Continuous‐Time Quantum Walk on the Homogeneous Tree

We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.

Let ? d be a homogeneous tree in which every vertex has d + 1 neighbours, where d≥ 2. The contact process on such a tree is known to have three distinct phases. We consider the process on a finite subtree, namely the rooted tree of depth h and branching factor d, and relate the behaviour of the process on the infinite tree to its behaviour on the finite tree for large h. In the phase of strong survival, we show that with probability ɛ independent of h, the process on the subtree starting from a single infection survives for a time which is doubly exponential in h and almost exponential in the number of vertices of the finite tree. In the phase of weak survival on the infinite tree, the survival time on the finite tree is approximately linear in h. In the phase of no survival, the survival time on the finite tree is linear in h if one starts with all vertices initially infected, and bounded by a random variable (independent of h) with an exponential tail if one starts from a single infection.

Adding self-loops at each vertex of a graph improves the performance of quantum walks algorithms over loopless algorithms. Many works approach quantum walks to search for a single marked vertex. In this article, we experimentally address several problems related to quantum walk in the hypercube with self-loops to search for multiple marked vertices. We first investigate the quantum walk in the loopless hypercube. We saw that neighbor vertices are also amplified and that approximately $1/2$ of the system energy is concentrated in them. We show that the optimal value of $l$ for a single marked vertex is not optimal for multiple marked vertices. We define a new value of $l = (n/N)\cdot k$ to search multiple marked vertices. Next, we use this new value of $l$ found to analyze the search for multiple marked vertices non-adjacent and show that the probability of success is close to $1$. We also use the new value of $l$ found to analyze the search for several marked vertices that are adjacent and show that the probability of success is directly proportional to the density of marked vertices in the neighborhood. We also show that, in the case where neighbors are marked, if there is at least one non-adjacent marked vertex, the probability of success increases to close to $1$. The results found show that the self-loop value for the quantum walk in the hypercube to search for several marked vertices is $l = (n / N) \cdot k $.

Adding self-loops at each vertex of a graph improves the performance of quantum walks algorithms over loopless algorithms. Many works approach quantum walks to search for a single marked vertex. In this article, we experimentally address several problems related to quantum walk in the hypercube with self-loops to search for multiple marked vertices. We first investigate the quantum walk in the loopless hypercube. We saw that neighbor vertices are also amplified and that approximately $1/2$ of the system energy is concentrated in them. We show that the optimal value of $l$ for a single marked vertex is not optimal for multiple marked vertices. We define a new value of $l = (n/N)\cdot k$ to search multiple marked vertices. Next, we use this new value of $l$ found to analyze the search for multiple marked vertices non-adjacent and show that the probability of success is close to $1$. We also use the new value of $l$ found to analyze the search for several marked vertices that are adjacent and show that the probability of success is directly proportional to the density of marked vertices in the neighborhood. We also show that, in the case where neighbors are marked, if there is at least one non-adjacent marked vertex, the probability of success increases to close to $1$. The results found show that the self-loop value for the quantum walk in the hypercube to search for several marked vertices is $l = (n / N) \cdot k $.

We study how quantum walks can be used to find structural anomalies in graphs\nvia several examples. Two of our examples are based on star graphs, graphs with\na single central vertex to which the other vertices, which we call external\nvertices, are connected by edges. In the basic star graph, these are the only\nedges. If we now connect a subset of the external vertices to form a complete\nsubgraph, a quantum walk can be used to find these vertices with a quantum\nspeedup. Thus, under some circumstances, a quantum walk can be used to locate\nwhere the connectivity of a network changes. We also look at the case of two\nstars connected at one of their external vertices. A quantum walk can find the\nvertex shared by both graphs, again with a quantum speedup. This provides an\nexample of using a quantum walk in order to find where two networks are\nconnected. Finally, we use a quantum walk on a complete bipartite graph to find\nan extra edge that destroys the bipartite nature of the graph.\n

Quantum walks have been proven to be a useful framework in designing new quantum algorithms, of which the search algorithm is the most notable. Besides a general search for a special vertex, recent researches have shown that quantum walks can also be used to find structural anomalies. Suppose a vertex of complete graph KN is attached to a second graph G, then the kind of structure anomaly will break the symmetry of the complete graph. The search algorithm based on scattering quantum walk model is presented to speed up locating this kind of structure anomaly. The concepts of scattering quantum walk model and collapsed graphs are presented. The definition of the evolutionary operator, which is different from that of a general search, is given. Based on the specific definition of evolutionary operator and the obvious symmetry of complete graph, it is shown that the search space is confined to a low-dimensional collapsed space, and the initial state is chosen to lie in this subspace. To illustrate the evolutionary process of the search algorithm, an example is given in the case that G is a single vertex. Taking advantage of our earlier study on the evolutionary operator of coined quantum walks with Grover coin, calculations of the unitary operator in the collapsed space are greatly simplified. To quantify the evolutionary process of the algorithm, we use the matrix perturbation theory involving a perturbative approach to find the eigenvalues and eigenstates. It is the degenerate zeroth-order eigenvalue 0 = 1 that leads to the interesting parts of the Hilbert space. Most of the recent researches of searching the structure anomalies focus on star graph SN with an unspecified graph G attached to one of its external vertices, where the overall graph is divided into two parts by the central vertex. It is shown that quantum speedup will occur if and only if the eigenvalues associated with these two parts in the N limit are the same. In this paper, we find that the collapsed graph of complete graphs can also be divided into two parts by a single collapsed vertex. As these two parts roughly correspond to the initial state and the desired state respectively, the techniques and results in star graphs can be generalized to the collapse graph on complete graph. What is more, under our definition of unitary evolution operator these two parts in the N limit will always share the same eigenvalue, i.e. 0 = 1, no matter what the structure of graph G is. Based on this, we prove that the search algorithm can find the target vertex in O(N) time steps with a success probability of 1-O(1N). That is to say, the quantum search algorithm gains a quadratic speedup over classical counterpart.

