+1},$$

$$\widetilde{S_{HN3}}=S_{HN3}+|\text{mbox}loop⟩⟨\text{mbox}loop|\otimes \sum _{x_v=1}^N|x_v⟩⟨x_v|,$$

$$\widetilde{S_{HN4}}=S_{HN4}+|\text{mbox}loop⟩⟨\text{mbox}loop|\otimes \sum _{x_v=1}^N|x_v⟩⟨x_v|.$$

$${\mathcal{U}}_{HN3}$$

$${\mathcal{U}}_{HN4}$$

$$|{\psi }_{HN3}⟩$$

$$|{\psi }_{HN4}⟩$$

$$T_{HN3}$$

$$T_{HN4}$$

Full text

I Introduction

Searching an unsorted database for a target state has an optimal speed of $\mathcal{O}(\sqrt{N})$ in quantum computations, which is achieved by the famous Grover search grover1 ; grover2 ; radha1 ; korepin1 ; giri ; zhang . In this algorithm elements of the database of size, $N$ are associated with an orthonormal basis states of an $N$-dimensional Hilbert space. An initial state is prepared which has typically a small overlap of $\mathcal{O}(1/\sqrt{N})$ with the target state. Then amplitude amplification technique is exploited to evolve the initial state to the target state in $\mathcal{O}(\sqrt{N})$ oracle queries, which is quadratically faster than the best known exhaustive classical search of time complexity $\mathcal{O}(N)$. The efficiency of quantum computations over its classical counterpart was anticipated over three and half decades ago by Richard Feynman and Paul Benioff. Although Grover algorithm does not have exponential speedup in comparison to the classical algorithm it has become very much popular since it can be used as a subroutine in numerous applications.

The transition from combinatorial search problem such as Grover to searching a spatially distributed database seemed difficult at the beginning. As pointed out by Paul Benioff beni , it will take time for a quantum robot to move from one vertex to the other eventually losing the efficiency over classical search. Naive implementation of Grover algorithm on a 2-dimensional lattice requires $\mathcal{O}(\sqrt{N})$ oracle calls, but each Grover iteration needs $\mathcal{O}(\sqrt{N})$ time steps for the reflection operation making the total running time $\mathcal{O}(N)$, which is not better than classical search. Usually, physical database is modeled as a graph where vertices childs ; amba1 ; amba2 ; meyer ; amba4 are considered as the elements of the database. It is possible to search this graph for a target state faster than the classical search.

For example, one-dimensional periodic lattice can be searched for a target state in $\mathcal{O}(N)$ time steps giri1 with $\mathcal{O}(1)$ success probability by lackadaisical quantum walk followed by amplitude amplification. Although, regular quantum walk(without laziness) search in one-dimensional lattice is not efficient. In two-dimensional periodic lattice recursive algorithm followed by amplitude amplification can search for a target state in $\mathcal{O}\left(\sqrt{N}\log^{2}N\right)$ time am ; amba1 . Regular quantum walk followed by amplitude amplification can improve this running time to $\mathcal{O}\left(\sqrt{N}\log N\right)$ and Tulsi’s controlled quantum walks tulsi , lackadaisical quantum walks wong1 or other method amba3 can further improve it to $\mathcal{O}(\sqrt{N\log N})$. For $d$-dimensional lattice($d\geq 3$) optimal speed of $\mathcal{O}(\sqrt{N})$ amba2 ; childs1 is achieved by quantum walk search algorithm. Other examples of graphs where search by quantum walks have been successfully exploited include a complete graph with directed self-loops portugal , Sierpinski gasket patel , Hanoi network(HN) portugal3 ; portugal4 , hypercubes, hexagonal network portugal1 , triangular lattice portugal2 etc.

