Quantum Walk Search on the Complete Bipartite Graph
The coined quantum walk is a discretization of the Dirac equation of relativistic quantum mechanics, and it is the basis of many quantum algorithms. We investigate how it searches the complete bipartite graph of vertices for one of marked vertices with different initial states. We prove intriguing dependence on the number of marked and unmarked vertices in each partite set. For example, when the graph is irregular and the initial state is the typical uniform superposition over the vertices, then the success probability can vary greatly from one timestep to the next, even alternating between 0 and 1, so the precise time at which measurement occurs is crucial. When the initial state is a uniform superposition over the edges, however, the success probability evolves smoothly. As another example, if the complete bipartite graph is regular, then the two initial states are equivalent. Then if two marked vertices are in the same partite set, the success probability reaches , but if they are in different partite sets, it instead reaches . This differs from the complete graph, which is the quantum walk formulation of Grover’s algorithm, where the success probability with two marked vertices is . This reveals a contrast to the continuous-time quantum walk, whose evolution is governed by Schrödinger’s equation, which asymptotically searches the regular complete bipartite graph with any arrangement of marked vertices in the same manner as the complete graph.
The coined quantum walk is a spatial and temporal discretization of the Dirac equation of relativistic quantum mechanics Meyer (1996a, b); Aharonov et al. (2001), and it is a fundamental method for designing quantum algorithms Ambainis (2003), such as for searching Shenvi et al. (2003), solving element distinctness Ambainis (2004), and evaluating boolean formulas Ambainis et al. (2010). They are universal for quantum computing Lovett et al. (2010), meaning any quantum algorithm can be formulated as a quantum walk, and they have been implemented in a variety of physical systems, including nuclear magnetic resonance Ryan et al. (2005), optical waveguides Perets et al. (2008), trapped atoms Karski et al. (2009), and trapped ions Schmitz et al. (2009).
In a coined quantum walk, the vertices of a graph label orthonormal basis states of an -dimensional vertex Hilbert space. The walker also has an internal coin or spin degree of freedom indicating which direction the particle is pointing. Together, or denotes a particle at vertex pointing to vertex . A step of the quantum walk is . Here, is the Grover diffusion coin Shenvi et al. (2003) that inverts the amplitudes of the coin states of each vertex about their average, and is the flip-flop shift Ambainis et al. (2005) that causes the particle to hop and turn around, so . For a detailed definition of the coined quantum walk for both regular and irregular graphs, see Wong (2017).
Quantum walks are often used to explore how quantum computers, or quantum cellular automata, search a graph or network for marked nodes by querying an oracle Shenvi et al. (2003); Ambainis et al. (2005); Childs and Goldstone (2004), which flips the sign of the amplitudes at marked vertices. So the system evolves by repeated applications of
Typically, the initial state is chosen to be a uniform superposition over the vertices, so if one measures the initial state, the walker is found at each vertex with equal probability. If the graph is complete, meaning each vertex is adjacent to every other, the graph is equivalent to the unordered database of Grover’s algorithm Grover (1996). As in Grover’s algorithm, search using a quantum walk is accomplished in time, or if there are marked vertices, in time Ambainis et al. (2005); Childs and Goldstone (2004); Wong (2015).
In this paper, we consider search on the complete bipartite graph, an example of which is shown in Fig. 1. A bipartite graph can be partitioned into two vertex sets and with and vertices, respectively, such that vertices in the same vertex set are not adjacent to each other. If the bipartite graph is complete, then each vertex in one partite set is adjacent to all vertices in the other vertex set.
Search on the complete bipartite graph using a continuous-time quantum walk, where the evolution is governed by Schrödinger’s equation, has been previously explored in various cases. If the graph is regular, meaning , then it is a strongly regular graph, and search on it with a single marked vertex was considered in Janmark et al. (2014). Since it is regular, the graph Laplacian and adjacency matrix effect the same walk. If the graph is irregular, meaning , then search on it with the adjacency matrix and a single marked vertex was considered in Novo et al. (2015). In Wong et al. (2016), search with multiple marked vertices and both the Laplacian and adjacency matrix were investigated, and they showed that when walking with the adjacency matrix, a non-uniform initial state is superior to the typical uniform one. Finally, search on the complete bipartite graph was also explored using a scattering quantum walk Reitzner et al. (2009), and while their choice of parameters makes it equivalent to the coined quantum walk considered here, their initial state places the particle uniformly across partite set , pointing to set , so it does not start in the vertices at all.
Here, we consider two initial states. The first is the typical uniform superposition over all vertices . Then, if the position of the walker is measured, it has an equal probability of of being found at each vertex. This expresses our initial lack of information as to which vertex is marked, so they are guessed with equal probability. Since the discrete-time quantum walk also contains a coin state, the amplitude at each vertex is uniformly distributed among its directions. That is,
When the graph is regular, applying the quantum walk (without the search query ) to this initial state leaves it unchanged. That is, if , then . This is what we would expect, as applying the quantum walk alone does not yield any information about the marked vertex since we did not query the oracle, so our initial equal guess over all the vertices is unchanged.
