We consider a quantum walk with two marked vertices, sender and receiver, and analyze its application to perfect state transfer on complete bipartite graphs. First, the situation with both the sender and the receiver vertex in the same part of the graph is considered. We show that in this case the dynamics of the quantum walk is independent of the size of the second part and reduces to the one for the star graph where perfect state transfer is achieved. Second, we consider the situation where the sender and the receiver vertex are in the opposite parts of the graph. In such a case the state transfer with unit fidelity is achieved only when the parts have the same size.
A distinguishability operator is defined for the continuous-time quantum walk (CTQW) of a bipartite quantum walker on two simply connected graphs, $W_{G_i,G_j} = U_{G_i}\left(t\right) \otimes U_{G_j}\left(t'\right) - U_{G_j}\left(t'\right) \otimes U_{G_i}\left(t\right)$, where $U_{G_i}\left(t\right)$ is the unitary CTQW operator for a labeled graph $G_i$ over a time interval $t$. The null space of $W_{G_i,G_j}$ defines the vector space of initial bipartite states whose time development is either constant or only dependent on $t + t'$ and is invariant to which quantum walker subsystem goes with…
This paper investigates continuous-time quantum walks on directed bipartite graphs based on a graph's adjacency matrix. We prove that on bipartite graphs, probability transport between the two node partitions can be completely suppressed by tuning a model parameter $α$. We provide analytic solutions to the quantum walks for the star and circulant graph classes that are valid for an arbitrary value of the number of nodes $N$, time $t$ and the model parameter $α$. We discuss quantitative and qualitative aspects of quantum walks based on directed graphs and their undirected counterparts. Numerica…
In this paper, we introduce a discrete quantum walk model called bipartite walks. Bipartite walks include many known discrete quantum walk models, like arc-reversal walks, vertex-face walks. For the transition matrix of a quantum walk, there is a Hamiltonian associated with it. We will study the Hamiltonians of the bipartite walks. Let $S$ be a skew-symmetric matrix. We are mainly interested in the Hamiltonians of the form $iS$. We show that the Hamiltonian can be written as $iS$ if and only if the adjacency matrix of the bipartite graph is invertible. We show that arc-reversal walks and verte…
In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover's walks. Any discrete quantum walk is given by the powers of a unitary matrix $U$ indexed by arcs or edges of the underlying graph. The walk is periodic if $U^k=I$ for some positive integer $k$. Kubota has given a characterization of periodicity of Grover's walk when the walk is defined on a regular bipartite graph with at most five eigenvalues. We extend Kubota's results--if a biregular graph $G$ has eigenvalues whose squares are algebraic integers with degree at most two, w…
We determine connected bipartite regular graphs with four distinct adjacency eigenvalues that induce periodic Grover walks, and show that it is only $C_6$. We also show that there are only three kinds of the second largest eigenvalues of bipartite regular periodic graphs with five distinct eigenvalues. Using walk-regularity, we enumerate feasible spectra for such graphs.
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 $N$ vertices for one of $k$ 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 be…
We consider the random walk of spin-zero bipartite spins in the classicalized holographic tensor network of the ground state of a strongly coupled two-dimensional conformal field theory. The bipartite-spin distribution induces a metric in this network. In the steady state of the random walk, the induced metric gives the two-dimensional anti-de Sitter (AdS$_2$) space metric. We consider this distribution as a microscopic statistical model of the AdS$_2$ space metric.
