Quantum divisibility test and its application in mesoscopic physics

We present a quantum algorithm to transform the cardinality of a set of charged particles flowing along a quantum wire into a binary number. The setup performing this task (for at most N particles) involves log_2 N quantum bits serving as counters and a sequential read out. Applications include a divisibility check to experimentally test the size of a finite train of particles in a quantum wire with a one-shot measurement and a scheme allowing to entangle multi-particle wave functions and generating Bell states, Greenberger-Horne-Zeilinger states, or Dicke states in a Mach-Zehnder interferometer.Comment: 9 pages, 5 figure

Quantum Abacus for counting and factorizing numbers

We generalize the binary quantum counting algorithm of Lesovik, Suslov, and Blatter [Phys. Rev. A 82, 012316 (2010)] to higher counting bases. The algorithm makes use of qubits, qutrits, and qudits to count numbers in a base 2, base 3, or base d representation. In operating the algorithm, the number n < N = d^K is read into a K-qudit register through its interaction with a stream of n particles passing in a nearby wire; this step corresponds to a quantum Fourier transformation from the Hilbert space of particles to the Hilbert space of qudit states. An inverse quantum Fourier transformation provides the number n in the base d representation; the inverse transformation is fully quantum at the level of individual qudits, while a simpler semi-classical version can be used on the level of qudit registers. Combining registers of qubits, qutrits, and qudits, where d is a prime number, with a simpler single-shot measurement allows to find the powers of 2, 3, and other primes d in the number n. We show, that the counting task naturally leads to the shift operation and an algorithm based on the quantum Fourier transformation. We discuss possible implementations of the algorithm using quantum spin-d systems, d-well systems, and their emulation with spin-1/2 or double-well systems. We establish the analogy between our counting algorithm and the phase estimation algorithm and make use of the latter's performance analysis in stabilizing our scheme. Applications embrace a quantum metrological scheme to measure a voltage (analog to digital converter) and a simple procedure to entangle multi-particle states.Comment: 23 pages, 15 figure

Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems

We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of given dimension n having m singular points. As a function of n,m, this bound turns out to be double exponential in the precise sense explained in the paper. As a corollary, we obtain a solution of the so called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeros of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko.Comment: Will appear in Annales de l'institut Fourier vol. 60 (2010

Histogram Tomography

In many tomographic imaging problems the data consist of integrals along lines or curves. Increasingly we encounter "rich tomography" problems where the quantity imaged is higher dimensional than a scalar per voxel, including vectors tensors and functions. The data can also be higher dimensional and in many cases consists of a one or two dimensional spectrum for each ray. In many such cases the data contain not just integrals along rays but the distribution of values along the ray. If this is discretized into bins we can think of this as a histogram. In this paper we introduce the concept of "histogram tomography". For scalar problems with histogram data this holds the possibility of reconstruction with fewer rays. In vector and tensor problems it holds the promise of reconstruction of images that are in the null space of related integral transforms. For scalar histogram tomography problems we show how bins in the histogram correspond to reconstructing level sets of function, while moments of the distribution are the x-ray transform of powers of the unknown function. In the vector case we give a reconstruction procedure for potential components of the field. We demonstrate how the histogram longitudinal ray transform data can be extracted from Bragg edge neutron spectral data and hence, using moments, a non-linear system of partial differential equations derived for the strain tensor. In x-ray diffraction tomography of strain the transverse ray transform can be deduced from the diffraction pattern the full histogram transverse ray transform cannot. We give an explicit example of distributions of strain along a line that produce the same diffraction pattern, and characterize the null space of the relevant transform.Comment: Small corrections from last versio