Hanoi network is interesting because it has one-dimensional periodic lattice as its main structure together with additional long-range edges. We know that quantum walks for search on the one-dimensional periodic lattice is not efficient. However, the additional long-range edges of the Hanoi network make the search by quantum walk efficient as shown in refs. portugal3 ; portugal4 . They used quantum walks followed by amplitude amplification with a modified Grover coin which can control the flow of probability flux on and outside of the main structure by a single parameter. Running times obtained by this modified method for HN3 and HN4 are $\mathcal{O}(N^{0.74})$ and $\mathcal{O}(N^{0.84})$ respectively. They also exploited Tulsi’s controlled quantum walks with regular Grover coin which provides running times of $\mathcal{O}(N^{0.62})$ and $\mathcal{O}(N^{0.65})$ respectively. Note that, the lackadaisical quantum walks followed by amplitude amplification can search for a target state on a one-dimensional periodic lattice. Therefore, it would be interesting to know how fast lackadaisical quantum walk can search for a target state on Hanoi network, which has more edges than the one-dimensional periodic lattice.

So, the purpose of this article is to study spatial search for a target state on Hanoi network with the help of recently introduced lackadaisical(lazy) quantum walks wong1 ; wong2 ; wong3 , which can improve the time complexity significantly. We also want to study quantum walk search with standard Grover coin. This article is arranged in the following fashion. First, we discuss the Hanoi network in section II. Then in section III we discuss quantum walk search with standard Grover coin in this network. In section IV lackadaisical quantum walk search on Hanoi network is discussed. We conclude in section V with a discussion.

II Hanoi network of degree three and four

Hanoi network(HN) has been described in detail in ref. boe as a network generated from the sequence of numbered disks of the Tower-of-Hanoi problem. It can have degree three and degree four. A one-dimensional lattice of size $N=2^{n}$ with periodic boundary conditions is the main structure of this network. Each vertex $1\leq x_{v}\leq 2^{n}$ has two edges connected to its two nearest neighbors. There are additional long-range edges, which give the small-world structure. In figure 1 the main structure with $N=16$ has been shown in black color. Long range edges are shown in blue color. We have added one directed self-loop(red) at each vertex of the Hanoi network for the purpose of spatial search by lackadaisical quantum walks.

Each vertex $x_{v}$ is located at position $j$ in a specific hierarchy denoted by $i$, which obeys the following relation

[TABLE]

where the pair $0\leq i\leq n$ and $0\leq j\leq j_{max}=\lfloor 2^{n-i-1}-1/2\rfloor$ uniquely identifies the vertex $x_{v}$. There are $N$ basis states

[TABLE]

expressed in terms of $x_{v}$ or alternatively in terms of $i,j$.

Hanoi network of degree three(HN3) has three edges at each vertex. Two of the edges which connect $|x_{v}\rangle$ to $|x_{v}-1\rangle$ and $|x_{v}+1\rangle$, come from the one-dimensional periodic lattice. Third edge connects vertices in a hierarchy, i.e., $|i,0\rangle$ is connected to $|i,1\rangle$, $|i,2\rangle$ is connected to $|i,3\rangle$, $\cdots$, and $|i,j_{max}-1\rangle$ is connected to $|i,j_{max}\rangle$. For $i=n-1$ and $i=n$ the third edge forms a directed self-loop, i.e., $|n-1,0\rangle$ and $|n,0\rangle$ connect to itself respectively. From fig. 1(a) we can see that there are two black edges and one blue long-range edge at each vertex. The long-range edge at [math] and $8$ forms a directed self-loop(blue).

Hanoi network of degree four(HN4) has four edges, out of which two come from the one-dimensional periodic lattice as the HN3 case. The rest of the two edges connecting $|i,j\rangle$ to $|i,j-1\rangle$ and $|i,j+1\rangle$ respectively, for $i\neq n-1,n$. There is one undirected self-loop at $|n-1,0\rangle$ and $|n,0\rangle$ respectively. From fig. 1(b) we can see that there are two black edges and two blue long-range edges at each vertex. Two long-range edges form an undirected blue self-loop at [math] and $8$ respectively.

