We consider a model of quantum computation in which the set of elementary operations is limited to Clifford unitaries, the creation of the state $\mid 0⟩$, and qubit measurement in the computational basis. In addition, we allow the creation of a one-qubit ancilla in a mixed state $\rho$, which should be regarded as a parameter of the model. Our goal is to determine for which $\rho$ universal quantum computation (UQC) can be efficiently simulated. To answer this question, we construct purification protocols that consume several copies of $\rho$ and produce a single output qubit with higher polarization. The protocols allow one to increase the polarization only along certain “magic” directions. If the polarization of $\rho$ along a magic direction exceeds a threshold value (about 65%), the purification asymptotically yields a pure state, which we call a magic state. We show that the Clifford group operations combined with magic states preparation are sufficient for UQC. The connection of our results with the Gottesman-Knill theorem is discussed.

Ideal quantum algorithms usually assume that quantum computing is performed continuously by a sequence of unitary transformations. However, there always exist idle finite time intervals between consecutive operations in a realistic quantum computing process. During these delays, coherent errors will accumulate from the dynamical phases of the superposed wave functions. Here we explore the sensitivity of Shor’s quantum factoring algorithm to such errors. Our results clearly show a severe sensitivity of Shor’s factorization algorithm to the presence of delay times between successive unitary transformations. Specifically, in the presence of these coherent errors, the probability of obtaining the correct answer decreases exponentially with the number of qubits of the work register. A particularly simple phase-matching approach is proposed in this paper to avoid or suppress these coherent errors when using Shor’s algorithm to factorize integers. The robustness of this phase-matching condition is evaluated analytically and numerically for the factorization of several integers: 4, 15, 21, and 33.

We present a tight-binding approach for the calculation of quasiparticle energy levels in confined systems such as molecules. The method is based on Hedin’s GW approximation, in which the self-energy is given as the product of the Green’s function $\left(G\right)$ and the screened Coulomb interaction $\left(W\right)$. Key quantities in the GW formalism such as the microscopic dielectric function are expressed in a minimal basis of spherically averaged atomic orbitals. All necessary integrals are either precalculated or approximated without resorting to empirical data. The method is validated against first-principles results for benzene and anthracene, where good agreement is found for levels close to the frontier orbitals. Further, the size dependence of the quasiparticle gap is studied for conformers of the polyacenes $\left({\mathrm{C}}_{4n+2}{\mathrm{H}}_{2n+4}\right)$ up to $n=30$.

The $3p$ photoelectron spectrum and the subsequent low kinetic energy Auger decay spectrum of atomic Mn have been recorded with high resolution. Experimental spectra have been compared with the results of relativistic multiconfiguration Dirac-Fock calculations. According to computations, correlations strongly affect the decay spectrum, which thus serves as the most stringent test of our theory.

The ground-state energy of three-body ions $(M^{+},M^{+},m^{-})$ evolves when the like-charge constituents are given different masses. The comparison of $(m_1^{+},m_2^{+},m^{-})$ with the average of $(m_1^{+},m_1^{+},m^{-})$ and $(m_2^{+},m_2^{+},m^{-})$ reveals a competition between the symmetric term and the antisymmetric one. The former dominates in the Born-Oppenheimer regime such as the $(p,t,e)$ case, while the latter wins for ${\mathrm{H}}^{-}$-like systems with two negative light particles surrounding a heavy nucleus. A comparison is also made with the case of baryons in simple quark models with flavor independence.

The statement by Loubenets [Phys. Rev. A 69, 042102 (2004)] that separable states can violate classical probabilistic constraints is based on a misleading definition of classicality, which is much narrower than Bell’s concept of local hidden variables. In a Bell-type setting, the notion of classicality used by Loubenets [Phys. Rev. A 69, 042102 (2004)] corresponds to the assumption of perfect correlations if the same observable is measured on both sides. While it is obvious that most separable states do not satisfy this assumption, this does not constitute “nonclassical” behavior in any usual sense of the word.

In this Comment, we question the recent analysis of $(e,3e)$ experimental data obtained on He presented by Dorn et al. [Phys. Rev. A 68, 012715 (2003)]. The authors claimed to observe “a new, somewhat unexpected effect,” namely the “disappearance” of the second Born effects at large momentum transfer $(q\sim 2\mathrm{a}.\mathrm{u}.)$ to the target. Here we show that their claim is not supported by their data. We argue that these data are consistent with strong deviations from the first Born regime, quite similar to the deviations observed at small momentum transfer $(q\sim 0.7-0.9\mathrm{a}.\mathrm{u}.)$.