The complete bipartite graph, however, can be irregular. In this case, is no longer stationary under the quantum walk. That is, when , then . So the system evolves, even though we have not gained any information about which vertices may be marked. So in this paper, we also consider the following initial state, which is a uniform superposition over the edges:
Here, the probability of pointing along each edge is the same. This state has the property that , which we desired. On the other hand, since vertices can have different numbers of edges, the probability of finding the particle at each vertex may not be uniform. A similar state was investigated for the continuous-time quantum walk in Wong et al. (2016).
In such spatial search problems Childs and Goldstone (2004), the underlying graph is assumed to be known. This is akin to knowing the physical arrangement of data, such as a tape drive’s data stored in a ribbon or a hard drive’s data arranged in cylinders. Given this, it is possible to construct the initial states and since they only require knowledge of the underlying graph; they can be prepared without knowledge of the marked vertices.
In the next Section, we explore search on the complete bipartite graph where the marked vertices are all in the same partite set, and we consider both starting states and . We find that for , the success probability at even and odd timesteps can vary greatly as the graph becomes more irregular. On the other hand, using , the success probability is smoother and consistently reaches . Following this, in Section III, we consider the case where the marked vertices lie in both partite sets. We will see that with either initial state, the success probability in a partite set only depends on the number of vertices (marked and unmarked) in that partite set. Again, can cause the success probability to vary greatly from one timestep to the next, whereas yields a smoother evolution that consistently reaches a success probability of in each partite set. Note when the graph is regular, the two initial states are equivalent, and we show that search behaves differently from the complete graph Wong (2015), which is Grover’s algorithm formulated as a quantum walk. This is different from the continuous-time quantum walk, which asymptotically searches the regular complete bipartite graph Wong et al. (2016) just like the complete graph Wong (2015).
Ii Marked Vertices in One Set
In this section, we consider the case where there are marked vertices, and they are all located in a single partite set, as shown in Fig. 2. Without loss of generality, we take them to be in set . With either initial state or , and evolution by (1), the system evolves in a four-dimensional (4D) subspace. This is because there are only three types of vertices, which we have labeled in Fig. 2: the marked vertices labeled , their adjacent vertices labeled , and their nonadjacent vertices labeled . A particle at an vertex can only point toward vertices, a particle at a vertex can point toward and/or vertices, and a particle at a vertex can only point toward vertices. Together, these yield the following orthonormal basis for the 4D subspace:
Using (9) from Prūsis et al. (2016), we can write the quantum walk operator in this 4D subspace. Furthermore, since only corresponds to a particle located at a marked vertex, the query is in the 4D subspace. Together, the search operator in the basis is
To find the evolution of the system for each starting state, we will need the eigenvectors and eigenvalues of , which are
where denotes transpose, and
Now let us consider each initial state separately.
ii.1 Uniform Initial State Over Vertices
Here, we consider the uniform state over the vertices (2) as the initial state. In the 4D subspace spanned by , it is
The system evolves by repeatedly multiplying this 4D vector by (4), and the success probability at time is given by . Calculating this numerically, is plotted in Fig. 3 with marked vertices and with various values of and . We see that when , as in Fig. (a)a, the success probability is somewhat smooth from one timestep to the next, reaching a maximum value of . But as the ratio of to becomes greater, as in Figs. (b)b and (c)c, the success probability from one timestep to the next becomes more and more jagged, with crests at even timesteps and troughs at odd timesteps. From Fig. (c)c, these crests and troughs can even alternate between (nearly) and . This indicates that the runtime must be known precisely—measuring the system one timestep later or one timestep earlier can result in a success probability of zero. Furthermore, Figs.a-c have the same values of and , so only is changing, yet they all peak in success probability after steps. Thus, their runtimes do not depend on . Next, Figs. (d)d and (e)e decrease the ratio of to , and again the success probability is jagged, but now it crests at odd timesteps and troughs at even timesteps. In Fig. (e)e, all the vertices in set are marked, and the success probability roughly alternates between and . Next, we prove these observations analytically.
To analytically determine the evolution, we express as a linear combination of the eigenvectors of (6):
Applying the search operator pulls out eigenvalues, so the state of the system at time is
Multiplying on the left by , substituting for the coefficients (, , , ), substituting for the eigenvalues (’s), and squaring, the success probability at even and odd timesteps is
Together, these analytical results agree perfectly with the numerical simulations in Fig. 3.