III Quantum walk search on Hanoi network

Quantum walks is useful to search for a target state on Hanoi network as briefly mentioned in the introduction. For a detail discussion on efficient search in this network using modified Grover coin and by Tulsi’s controlled quantum walk see refs. portugal3 ; portugal4 . The $N$-dimensional Hilbert space, $\mathcal{H}_{V}$, of the vertices of Hanoi network is the database in quantum walks. We impose periodic boundary conditions on each hierarchy of the network. We assume that one of the basis states of the Hilbert space $\mathcal{H}_{V}$ is the target state. Our task is to find out the target state with the help of quantum walk search. There are $d$-directions in which the quantum walker can move, which form the basis states for the coin space $\mathcal{H}_{C}$. For HN3 $d=3$ and for HN4 $d=4$. The quantum walk, which is the rotation of the coin state followed by the shift operation on the vertex space, takes place on the tensor product space $\mathcal{H}=\mathcal{H}_{C}\times\mathcal{H}_{V}$. An initial state of the form

[TABLE]

is considered for the purpose of search by quantum walks, where $|x_{c}\rangle$ and $|x_{v}\rangle$ are the basis states of the coin space and vertex space respectively. In analogy with the classical random walks, in quantum case, the evolution operator as stated above has two components. One for the toss of the coin, which can be simulated by unitary operator $C$ acted on the coin space and the other for the shift of the walker from one vertex to the next adjacent vertex, which can be done by the shift operator $S$. Formally we can write the unitary evolution operator as

[TABLE]

which acts repeatedly on the initial state to let it evolve. To exploit this quantum walks to search for a target state we need to have a way to distinguish the target state $|t_{v}\rangle$ from the rest of the basis states. Usually it is done by using two different coins for the target state and for the rest of the non-target states respectively. So, let us use $C$ for the non-target states and $-C$ for the target state, which can be compactly expressed as

[TABLE]

Note that $\mathcal{C}$ acts on the tensor product space $\mathcal{H}$ as opposed to $C$ which only acts on the Hilbert space of the coin $\mathcal{H}_{C}$. We choose $C$ to be the Grover coin prepared from the basis states $|x_{c}\rangle$ of the coin space.

For the quantum search on the spacial region, we use flip-flop shift operator, which moves the walker from one vertex to the other and inverts the direction of the walker at the same time. For HN3 it is of the form

[TABLE]

and for HN4 one more edge is added to each vertex, which is incorporated into the shift operator as

[TABLE]

Note that states $|0\rangle,|1\rangle$ are the regular edges and $|2\rangle,|3\rangle$ are the long range edges of the networks, which form the coin space. HN3 has only one long range edge $|2\rangle$ and HN4 has two long range edges $|2\rangle,|3\rangle$.

The initial state $|\psi_{in}\rangle$ is evolved by repeated application of the evolution operator $\mathcal{U}=S\mathcal{C}$ until the probability of the target state is reached to its maximum. The final state thus obtained

[TABLE]

is not aligned with the target state with high fidelity for the quantum search on Hanoi network. In fact, $|\langle t_{v}|\psi_{f}\rangle|^{2}<<1$, which implies that we need amplitude amplification technique of Grover to amplify the probability amplitude of the target state to $\mathcal{O}(1)$ in the evolving state.