In this Reply, we demonstrate that at large momentum transfer $(\mid q\mid =2\mathrm{a.u.})$, within the experimentally accessible angular range of the ionized electrons, there is good agreement between experiment and the first Born calculation concerning the shape of the cross section. Furthermore, we show that the fact that our results for $(e,3e)$ double ionization seemingly “contradict well-accepted results in $(e,2e)$ single ionization” can be traced to a second Born process in which the projectile interacts with both target electrons in sequence. This process competes effectively with other double ionization mechanisms but is relatively unimportant for single ionization.

In several recent papers on entanglement in relativistic quantum systems and relativistic Bell’s inequalities, relativistic Bell-type two-particle states have been constructed in analogy to nonrelativistic states. These constructions do not have the form suggested by relativistic invariance of the dynamics. Two relativistic formulations of Bell-type states are shown for massive particles, one using the standard Wigner spin basis and one using the helicity basis. The construction hinges on the use of Clebsch-Gordan coefficients of the Poincaré group to reduce the direct product of two unitary irreducible representations (UIRs) into a direct sum of UIRs.

We study theoretically and experimentally the quantum properties of a type-II frequency-degenerate optical parametric oscillator below threshold with a quarter-wave plate inserted inside the cavity which induces a linear coupling between the orthogonally polarized signal and idler fields. This original device provides a good insight into general properties of two-mode Gaussian states, illustrated in terms of the covariance matrix. We report on the experimental generation of a two-mode squeezed vacuum on nonorthogonal quadratures depending on the plate angle. After a simple operation, the entanglement is maximized and put into standard form, i.e., quantum correlations and anticorrelations on orthogonal quadratures. A half sum of squeezed variances as low as $0.33\pm 0.02$, well below the unit limit for inseparability, is obtained and the entanglement measured by the entropy of formation.

Entanglement swapping of two-mode Gaussian states is investigated by use of quantum teleportation protocol of continuous variables. In the entanglement swapping protocol, we take the two-mode squeezed vacuum which is exposed in a thermal environment as a mixed-quantum-channel state and a two-mode standard form Gaussian state as an input state to be teleported. When the input state is pure, we find that the swapped state is inseparable if and only if both the input state and the quantum-channel state are entangled. When the input state is mixed and entangled, we find that the swapped state is inseparable only if the quantum-channel state is sufficiently much entangled. We also notice that the required minimum entanglement in the mixed quantum channel to maintain entanglement in the swapped state strongly depends on the global purity of the input state.

We observed rubidium dimer destruction by excitation of rubidium vapor with diode laser light tuned across the Rb ${\mathrm{D}}_2$ resonance line in a 2400 GHz tuning interval. The destruction was measured for rubidium atom concentrations in the $\left(1\text{–}9\right)\times {10}^{16}{\mathrm{cm}}^{-3}$ range, pump beam power up to 43 mW, and with a 5 Torr of the helium buffer gas. We discuss the physical mechanisms involved and specify the molecular pathways which may effectively lead to the observed dimer destruction.

Cross sections at low energies for vibrationally elastic and inelastic scattering, as well as electron attachment to $\mathrm{S}{\mathrm{F}}_6$, have been calculated using a multichannel effective range theory (ERT) with complex boundary conditions. The most active vibrational modes, the totally symmetric mode ${\nu }_1$ and the infrared active mode ${\nu }_3$, have been included in the calculation. The ERT parameters were fitted to reproduce the experimental total and attachment cross sections. Differential elastic and vibrational excitation cross sections have been measured at 30° and 135° using a spectrometer with hemispherical analyzers. The calculation reproduces correctly the magnitudes and shapes of the differential elastic and ${\nu }_1$, ${\nu }_3$, and $2{\nu }_1$ excitation cross sections, in particular the sharp structures at vibrational thresholds. The $s$- and $p$-wave phase shifts obtained in the present analysis differ from those recently derived by Field et al. [Phys. Rev. A, 69, 052716 (2004)].

Translational energy spectroscopy has been used to study one-electron capture by $200\text{–}2000\mathrm{eV}{\mathrm{amu}}^{-1}$ ${\mathrm{He}}^{2+}$ ions in ${\mathrm{CH}}_4$, ${\mathrm{C}}_2{\mathrm{H}}_4$, and ${\mathrm{C}}_2{\mathrm{H}}_6$. The main ${\mathrm{He}}^{+}(n,l)$ excited product channels have been identified and their relative importance assessed. Although significantly different patterns of behavior are observed, all three cases exhibit the highly selective nature of the electron capture process in spite of the large number of possible nondissociative and dissociative product channels associated with a wide spread in energy defects $\mathrm{\Delta }E$.

