Effects of non-local initial conditions in the Quantum Walk on the line

We report an enhancement of the decay rate of the survival probability when non-local initial conditions in position space are considered in the Quantum Walk on the line. It is shown how this interference effect can be understood analytically by using previously derived results. Within a restricted position subspace, the enhanced decay is correlated with a maximum asymptotic entanglement level while the normal decay rate corresponds to initial relative phases associated to a minimum entanglement level.Comment: 5 pages, 1 figure, Elsevier style, to appear in Physica

Quantum walks on two-dimensional grids with multiple marked locations

The running time of a quantum walk search algorithm depends on both the structure of the search space (graph) and the configuration of marked locations. While the first dependence have been studied in a number of papers, the second dependence remains mostly unstudied. We study search by quantum walks on two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [AKR05]. The original paper analyses one and two marked location cases only. We move beyond two marked locations and study the behaviour of the algorithm for an arbitrary configuration of marked locations. In this paper we prove two results showing the importance of how the marked locations are arranged. First, we present two placements of $k$ marked locations for which the number of steps of the algorithm differs by $\Omega(\sqrt{k})$ factor. Second, we present two configurations of $k$ and $\sqrt{k}$ marked locations having the same number of steps and probability to find a marked location

The tensor hypercontracted parametric reduced density matrix algorithm: coupled-cluster accuracy with O(r^4) scaling

Tensor hypercontraction is a method that allows the representation of a high-rank tensor as a product of lower-rank tensors. In this paper, we show how tensor hypercontraction can be applied to both the electron repulsion integral (ERI) tensor and the two-particle excitation amplitudes used in the parametric reduced density matrix (pRDM) algorithm. Because only O(r) auxiliary functions are needed in both of these approximations, our overall algorithm can be shown to scale as O(r4), where r is the number of single-particle basis functions. We apply our algorithm to several small molecules, hydrogen chains, and alkanes to demonstrate its low formal scaling and practical utility. Provided we use enough auxiliary functions, we obtain accuracy similar to that of the traditional pRDM algorithm, somewhere between that of CCSD and CCSD(T).Comment: 11 pages, 1 figur

Controlling discrete quantum walks: coins and intitial states

In discrete time, coined quantum walks, the coin degrees of freedom offer the potential for a wider range of controls over the evolution of the walk than are available in the continuous time quantum walk. This paper explores some of the possibilities on regular graphs, and also reports periodic behaviour on small cyclic graphs.Comment: 10 (+epsilon) pages, 10 embedded eps figures, typos corrected, references added and updated, corresponds to published version (except figs 5-9 optimised for b&w printing here

Multiquantum vibrational excitation of NO scattered from Au(111): quantitative comparison of benchmark data to Ab initio theories of nonadiabatic molecule-surface interactions.

Measurements of absolute probabilities are reported for the vibrational excitation of NO(v=0→1,2) molecules scattered from a Au(111) surface. These measurements were quantitatively compared to calculations based on ab initio theoretical approaches to electronically nonadiabatic molecule–surface interactions. Good agreement was found between theory and experiment (see picture; Ts=surface temperature, P=excitation probability, and E=incidence energy of translation)

Efficient quantum algorithms for simulating sparse Hamiltonians

We present an efficient quantum algorithm for simulating the evolution of a sparse Hamiltonian H for a given time t in terms of a procedure for computing the matrix entries of H. In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and |H| is bounded by a constant, we may select any positive integer $k$ such that the simulation requires O((\log^*n)t^{1+1/2k}) accesses to matrix entries of H. We show that the temporal scaling cannot be significantly improved beyond this, because sublinear time scaling is not possible.Comment: 9 pages, 2 figures, substantial revision

Quantum walks can find a marked element on any graph

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set $M$ consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time $HT(P,M)$ of any reversible random walk $P$ on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity $HT^+(\mathit{P,M})$ which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk $P$ and the absorbing walk $P'$, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk $P$ is not state-transitive. We also provide algorithms in the cases when only approximations or bounds on parameters $p_M$ (the probability of picking a marked vertex from the stationary distribution) and $HT^+(\mathit{P,M})$ are known.Comment: 50 page

Electron spin as a spectrometer of nuclear spin noise and other fluctuations

This chapter describes the relationship between low frequency noise and coherence decay of localized spins in semiconductors. Section 2 establishes a direct relationship between an arbitrary noise spectral function and spin coherence as measured by a number of pulse spin resonance sequences. Section 3 describes the electron-nuclear spin Hamiltonian, including isotropic and anisotropic hyperfine interactions, inter-nuclear dipolar interactions, and the effective Hamiltonian for nuclear-nuclear coupling mediated by the electron spin hyperfine interaction. Section 4 describes a microscopic calculation of the nuclear spin noise spectrum arising due to nuclear spin dipolar flip-flops with quasiparticle broadening included. Section 5 compares our explicit numerical results to electron spin echo decay experiments for phosphorus doped silicon in natural and nuclear spin enriched samples.Comment: Book chapter in "Electron spin resonance and related phenomena in low dimensional structures", edited by Marco Fanciulli. To be published by Springer-Verlag in the TAP series. 35 pages, 9 figure

Quantum-dot spin qubit and hyperfine interaction

We review our investigation of the spin dynamics for two electrons confined to a double quantum dot under the influence of the hyperfine interaction between the electron spins and the surrounding nuclei. Further we propose a scheme to narrow the distribution of difference in polarization between the two dots in order to suppress hyperfine induced decoherence.Comment: 12 pages, 3 figures; Presented as plenary talk at the annual DPG meeting 2006, Dresden (to appear in Advances in Solid State Physics vol. 46, 2006

Fractional recurrence in discrete-time quantum walk

Quantum recurrence theorem holds for quantum systems with discrete energy eigenvalues and fails to hold in general for systems with continuous energy. We show that during quantum walk process dominated by interference of amplitude corresponding to different paths fail to satisfy the complete quantum recurrence theorem. Due to the revival of the fractional wave packet, a fractional recurrence characterized using quantum P\'olya number can be seen.Comment: 10 pages, 11 figure : Accepted to appear in Central European Journal of Physic