Quantum walks on Cayley graphs

We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterise quantum walks on free groups and present partial results on more general cases. Some examples are given, including a family of quantum walks on the hypercube involving a Clifford Algebra.Comment: J. Phys. A (accepted for publication

Separable states are more disordered globally than locally

A remarkable feature of quantum entanglement is that an entangled state of two parties, Alice (A) and Bob (B), may be more disordered locally than globally. That is, S(A) > S(A,B), where S(.) is the von Neumann entropy. It is known that satisfaction of this inequality implies that a state is non-separable. In this paper we prove the stronger result that for separable states the vector of eigenvalues of the density matrix of system AB is majorized by the vector of eigenvalues of the density matrix of system A alone. This gives a strong sense in which a separable state is more disordered globally than locally and a new necessary condition for separability of bipartite states in arbitrary dimensions. We also investigate the extent to which these conditions are sufficient to characterize separability, exhibiting examples that show separability cannot be characterized solely in terms of the local and global spectra of a state. We apply our conditions to give a simple proof that non-separable states exist sufficiently close to the completely mixed state of $n$ qudits.Comment: 4 page

Fault-Tolerant Quantum Computation via Exchange interactions

Quantum computation can be performed by encoding logical qubits into the states of two or more physical qubits, and controlling a single effective exchange interaction and possibly a global magnetic field. This "encoded universality" paradigm offers potential simplifications in quantum computer design since it does away with the need to perform single-qubit rotations. Here we show that encoded universality schemes can be combined with quantum error correction. In particular, we show explicitly how to perform fault-tolerant leakage correction, thus overcoming the main obstacle to fault-tolerant encoded universality.Comment: 5 pages, including 1 figur

Hardness of approximation for quantum problems

The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of approximation for include, among others, a quantum version of the Succinct Set Cover problem, and a variant of the local Hamiltonian problem with hybrid classical-quantum ground states.Comment: 21 pages, 1 figure, extended abstract appeared in Proceedings of the 39th International Colloquium on Automata, Languages and Programming (ICALP), pages 387-398, Springer, 201

Parsimonious Segmentation of Time Series' by Potts Models

Typical problems in the analysis of data sets like time-series or images crucially rely on the extraction of primitive features based on segmentation. Variational approaches are a popular and convenient framework in which such problems can be studied. We focus on Potts models as simple nontrivial instances. The discussion proceeds along two data sets from brain mapping and functional genomics

Quantum walks on general graphs

Quantum walks, both discrete (coined) and continuous time, on a general graph of N vertices with undirected edges are reviewed in some detail. The resource requirements for implementing a quantum walk as a program on a quantum computer are compared and found to be very similar for both discrete and continuous time walks. The role of the oracle, and how it changes if more prior information about the graph is available, is also discussed.Comment: 8 pages, v2: substantial rewrite improves clarity, corrects errors and omissions; v3: removes major error in final section and integrates remainder into other sections, figures remove

Bounds for mixing time of quantum walks on finite graphs

Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov, Ambainis, Kempe and Vazirani (2001) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speed-up in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e., walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.Comment: This is the journal version (except formatting); it is a significant revision of the previous version, in particular, it contains a new result about the convergence of quantum walks with decoherence; 16 page

Rheological study of structural transitions in triblock copolymers in a liquid crystal solvent

Rheological properties of triblock copolymers dissolved in a nematic liquid crystal (LC) solvent demonstrate that their microphase separated structure is heavily influenced by changes in LC order. Nematic gels were created by swelling a well-defined, high molecular weight ABA block copolymer with the small-molecule nematic LC solvent 4-pentyl-4-cyanobiphenyl (5CB). The B midblock is a side-group liquid crystal polymer (SGLCP) designed to be soluble in 5CB and the A endblocks are polystyrene, which is LC-phobic and microphase separates to produce a physically cross-linked, thermoreversible, macroscopic polymer network. At sufficiently low polymer concentration a plateau modulus in the nematic phase, characteristic of a gel, abruptly transitions to terminal behavior when the gel is heated into its isotropic phase. In more concentrated gels, endblock aggregates persist into the isotopic phase. Dramatic changes in network structure are observed over small temperature windows (as little as 1 °C) due to tccche rapidly changing LC order near the isotropization point. The discontinuous change in solvent quality produces an abrupt change in viscoelastic properties for three polymers having different pendant mesogenic groups and matched block lengths