We study the Levinson theorem for a Dirac particle in an $n$-dimensional central field by use of the Green function approach, based on an analysis of the $n$-dimensional radial Dirac equation obtained through a simple algebraic derivation. We show that the zero-momentum phase shifts are related to the number of bound states with $\mid E\mid <m$ plus the number of half-bound states of zero momenta—i.e., $\mid E\mid =m$—which are denoted by finite, but not square-integrable, wave functions.

Atomic manganese ${}^{55}\mathrm{Mn}$ and chromium ${}^{52}\mathrm{Cr}$ are simultaneously loaded and confined in a magnetic trap. Using a cryogenic ${}^3\mathrm{He}$ buffer gas, ${10}^{11}$ manganese and ${10}^{12}$ chromium atoms are trapped at an initial temperature of 600 mK. The buffer gas is then pumped away, thermally isolating the sample. The Mn-Cr interspecies inelastic rate constant is measured to be ${\Gamma }_{\mathrm{Mn},\mathrm{Cr}}=1.5(\pm 0.2)\times {10}^{-13}{\mathrm{cm}}^3∕\mathrm{s}$.

The effect of intersubsystem coupling on the adiabaticity of composite systems and that of its subsystems is investigated. Similar to the adiabatic evolution defined for pure states, nontransitional evolution for mixed states is introduced; conditions for the nontransitional evolution are derived and discussed. An example that describes two coupled qubits is presented to detail the general presentation. The effects due to nonadiabatic evolution on the geometric phase are also presented and discussed.

We study the radial Schrödinger equation for a particle of mass $m$ in the field of a singular attractive $g^2∕r^4$ potential with particular emphasis on the bound-states problem. Using the regularization method of Beane et al. [Phys. Rev. A 64, 042103 (2001)], we solve analytically the corresponding “renormalization-group flow” equation. We find in agreement with previous studies that its solution exhibits a limit cycle behavior and has infinitely many branches. We show that a continuous choice for the solution corresponds to a given fixed number of bound states and to low-energy phase shifts that vary continuously with energy. We study in detail the connection between this regularization method and a conventional method modifying the short-range part of the potential with an infinitely repulsive hard core. We show that both methods yield bound-states results in close agreement even though the regularization method of Beane et al. does not include explicitly any new scale in the problem. We further illustrate the use of the regularization method in the computation of electron bound states in the field of neutral polarizable molecules without dipole moment. We find the binding energy of $s$-wave polarization bound electrons in the field of ${\mathrm{C}}_{60}$ molecules to be 17 meV for a scattering length corresponding to a hard-core radius of the size of the molecule radius $(\sim 3.37\mathrm{\AA })$. This result can be further compared with recent two-parameter fits using the Lennard-Jones potential yielding binding energies ranging from 3 to 25 meV.

A long-lived coherent state and nonlinear interaction have been experimentally demonstrated for the vibrational mode of a trapped ion. We propose an implementation of quantum computation using coherent states of the vibrational modes of trapped ions. Differently from earlier experiments, we consider a far-off resonance for the interaction between external fields and the ion in a bidimensional trap. By appropriate choices of the detunings between the external fields, the adiabatic elimination of the ionic excited level from the Hamiltonian of the system allows for beam splitting between orthogonal vibrational modes, production of coherent states, and nonlinear interactions of various kinds. In particular, this model enables the generation of the four coherent Bell states. Furthermore, all the necessary operations for quantum computation, such as preparation of qubits and one-qubit and controlled two-qubit operations, are possible. The detection of the state of a vibrational mode in a Bell state is made possible by the combination of resonant and off-resonant interactions between the ion and some external fields. We show that our read-out scheme provides highly efficient discrimination between all the four Bell states. We extend this to a quantum register composed of many individually trapped ions. In this case, operations on two remote qubits are possible through a cavity mode. We emphasize that our remote-qubit operation scheme does not require a high-quality factor resonator: the cavity field acts as a catalyst for the gate operation.

The results of diabatic hyperspherical close-coupling calculations are presented for the charge exchange of a negative muon from muonic hydrogen to oxygen and nitrogen for collision energies from ${10}^{-3}$ to ${10}^3\mathrm{eV}$. It is shown that converged results can be obtained using a much smaller number of channels than in the traditional adiabatic approach. For the energy range below 10 eV our results for nitrogen are in good agreements with the available experimental data and the recent calculations within hyperspherical elliptic coordinates. However, discrepancies were found in the case of oxygen, where a $p$-wave shape resonance is shown to contribute significantly to the cross sections. We show that for oxygen the $p$-wave resonance extends to a large volume and is sensitive to the many-body effect. Calculations including outer screening of the oxygen atom have been performed to illustrate the importance of this effect.