Now the runtime of the algorithm is the time at which and (8) reach their first maxima. They are
So the maximum success probability at even and odd timesteps occur right after each other. Furthermore, recall from (7) and (5) that only depends on and , so the runtime only depends on the number of marked and unmarked vertices in set , in agreement with Figs. 3a-c. Plugging these runtimes into (8), the corresponding maximum success probabilities are
and the corresponding success probabilities (10) are
This is summarized in the first row of Table 1, and it agrees with Fig. (a)a, where the success probability reaches at time . Note search on the complete graph, which is Grover’s algorithm formulated as a quantum walk, also reaches its maximum success probability at time Wong (2015). But its maximum success probability is when and when Wong (2015), so it only matches the regular complete bipartite graph for a single marked vertex. This result differs from the continuous-time quantum walk, where the complete graph and the regular complete bipartite graph behave the same way for large and , even with multiple marked vertices Wong et al. (2016).
Next, when , as in Figs. (b)b and (c)c, the success probability reaches a maximum value of and at respective runtimes and (11). These results agree numerically with Fig. (b)b, where the success probability reaches and at times and , respectively. Similarly, in Fig. (c)c, the success probability reaches and at times and , respectively. Since , , so we use the even success probability in the second row of Table 1.
As these cases demonstrate, large differences between and create large differences in the success probability at even and odd steps, and therefore, it becomes important to account for this when measuring the system.
ii.2 Uniform Initial State Over Edges
Now we instead consider the initial state (3) that is a uniform superposition over the edges. In the 4D subspace spanned by , it is
Then the success probability at time is . This is plotted in Fig. 4 with the same choices of , , and as Fig. 3, allowing for a direct comparison between the two initial states. In Figs. (a)a and (a)a, the graph is regular, so and are equal, and we get the same evolution for both initial states. For the remaining subfigures, the graph is irregular, and we get a different evolutions with from . First, the evolution with is much smoother compared to the jaggedness with . Second, the success probability consistently reaches a maximum of 1/2. Third, the evolutions in Figs. 4a-c are identical, so if the marked vertices are in set , the evolution does not depend on the number of vertices in . If we change the number of vertices in set , however, then the evolution does change, as shown in Figs. 4d-e.
To prove this analytically, we again express the initial state as a linear combination of the eigenvectors of the search operator :
Then the success probability at time is given by , and breaking it into even and odd timesteps, we get
The runtime of the algorithm is the time at which reaches its first maximum, which at even and odd timesteps is
So the maximum success probability is always , no matter which values and take. Even in the extreme case where there are only marked vertices in one set, the probability is always , as shown in Fig. (e)e instead of alternating between zero and one as seen in Fig. (e)e. These results are summarized in the first three rows of Table 1. Thus, if the graph is regular, the two initial states are identical, but if the graph is irregular, yields a better success probability than , although one must be careful to measure at an even timestep when and an odd timestep when .
Iii Marked Vertices in Both Sets
Now we move on to the case where there are marked vertices in partite set and marked vertices in partite set , as shown in Fig. 5. Then there are four types of vertices, and vertices can point to both marked and unmarked vertices in the other set. This leads to an 8D subspace spanned by the orthonormal basis states
In this basis, the quantum walk operator can be obtained using Eq.  from Prūsis et al. (2016), and since both and vertices are marked, the oracle is in this basis. Combining them, the search operator (1) is
where and .
In order to find the evolution of the system, we want to find the eigenvalues and eigenvectors of and express the initial state as a linear combination of these eigenvectors. The exact eigenvectors are complicated and make it difficult to discern the behavior of the algorithm, so we assume that and , otherwise we can classically sample for a marked vertex in each set in constant time, eliminating the need for a quantum search algorithm. With this assumption, the eigenvectors and eigenvalues of are asymptotically
The details of this derivation are given in Appendix A. Next, let us consider search with each of the two starting states, beginning with .
iii.1 Uniform Initial State Over Vertices
In the 8D basis, the typical initial state (2), which is a uniform superposition over the vertices, is
This exactly evolves by repeated applications of (12), and at time , the success probability in set is , and the success probability in set is , so the total success probability is . These are plotted in Fig. 6 with , , and various values of and . We see in Fig. (a)a that the peak success probability in and do not coincide, so the total success probability does not reach . In Fig. (b)b, however, they do align, and the success probability does reach . Finally, in Fig. (c)c, all the vertices in set are marked, and the graph is highly irregular. The total success probability is jagged, roughly at odd timesteps, and gradually increasing at even timesteps until it is also .
Now let us investigate these results analytically for large and . Then we can explain Figs. (a)a and (b)b, but unfortunately, Fig. (c)c has , so it does not satisfy large compared to . For large and , the initial state is asymptotically
Expressing this as a linear combination of the asymptotic eigenvectors of (13),
Applying multiplies each eigenvector by its eigenvalue, and adding the squares of the amplitudes in , , , and , the success probability at time is
To find the runtime and maximum success probability, we take the derivative of , set it equal to zero, and solve for the runtime. This does not have a closed-form solution, however. So instead, we evaluate the success probability in each partite set independently. The success probability in is