In figure 2 we present the result of the numerical simulation in log-log-scale for the quantum search of the target state $|4\rangle$ on Hanoi network by regular quantum walks(without self-loop). Running time to obtain the first peak of the success probability as a function of the number of vertices $N$ is shown in fig. 2(a) and its corresponding success probability of the first peak is plotted in fig. 2(b). Green and blue squares are the numerical data for HN3 and HN4 network respectively. From the curve(red dash) fitting, we see that for HN3 the running time and the success probability scale respectively as $\mathcal{O}(N^{0.57913867}\sqrt{\log N})$ and $\mathcal{O}(N^{-0.41444274})$. And for HN4 the running time and the success probability scale as $\mathcal{O}(N^{0.50823082}\sqrt{\log N})$ and $\mathcal{O}(N^{-0.28004801})$. After an appropriate number of amplitude amplifications, the running time for HN3 and HN4 are respectively given by

[TABLE]

This result indicates that the long-range edges facilitate the search by the quantum walks on Hanoi network. In the above analysis we assumed $aN^{b}\sqrt{\log N}$ and $c/N^{d}$ as the form of the functions for the running time and the success probability respectively for the curve fitting, where the constants $a,b,c,d$ are evaluated by the curve fit. If we consider $aN^{b}$ as the form for the running time, then we obtain green dash curves in figure 2(a) as the curve fit, which although does not fit well for small network sizes but for large network sizes fit well. In this case $T_{HN3}\approx\mathcal{O}(N^{0.83})$ and $T_{HN4}\approx\mathcal{O}(N^{0.69})$ are the running times with $\mathcal{O}(1)$ success probability.

IV LQW search on Hanoi network

In this section, we discuss how lackadaisical quantum walks can be exploited to search a target state on Hanoi network. It has been observed that the lackadaisical quantum walks can search target states in two-dimensional lattice in $\mathcal{O}(\sqrt{\frac{N}{M}\log\frac{N}{M}})$ time steps giri1 . In one-dimensional periodic lattice also it can be efficiently used for the search algorithm.

In this approach, a self-loop of weight $l$ is attached to each vertex of the Hanoi network as shown by the red self-loops in figure 1. The initial state becomes

[TABLE]

where the initial state of the coin is of the form

[TABLE]

Note that the state $|\mbox{loop}\rangle$ in the coin space is responsible for the laziness of the quantum walk. In order to search by lackadaisical quantum walks we first need to rotate the coin state $|\psi_{c}\rangle$ by a coin operator, which is followed by flip-flop shift operator acting on the vertex state. The Grover diffusion operator for the rotation of the coin state is given by

[TABLE]

Note that for the spatial search the modified coin (5) can be obtained replacing $C$ by eq. (12). The shift operators are also modified because of the self-loop term. For the Hanoi network of degree three and four, it can be respectively written as

[TABLE]

Note that the self-loop terms in the shift operator retain the probability amplitude at the same vertex. It works in favor of increasing the efficiency of the search algorithm when the shift operator acts on the target state. However, it works against the increase of efficiency of the search algorithm when the shift operator acts on the non-target states. Therefore we need to balance it in such a way that we obtain an overall increase of the efficiency of the search algorithm, which can be done by suitably controlling the weight $l$ of the self-loop.

The initial states are evolved by the repeated application of the unitary operators

[TABLE]

and we obtain the final states as

[TABLE]

In order to exploit the effect of the self-loop in search algorithm, we need to obtain the optimum values of the self-loop parameter $l$ for which the first peak of the success probability is maximum. We plot the success probability as a function of $Nl$ in figure 3(a) and the corresponding running time in figure 3(b) for two different lattice sizes $N=1024$(blue curves) and $N=4096$(red curves). Star curves refer to the data for HN3 network and thick cross curves refer to the data for HN4 network. We observe that the curves for the success probability maximize at $l=2.52/N$ for HN3 and at $l=3.48/N$ for HN4, which will be held fixed from now onward for the analysis of searching for a target state.