Quantum walk is a widely used method in designing quantum algorithms. In this article, we consider the lackadaisical discrete-time quantum walk (DTQW) on strongly regular graphs (SRG). When there is a single marked vertex in a SRG, we prove that lackadaisical DTQW can find the marked vertex with asymptotic success probability 1, with a quadratic speedup compared to classical random walk. This paper deals with any parameter family of SRG and argues that, by adding self-loops with proper weights, the asymptotic success probability can reach 1. The running time and asymptotic success probability matches the result of continuous-time quantum walk, and improves the result of DTQW.

Consider a branching random walk on the ball of radius N in a homogeneous tree. We obtain precise asymptotics on the critical value and on the extinction time (in critical and subcritical cases) as N → ∞.

We make and generalize the observation that summing of probability amplitudes of a discrete-time quantum walk over partitions of the walking graph consistent with the step operator results in a unitary evolution on the reduced graph which is also a quantum walk. Since the effective walking graph of the projected walk is not necessarily simpler than the original, this may bring new insights into the dynamics of some kinds of quantum walks using known results from thoroughly studied cases like Euclidean lattices. We use abstract treatment of the walking space and walker displacements in aim for a generality of the presented statements. Using this approach we also identify some pathological cases in which the projection mapping breaks down. For walks on lattices, the operation typically results in quantum walks with hyper-dimensional coin spaces. Such walks can, conversely, be viewed as projections of walks on inaccessible, larger spaces, and their properties can be inferred from the parental walk. We show that this is is the case for a lazy quantum walk, a walk with large coherent jumps and a walk on a circle with a twisted boundary condition. We also discuss the relation of this theory to the time-multiplexing optical implementations of quantum walks. Moreover, this manifestly irreversible operation can, in some cases and with a minor adjustment, be undone, and a quantum walk can be reconstructed from a set of its projections.

Quantum walks are roughly analogous to classical random walks, and similar to classical walks they have been used to find new (quantum) algorithms. When studying the behavior of large graphs or combinations of graphs, it is useful to find the response of a subgraph to signals of different frequencies. In doing so, we can replace an entire subgraph with a single vertex with variable scattering coefficients. In this paper, a simple technique for quickly finding the scattering coefficients of any discrete-time quantum graph will be presented. These scattering coefficients can be expressed entirely in terms of the characteristic polynomial of the graph’s time step operator. This is a marked improvement over previous techniques which have traditionally required finding eigenstates for a given eigenvalue, which is far more computationally costly. With the scattering coefficients we can easily derive the “impulse response” which is the key to predicting the response of a graph to any signal. This gives us a powerful set of tools for rapidly understanding the behavior of graphs or for reducing a large graph into its constituent subgraphs regardless of how they are connected.

A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk, resolving a question that had been open for 15 years.

We prove that any two locally finite homogeneous trees with valency greater than 3 are bilipschitz equivalent. This implies that the quotienth1(G)/h k (G), whereh k (G) is thekthL2-Betti number ofG, is not a quasi-isometry invariant.

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set \(M\) consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time \({{\mathrm{HT}}}(P,M)\) of any reversible random walk \(P\) on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity \({\hbox {HT}}^{+}(P,M)\) which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk \(P\) and the absorbing walk \(P'\), whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk \(P\) is not state-transitive. We also provide algorithms in the cases when only approximations or bounds on parameters \(p_M\) (the probability of picking a marked vertex from the stationary distribution) and \({\hbox {HT}}^{+}(P,M)\) are known.

Finding or estimating the lowest eigenstate of quantum system Hamiltonians is an important problem for quantum computing, quantum physics, quantum chemistry, and material science. Several quantum computing approaches have been developed to address this problem. The most frequently used method is variational quantum eigensolver (VQE). Many quantum systems, and especially nanomaterials, are described using tight-binding Hamiltonians, but until now no quantum computation method has been developed to find the lowest eigenvalue of these specific, but very important, Hamiltonians. We address the problem of finding the lowest eigenstate of tight-binding Hamiltonians using quantum walks. Quantum walks is a universal model of quantum computation equivalent to the quantum gate model. Furthermore, quantum walks can be mapped to quantum circuits comprising qubits, quantum registers, and quantum gates and, consequently, executed on quantum computers. In our approach, probability distributions, derived from wave function probability amplitudes, enter our quantum algorithm as potential distributions in the space where the quantum walk evolves. Our results showed the quantum walker localization in the case of the lowest eigenvalue is distinctive and characteristic of this state. Our approach will be a valuable computation tool for studying quantum systems described by tight-binding Hamiltonians.

We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle, expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantum walk. As a corollary, we give another proof for localization of the quantum walk on homogeneous trees shown by Chisaki et al.