Edgar Feldman

Quantum searches on highly symmetric graphs

We study scattering quantum walks on highly symmetric graphs and use the walks to solve search problems on these graphs. The particle making the walk resides on the edges of the graph, and at each time step scatters at the vertices. All of the vertices have the same scattering properties except for a subset of special vertices. The object of the search is to find a special vertex. A quantum circuit implementation of these walks is presented in which the set of special vertices is specified by a quantum oracle. We consider the complete graph, a complete bipartite graph, and an M-partite graph. In all cases, the dimension of the Hilbert space in which the time evolution of the walk takes place is small (between three and six), so the walks can be completely analyzed analytically. Such dimensional reduction is due to the fact that these graphs have large automorphism groups. We find the usual quadratic quantum speedups in all cases considered.

Programmable quantum state discriminators with simple programs

We describe a class of programmable devices that can discriminate between two quantum states. We consider two cases. In the first, both states are unknown. One copy of each of the unknown states is provided as an input, or program, for the two program registers, and the data state, which is guaranteed to be prepared in one of the program states, is fed into the data register of the device. This device will then tell us, in an optimal way, which of the templates stored in the program registers the data state matches. In the second case, we know one of the states while the other is unknown. One copy of the unknown state is fed into the single program register, and the data state which is guaranteed to be prepared in either the program state or the known state, is fed into the data register. The device will then tell us, again optimally, whether the data state matches the template or is the known state. We determine two types of optimal devices. The first performs discrimination with minimum error, and the second performs optimum unambiguous discrimination. In all cases we first treat the simpler problem of only one copy of the data state and then generalize the treatment to n copies. In comparison to other works we find that providing n>1 copies of the data state yields higher success probabilities than providing n>1 copies of the program states.

Optimal unambiguous discrimination of two subspaces as a case in mixed-state discrimination

We show how to optimally unambiguously discriminate between two subspaces of a Hilbert space. In particular we suppose that we are given a quantum system in either the state [{psi}{sub 1}>, where [{psi}{sub 1}> can be any state in the subspace S{sub 1}, or [{psi}{sub 2}>, where [{psi}{sub 2}> can be any state in the subspace S{sub 2}, and our task is to determine in which of the subspaces the state of our quantum system lies. We do not want to make any error, which means that our procedure will sometimes fail if the subspaces are not orthogonal. This is a special case of the unambiguous discrimination of mixed states. We present the positive operator valued measures that solve this problem and several applications of this procedure, including the discrimination of multipartite states without classical communication.

Scattering theory and discrete-time quantum walks

We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each, and consider walks that proceed from one half line, through the graph, to the other. The probability of starting on one line and reaching the other after n steps can be expressed in terms of the transmission amplitude for the graph.

Modifying quantum walks : a scattering theory approach

We show how to construct discrete-time quantum walks on directed, Eulerian graphs. These graphs have tails on which the particle making the walk propagates freely, and this makes it possible to analyze the walks in terms of scattering theory. The probability of entering a graph from one tail and leaving from another can be found from the scattering matrix of the graph. We show how the scattering matrix of a graph that is an automorphic image of the original is related to the scattering matrix of the original graph, and we show how the scattering matrix of the reverse graph is related to that of the original graph. Modifications of graphs and the effects of these modifications are then considered. In particular we show how the scattering matrix of a graph is changed if we remove two tails and replace them with an edge or cut an edge and add two tails. This allows us to combine graphs, that is if we connect two graphs we can construct the scattering matrix of the combined graph from those of its parts. Finally, using these techniques, we show how two graphs can be compared by constructing a larger graph in which the two original graphs are in parallel, and performing a quantum walk on the larger graph. This is a kind of quantum walk interferometry.

Isoperimetric inequalities on curved surfaces

In this paper we extend the solutions of Lord Rayleigh’s and St. Venant’s conjectures to bounded simply connected domains on curved 2-dimensional Riemannian manifolds. The conjectures investigate the relation of the fundamental tone of a vibrating membrane with fixed boundary and the torsional rigidity of cylindrical beams to the respective areas of the membrane and the cross section of the beam. Both problems are related to the isoperimetric inequality relating the area of a bounded domain to the length of its boundary, and indeed the isoperimetric inequality will be the starting point of our work. To state our results we require some definitions. M will denote a 2-dimensional manifold with complete Ck, K > 2, Riemannian metric, {In, ,..., Q,} will be a collection of pairwise disjoint bounded simply connected domains in M, such that for each j = l,..., m the boundary of Qj , r, , is a simply closed continuous, piecewise regular curve in M (by regular we mean Ck, K > 1, and of maximal rank). We let Q = UE, szj , and F = uj”=, I’j be the boundary of 8. Let A denote the area of B and L the length of r with respect to the given Riemannian metric. We denote the Gauss curvature function of the Riemannian metric by K: M + R.

Cylinders on surfaces

(Note that when ~ = 1 the two inequalities coincide.) The proof will consist of two parts: (i) we show the validity in the universal covering of M, /~7/, of the construction given in Figure 3 in [2] (without the symmetry about the vertical geodesic) and then show, as in [2], that the top lateral geodesic in Figure 3 can intersect at most one of the side geodesics; (ii) will then consist of a comparison argument in the universal covering/(/ .

Quantum walks as a probe of structural anomalies in graphs

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 at the case of two stars connected at one of their external vertices. A quantum walk can find the vertex shared by both graphs, again with a quantum speedup. This provides an example of using a quantum walk in order to find where two networks are connected. Finally, we use a quantum walk on a complete bipartite graph to find an extra edge that destroys the bipartite nature of the graph.

Extracting Information from a Qubit by Multiple Observers: Toward a Theory of Sequential State Discrimination

We discuss sequential unambiguous state-discrimination measurements performed on the same qubit. Alice prepares a qubit in one of two possible states. The qubit is first sent to Bob, who measures it, and then on to Charlie, who also measures it. The object in both cases is to determine which state Alice sent. In an unambiguous state discrimination measurement, we never make a mistake, i.e., misidentify the state, but the measurement may fail, in which case we gain no information about which state was sent. We find that there is a nonzero probability for both Bob and Charlie to identify the state, and we maximize this probability. The probability that Charlies measurement succeeds depends on how much information about the state Alice sent is left in the qubit after Bobs measurement, and this information can be quantified by the overlap between the two possible states in which Bobs measurement leaves the qubit. This Letter is a first step toward developing a theory of nondestructive sequential quantum measurements, which could be useful in quantum communication schemes.

A geometric approach to quantum state separation

Probabilistic quantum state transformations can be characterized by the degree of state separation they provide. This, in turn, sets limits on the success rate of these transformations. We consider optimum state separation of two known pure states in the general case where the known states have arbitrary a priori probabilities. The problem is formulated from a geometric perspective and shown to be equivalent to the problem of finding tangent curves within two families of conics that represent the unitarity constraints and the objective functions to be optimized, respectively. We present the corresponding analytical solutions in various forms. In the limit of perfect state separation, which is equivalent to unambiguous state discrimination, the solution exhibits a phenomenon analogous to a second order symmetry breaking phase transition. We also propose a linear optics implementation of separation which is based on the dual rail representation of qubits and single-photon multiport interferometry.