In figure 4 the numerical data for the search of the target state $|4\rangle$ on Hanoi network of degree three(HN3) and degree four(HN4) by lackadaisical quantum walks has been presented in log-log-scale. Green and blue square curves represent HN3 and HN4 respectively. The success probabilities(fig. 3(b)) approach to a constant $\mathcal{O}(0.94820898)$ and $\mathcal{O}(0.99439784)$ for HN3 and HN4 respectively for large $N$. From the best fit(red dashes) of the running time curves(fig. 2(a)) we obtain

[TABLE]

as the running time needed to perform the quantum spatial search on HN3 and HN4 respectively. The results are valid for any choices of the target vertex except for $x_{v}=2^{n-1}$ and $2^{n}$, which can not be found by (lackadaisical) quantum walk search. These two vertices are special, because the long-range edges form a self-loop, which prohibits the flow of probability flux from other neighboring vertices to these two vertices through the long-range edges.

Note that in the above analysis we assumed $aN^{b}\log N$ as the form of the function for the running time for curve fitting, where the constants $a,b$ are evaluated by the curve fit as done in the previous section. If we instead consider $aN^{b}$ as the form for the running time, then we obtain green dash curves in figure 4(a) as the curve fit, which does not fit well for small network sizes but for large network sizes fit well. In this case $T_{HN3}\approx\mathcal{O}(N^{0.66})$ and $T_{HN4}\approx\mathcal{O}(N^{0.57})$ are the running times with $\mathcal{O}(1)$ success probability.

V Conclusions

Quantum algorithm for search on a physical database is usually based on quantum walks. In $d\geq 3$-dimensional periodic lattice the time complexity of the quantum walk search algorithm can reach to the optimum value of $\mathcal{O}(\sqrt{N})$. However, for $d\leq 2$ the situation is different. For example, on $2$-dimensional periodic Cartesian lattice with degree four one can find a target state in $\mathcal{O}(\sqrt{N\log N})$ time steps. In one-dimensional periodic lattice, regular quantum walk search is not efficient. However using lackadaisical quantum walk followed by the amplitude amplification one can obtain a target state in $\mathcal{O}\left(N\right)$ time steps with $\mathcal{O}(1)$ success probability. Lackadaisical quantum walks is also used in two-dimensional periodic lattice to search for $M$ targets, which needs $\mathcal{O}\left(\sqrt{\frac{N}{M}\log\frac{N}{M}}\right)$ time steps. This numerical result has been tested upto $M=6$ in ref. giri1 .

Given the success of the lackadaisical quantum walks in searching target states in one- and two-dimensional periodic lattice we in this article considered another graph known as Hanoi network. We also studied the quantum search on this network using regular quantum walk followed by amplitude amplification. Hanoi network is different from the one-dimensional periodic lattice, because it has extra long-range edges on top of the regular two-edges at each vertex. It has been observed that these extra long-range edges enhance the efficiency of the quantum walk search. Our numerical simulation suggests that the running time for the regular quantum walks followed by amplitude amplification for the Hanoi network of degree three and four scales as $\mathcal{O}\left(N^{0.79}\sqrt{\log N}\right)$ and $\mathcal{O}\left(N^{0.65}\sqrt{\log N}\right)$, which are in the class of functions $\mathcal{O}(N^{0.83})$ and $\mathcal{O}(N^{0.69})$ respectively. Using the lackadaisical quantum walks as the search algorithm is even more efficient. It has been shown by the curve fitting in figure 4(a) that the running time scales as $\mathcal{O}\left(N^{0.57}\log N\right)$ and $\mathcal{O}\left(N^{0.50}\log N\right)$ for HN3 and HN4, which are in the class of functions $\mathcal{O}(N^{0.66})$ and $\mathcal{O}(N^{0.57})$ respectively. The class of function $\mathcal{O}(N^{0.66})$ is in agreement with the result $N^{d_{s}}$ obtained in ref. boe1 for the running time of HN3 network, where $d_{s}=0.652879$ is the spectral dimension of the network.

Acknowledgements

P. R. Giri is supported by International Institute of Physics, UFRN, Natal, Brazil. V. Korepin is grateful to SUNY Center of Quantum Information Science at Long Island for support. We thank Chen-Fu Chiang for his comments/suggestions.