We address perfect discrimination of two separable states. When available states are restricted to separable states, we can theoretically consider a larger class of measurements than the class of measurements allowed in quantum theory. The framework composed of the class of separable states and the above extended class of measurements is a typical example of general probabilistic theories. In this framework, we give a necessary and sufficient condition to discriminate two separable pure states perfectly. In particular, we derive measurements explicitly to discriminate two separable pure states…
Quantum walks on graphs have shown prioritized benefits and applications in wide areas. In some scenarios, however, it may be more natural and accurate to mandate high-order relationships for hypergraphs, due to the density of information stored inherently. Therefore, we can explore the potential of quantum walks on hypergraphs. In this paper, by presenting the one-to-one correspondence between regular uniform hypergraphs and bipartite graphs, we construct a model for quantum walks on bipartite graphs of regular uniform hypergraphs with Szegedy's quantum walks, which gives rise to a quadratic …
In this work, we investigate what kinds of quantum states are feasible to perform perfectly secure secret sharing, and present its necessary and sufficient conditions. We also show that the states are bipartite distillable for all bipartite splits, and hence the states could be distillable into the Greenberger-Horne-Zeilinger state. We finally exhibit a class of secret-sharing states, which have an arbitrarily small amount of bipartite distillable entanglement for a certain split.
Perfect state transfer between two marked vertices of a graph by means of discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that for the coined quantum walks on star graph and complete graph with self-loops perfect state transfer between the sender and receiver vertex is achieved for arbitrary number of vertices $N$ in $O(\sqrt{N})$ steps of the walk. Finally, we show that Szegedy's walk with queries on complete graph allows for state transfer with unit fidelity in the limit o…
In this paper, we consider a continuous-time quantum walk based search algorithm. We introduce equitable partition of the graph and perfect state transfer on it. By these two methods, we can calculate the success probability and the finding time of the search algorithm. In addition, we gave some examples of graphs that we can calculate the success probability and the finding time.
High-fidelity quantum state transfer is critical for quantum communication and scalable quantum computation. Current quantum state transfer algorithms on the complete bipartite graph, which are based on discrete-time quantum walk search algorithms, suffer from low fidelity in some cases. To solve this problem, in this paper we propose a two-stage quantum state transfer algorithm on the complete bipartite graph. The algorithm is achieved by the generalized Grover walk with one marked vertex. The generalized Grover walk's coin operators and the query oracles are both parametric unitary matrices,…
We show a perfect state transfer of an arbitrary unknown two-qubit state can be achieved via a discrete-time quantum walk with various settings of coin flippings, and extend this method to distribution of an arbitrary unknown multi-qubit entangled state between every pair of sites in the multi-dimensional network. Furthermore, we study the routing of quantum information on this network in a quantum walk architecture, which can be used as quantum information processors to communicate between separated qubits.
We systematically investigated perfect state transfer between antipodal nodes of discrete time quantum walks on variants of the cycles C_4, C_6 and C_8 for three choices of coin operator. Perfect state transfer was found, in general, to be very rare, only being preserved for a very small number of ways of modifying the cycles. We observed that some of our useful modifications of C_4 could be generalised to an arbitrary number of nodes, and present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for ap…
We propose a scheme for perfect transfer of an unknown qubit state via the discrete-time quantum walk on a line or a circle. For this purpose, we introduce an additional coin operator which is applied at the end of the walk. This operator does not depend on the state to be transferred. We show that perfect state transfer over an arbitrary distance can be achieved only if the walk is driven by an identity or a flip coin operator. Other biased coin operators and Hadamard coin allow perfect state transfer over finite distances only. Furthermore, we show that quantum walks ending with a perfect st…
Bell inequalities and nonlocality have been widely studied in one-dimensional quantum systems. As a kind of quantum correlation, it is expected that bipartite nonlocaity should be present in quantum systems, just as bipartite entanglement does. Surprisingly, for various models, two-qubit states do not violate Bell inequalities, i.e., they are local. Recently, it is realized that the results are related to the monogamy trade-off obeyed by bipartite Bell correlations, thus it is believed that for general translation invariant systems, two-qubit states should not violate the Bell inequality(Olive…
