A lower bound on the probability of error in quantum state discrimination

We give a lower bound on the probability of error in quantum state discrimination. The bound is a weighted sum of the pairwise fidelities of the states to be distinguished.Comment: 4 pages; v2 fixes typos and adds remarks; v3 adds a new referenc

On discontinuity waves and smooth waves in thermo-piezoelectric bodies

The solid body B under consideration is composed of a linear thermo-piezoelectric medium, i.e., a non-magnetizable linearly elastic dielectric medium that is heat conducting and not electric conducting; B has a natural conguration, say a placement of the three-dimensional Euclidean space that B can occupy with zero stress, uniform temperature and uniform electric field. Such natural conguration and state will be used as reference. We consider processes of B constituted by small displacements, thermal deviations and small electric fields (u; T; E) superposed to the reference state. A smooth singular surface (or discontinuity surface) of order r in the triple of fields (u; T; E) is referred to as a weak (thermo-piezoelectric) wave if r >=2. Any singular surface of order r>= 2 is characteristic (for the linear thermo-piezoelectric partial dierential equations). Then smooth waves are considered. (i) It is shown that the wavefront of a plane progressive wave is characteristic if and only if the wave is isothermal. (ii) The differential equations are characterized for standing waves of a general type and for the standing waves which are sinusoidal. The latter are isothermal, isentropic, have wavefronts which are characteristic, and their directions of propagation satisfy certain constitutive conditions. (iii) The differential equations for plane progressive waves which are reversible in time are characterized

Quantum walk speedup of backtracking algorithms

We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on n variables, or outputs that none exists, and whose corresponding tree contains T vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using O(sqrt(T) n^(3/2) log n) tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree. We also discuss how, for certain distributions on the inputs, the algorithm can lead to an exponential reduction in expected runtime.Comment: 23 pages; v2: minor changes to presentatio

A Green-Naghdi approach for thermo-electroelasticity

The constitutive relations of piezoelectric ceramics are essentially nonlinear since the so-called piezoelectric moduli depend on the induced strains. Pioneering papers in these topics dealt mainly with the isothermal case. In view of applications, however, thermal effects have to be taken into account in connection with thermo-electric behaviors. Here we briefly compare continuum theories for nonlinear thermoelettroelasticity. In particular we describe an extension of Green-Naghdi thermoelasticity theory for an electrically polarizable and finitely deformable heat conducting elastic continuumn, which interacts with the electric field. In this theory, unlike other, thermal waves propagate at a finite speed

The quantum complexity of approximating the frequency moments

The $k$'th frequency moment of a sequence of integers is defined as $F_k = \sum_j n_j^k$, where $n_j$ is the number of times that $j$ occurs in the sequence. Here we study the quantum complexity of approximately computing the frequency moments in two settings. In the query complexity setting, we wish to minimise the number of queries to the input used to approximate $F_k$ up to relative error $\epsilon$. We give quantum algorithms which outperform the best possible classical algorithms up to quadratically. In the multiple-pass streaming setting, we see the elements of the input one at a time, and seek to minimise the amount of storage space, or passes over the data, used to approximate $F_k$. We describe quantum algorithms for $F_0$, $F_2$ and $F_\infty$ in this model which substantially outperform the best possible classical algorithms in certain parameter regimes.Comment: 22 pages; v3: essentially published versio

On the propagation of plane waves in piezoelectromagnetic monoclinic crystals

In a piezoelectromagnetic crystalline medium belonging to the class 2 of the monoclinic crystallographic system we find some classes of piezoelectricity-induced electromagnetic waves. These are time harmonic plane waves propagating along the symmetry axis and depending only on the axial coordinate. There are two indepen- dent modes of propagation, one longitudinal and one transverse, with mechanical and electromagnetical couplings. The transverse mode admits as a particular case an electromagnetic wave with no associated elastic deformation