A State Distillation Protocol to Implement Arbitrary Single-qubit Rotations

An important task required to build a scalable, fault-tolerant quantum computer is to efficiently represent an arbitrary single-qubit rotation by fault-tolerant quantum operations. Traditionally, the method for decomposing a single-qubit unitary into a discrete set of gates is Solovay-Kitaev decomposition, which in practice produces a sequence of depth O(\log^c(1/\epsilon)), where c~3.97 is the state-of-the-art. The proven lower bound is c=1, however an efficient algorithm that saturates this bound is unknown. In this paper, we present an alternative to Solovay-Kitaev decomposition employing state distillation techniques which reduces c to between 1.12 and 2.27, depending on the setting. For a given single-qubit rotation, our protocol significantly lowers the length of the approximating sequence and the number of required resource states (ancillary qubits). In addition, our protocol is robust to noise in the resource states.Comment: 10 pages, 18 figures, 5 table

Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains

We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.Comment: to appear in SIAM Journal on Mathematical Analysi

Weakly regular Floquet Hamiltonians with pure point spectrum

We study the Floquet Hamiltonian: -i omega d/dt + H + V(t) as depending on the parameter omega. We assume that the spectrum of H is discrete, {h_m (m = 1..infinity)}, with h_m of multiplicity M_m. and that V is an Hermitian operator, 2pi-periodic in t. Let J > 0 and set Omega_0 = [8J/9,9J/8]. Suppose that for some sigma > 0: sum_{m,n such that h_m > h_n} mu_{mn}(h_m - h_n)^(-sigma) < infinity where mu_{mn} = sqrt(min{M_m,M_n)) M_m M_n. We show that in that case there exist a suitable norm to measure the regularity of V, denoted epsilon, and positive constants, epsilon_* & delta_*, such that: if epsilon |Omega_0| - delta_* epsilon and the Floquet Hamiltonian has a pure point spectrum for all omega in Omega_infinity.Comment: 35 pages, Latex with AmsAr

Effective Hamiltonians for atoms in very strong magnetic fields

We propose three effective Hamiltonians which approximate atoms in very strong homogeneous magnetic fields $B$ modelled by the Pauli Hamiltonian, with fixed total angular momentum with respect to magnetic field axis. All three Hamiltonians describe $N$ electrons and a fixed nucleus where the Coulomb interaction has been replaced by $B$-dependent one-dimensional effective (vector valued) potentials but without magnetic field. Two of them are solvable in at least the one electron case. We briefly sketch how these Hamiltonians can be used to analyse the bottom of the spectrum of such atoms.Comment: 43 page

Bound states and scattering in quantum waveguides coupled laterally through a boundary window

We consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width $\ell$ in the common boundary. We show that such a system has at least one bound state for any $\ell>0$. We find the corresponding eigenvalues and eigenfunctions numerically using the mode--matching method, and discuss their behavior in several situations. We also discuss the scattering problem in this setup, in particular, the turbulent behavior of the probability flow associated with resonances. The level and phase--shift spacing statistics shows that in distinction to closed pseudo--integrable billiards, the present system is essentially non--chaotic. Finally, we illustrate time evolution of wave packets in the present model.Comment: LaTeX text file with 12 ps figure

The Effects of Additives on the Physical Properties of Electroformed Nickel and on the Stretch of Photoelectroformed Nickel Components

The process of nickel electroforming is becoming increasingly important in the manufacture of MST products, as it has the potential to replicate complex geometries with extremely high fidelity. Electroforming of nickel uses multi-component electrolyte formulations in order to maximise desirable product properties. In addition to nickel sulphamate (the major electrolyte component), formulation additives can also comprise nickel chloride (to increase nickel anode dissolution), sulphamic acid (to control pH), boric acid (to act as a pH buffer), hardening/levelling agents (to increase deposit hardness and lustre) and wetting agents (to aid surface wetting and thus prevent gas bubbles and void formation). This paper investigates the effects of some of these variables on internal stress and stretch as a function of applied current density.Comment: Submitted on behalf of TIMA Editions (http://irevues.inist.fr/tima-editions