Previously it was shown that (almost) perfect state transfer can be achieved on the complete bipartite graph by a discrete-time coined quantum walk based algorithm when both the sender and receiver vertices are in the same partition of the graph and when the sender and receiver are in opposite partitions of the same size. By changing the coin operator, we analyze the state transfer problem and we show that it is still possible to achieve state transfer with high fidelity even when the sender and receiver are in different partitions with different sizes. Moreover, it is also possible to use an …
The no-signaling constraint on bi-partite correlations is reviewed. It is shown that in order to obtain non-trivial Bell-type inequalities that discern no-signaling correlations from more general ones, one must go beyond considering expectation values of products of observables only. A new set of nontrivial no-signaling inequalities is derived which have a remarkably close resemblance to the CHSH inequality, yet are fundamentally different. A set of inequalities by Roy and Singh and Avis et al., which is claimed to be useful for discerning no-signaling correlations, is shown to be trivially sa…
We introduce some new perfect state transfer and teleportation schemes by quantum walks with two coins. Encoding the transferred information in coin 1 state and alternatively using two coin operators, we can perfectly recover the information on coin 1 state at target position only by at most two times of flipping operation. Based on quantum walks with two coins either on the line or on the $N$-circle, we can perfectly transfer any qubit state. In addition, using quantum walks with two coins either on complete graphs or regular graphs, we can first implement perfect qudit state transfer by quan…
It is proven that recently introduced states with perfectly secure bits of cryptographic key (private states representing secure bit) [K. Horodecki et al., Phys. Rev. Lett. 94, 160502 (2005)] as well as its multipartite and higher dimension generalizations always represent distillable entanglement. The corresponding lower bounds on distillable entanglement are provided. We also present a simple alternative proof that for any bipartite quantum state entanglement cost is an upper bound on distillable cryptographic key in bipartite scenario.
We study how quantum walks can be used to find structural anomalies in graphs via several examples. Two of our examples are based on star graphs, graphs with a single central vertex to which the other vertices, which we call external vertices, are connected by edges. In the basic star graph, these are the only edges. If we now connect a subset of the external vertices to form a complete subgraph, a quantum walk can be used to find these vertices with a quantum speedup. Thus, under some circumstances, a quantum walk can be used to locate where the connectivity of a network changes. We also look…
We study perfect state transfer in Kendon's model of discrete quantum walks. In particular, we give a characterization of perfect state transfer purely in terms of the graph spectra, and construct an infinite family of $4$-regular circulant graphs that admit perfect state transfer. Prior to our work, the only known infinite families of examples were variants of cycles and diamond chains.
We study perfect state transfer in Grover walks, which are typical discrete-time quantum walk models. In particular, we focus on states associated to vertices of a graph. We call such states vertex type states. Perfect state transfer between vertex type states can be studied via Chebyshev polynomials. We derive a necessary condition on eigenvalues of a graph for perfect state transfer between vertex type states to occur. In addition, we perfectly determine the complete multipartite graphs whose partite sets are the same size on which perfect state transfer occurs between vertex type states, to…
We study electric quantum walks in two dimensions considering Grover, Alternate, Hadamard, and DFT quantum walks. In the Grover walk the behaviour under an electric field is easy to summarize: when the field direction coincides with the x or y axes, it produces a transient trapping of the probability distribution along the direction of the field, while when it is directed along the diagonals, a perfect 2D trapping is frustrated. The analysis of the alternate walk helps to understand the behaviour of the Grover walk as both walks are partially equivalent; in particular, it helps to understand t…
When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the n…
Coined Quantum Walks (QWs) are being used in many contexts with the goal of understanding quantum systems and building quantum algorithms for quantum computers. Alternative models such as Szegedy's and continuous-time QWs were proposed taking advantage of the fact that quantum theory seems to allow different quantized versions based on the same classical model, in this case, the classical random walk. In this work, we show the conditions upon which coined QWs are equivalent to Szegedy's QWs. Those QW models have in common a large class of instances, in the sense that the evolution operators ar…
We carryout a comparative study of spin distributions defined over the sphere for bipartite quantum spin assemblies. We analyse Einstein-Podolsky-Rosen-Bohm (EPRB) spin correlations in a spin-s singlet state using these distributions. We observe that in the classical limit EPRB spin distributions turn out to be delta functions, thus reflecting the perfect anticorrelation property of two spin vectors associated with a spin-s singlet state.
A discrete-time quantum walk is the quantum analogue of a Markov chain on a graph. Zhan [J. Algebraic Combin. 53(4):1187-1213, 2020] proposes a model of discrete-time quantum walk whose transition matrix is given by two reflections, using the face and vertex incidence relations of a graph embedded in an orientable surface. We show that the evolution of a general discrete-time quantum walk that consists of two reflections satisfies a Chebyshev recurrence, under a projection. For the vertex-face walk, we prove theorems about perfect state transfer and periodicity and give infinite families of ex…
Quantum walks are an analog of classical random walks in quantum systems. Quantum walks have smaller hitting times compared to classical random walks on certain types of graphs, leading to a quantum advantage of quantum-walks-based algorithms. An important feature of quantum walks is that they are accompanied by the excitation transfer from one site to another, and a moment of hitting the destination site is characterized by the maximum probability amplitude of observing the excitation on this site. It is therefore prospective to consider such problems as candidates for quantum advantage demon…
We provide a characterization of perfect state transfer in a quantum walk whose Hamiltonian is given by the normalized Laplacian. We discuss a connection between classical random walks and quantum walks only present in this model, and we also rule out several trees as candidates to host perfect state transfer.
In a general setting, we introduce a new bipartite state property sufficient for the validity of the perfect correlation form of the original Bell inequality for any three bounded quantum observables. A bipartite quantum state with this property does not necessarily exhibit perfect correlations. The class of bipartite states specified by this property includes both separable and nonseparable states. We prove analytically that, for any dimension d>2, every Werner state, separable or nonseparable, belongs to this class.
For a nonseparable bipartite quantum state violating the Clauser-Horne-Shimony-Holt (CHSH) inequality, we evaluate amounts of noise breaking the quantum character of its statistical correlations under any generalized quantum measurements of Alice and Bob. Expressed in terms of the reduced states, these new threshold bounds can be easily calculated for any concrete bipartite state. A noisy bipartite state, satisfying the extended CHSH inequality and the perfect correlation form of the original Bell inequality for any quantum observables, neither necessarily admits a local hidden variable model …
For linear combinations of quantum product averages in an arbitrary bipartite state, we derive new quantum Bell-form and CHSH-form inequalities with the right-hand sides expressed in terms of a bipartite state. This allows us to specify in a general setting bipartite state properties sufficient for the validity of a classical CHSH-form inequality and the perfect correlation form of the original Bell inequality for any bounded quantum observables. We also introduce a new general condition on a bipartite state and quantum observables sufficient for the validity of the original Bell inequality, i…
We characterize a close connection between the continuous-time quantum-walk model and a discrete-time quantum-walk version, based on the staggered model with Hamiltonians in a class of Cayley graphs, which can be considered as a discretization of continuous-time quantum walks. This connection provides examples of perfect state transfer and instantaneous uniform mixing in the staggered model. On the other hand, we provide some more examples of perfect state transfer and instantaneous uniform mixing in the staggered model that cannot be reproduced by the continuous-time model.
For a graph $G$ and a related symmetric matrix $M$, the continuous-time quantum walk on $G$ relative to $M$ is defined as the unitary matrix $U(t) = \exp(-itM)$, where $t$ varies over the reals. Perfect state transfer occurs between vertices $u$ and $v$ at time $τ$ if the $(u,v)$-entry of $U(τ)$ has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer: (1) If a $n$-vertex graph has perfect state transfer at time $τ$ relative to the Laplacian, then so does its complement if $nτ$ …
In this paper, we present an abstract model of continuous-time quantum walk (CTQW) based on Bernoulli functionals and show that the model has perfect state transfer (PST), among others. Let $\mathfrak{h}$ be the space of square integrable complex-valued Bernoulli functionals, which is infinitely dimensional. First, we construct on a given subspace $\mathfrak{h}_L \subset \mathfrak{h}$ a self-adjoint operator $Δ_L$ via the canonical unitary involutions on $\mathfrak{h}$, and by analyzing its spectral structure we find out all its eigenvalues. Then, we introduce an abstract model of CTQW with $\…
