A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem

A quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. This quantum adiabatic behavior is the basis of a new class of algorithms for quantum computing. We test one such algorithm by applying it to randomly generated, hard, instances of an NP-complete problem. For the small examples that we can simulate, the quantum adiabatic algorithm works well, and provides evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.Comment: 15 pages, 6 figures, email correspondence to [email protected] ; a shorter version of this article appeared in the April 20, 2001 issue of Science; see http://www.sciencemag.org/cgi/content/full/292/5516/47

Revisiting Critical Vortices in Three-Dimensional SQED

We consider renormalization of the central charge and the mass of the ${\cal N}=2$ supersymmetric Abelian vortices in 2+1 dimensions. We obtain ${\cal N}=2$ supersymmetric theory in 2+1 dimensions by dimensionally reducing the ${\cal N}=1$ SQED in 3+1 dimensions with two chiral fields carrying opposite charges. Then we introduce a mass for one of the matter multiplets without breaking N=2 supersymmetry. This massive multiplet is viewed as a regulator in the large mass limit. We show that the mass and the central charge of the vortex get the same nonvanishing quantum corrections, which preserves BPS saturation at the quantum level. Comparison with the operator form of the central extension exhibits fractionalization of a global U(1) charge; it becomes 1/2 for the minimal vortex. The very fact of the mass and charge renormalization is due to a "reflection" of an unbalanced number of the fermion and boson zero modes on the vortex in the regulator sector.Comment: 24 pages, 2 figures Minor modifications, reference adde

Improved Error-Scaling for Adiabatic Quantum State Transfer

We present a technique that dramatically improves the accuracy of adiabatic state transfer for a broad class of realistic Hamiltonians. For some systems, the total error scaling can be quadratically reduced at a fixed maximum transfer rate. These improvements rely only on the judicious choice of the total evolution time. Our technique is error-robust, and hence applicable to existing experiments utilizing adiabatic passage. We give two examples as proofs-of-principle, showing quadratic error reductions for an adiabatic search algorithm and a tunable two-qubit quantum logic gate.Comment: 10 Pages, 4 figures. Comments are welcome. Version substantially revised to generalize results to cases where several derivatives of the Hamiltonian are zero on the boundar

RPAE versus RPA for the Tomonaga model with quadratic energy dispersion

Recently the damping of the collective charge (and spin) modes of interacting fermions in one spatial dimension was studied. It results from the nonlinear correction to the energy dispersion in the vicinity of the Fermi points. To investigate the damping one has to replace the random phase approximation (RPA) bare bubble by a sum of more complicated diagrams. It is shown here that a better starting point than the bare RPA is to use the (conserving) linearized time dependent Hartree-Fock equations, i.e. to perform a random phase approximation (with) exchange (RPAE) calculation. It is shown that the RPAE equation can be solved analytically for the special form of the two-body interaction often used in the Luttinger liquid framework. While (bare) RPA and RPAE agree for the case of a strictly linear disperson there are qualitative differences for the case of the usual nonrelativistic quadratic dispersion.Comment: 6 pages, 3 figures, misprints corrected; to appear in PRB7

Spontaneously Broken Spacetime Symmetries and Goldstone's Theorem

Goldstone's theorem states that there is a massless mode for each broken symmetry generator. It has been known for a long time that the naive generalization of this counting fails to give the correct number of massless modes for spontaneously broken spacetime symmetries. We explain how to get the right count of massless modes in the general case, and discuss examples involving spontaneously broken Poincare and conformal invariance.Comment: 4 pages; 1 figure; v2: minor corrections. version to appear on PR

Emergence of a confined state in a weakly bent wire

In this paper we use a simple straightforward technique to investigate the emergence of a bound state in a weakly bent wire. We show that the bend behaves like an infinitely shallow potential well, and in the limit of small bending angle and low energy the bend can be presented by a simple 1D delta function potential.Comment: 4 pages, 3 Postscript figures (uses Revtex); added references and rewritte

Brueckner-Goldstone perturbation theory for the half-filled Hubbard model in infinite dimensions

We use Brueckner-Goldstone perturbation theory to calculate the ground-state energy of the half-filled Hubbard model in infinite dimensions up to fourth order in the Hubbard interaction. We obtain the momentum distribution as a functional derivative of the ground-state energy with respect to the bare dispersion relation. The resulting expressions agree with those from Rayleigh-Schroedinger perturbation theory. Our results for the momentum distribution and the quasi-particle weight agree very well with those obtained earlier from Feynman-Dyson perturbation theory for the single-particle self-energy. We give the correct fourth-order coefficient in the ground-state energy which was not calculated accurately enough from Feynman-Dyson theory due to the insufficient accuracy of the data for the self-energy, and find a good agreement with recent estimates from Quantum Monte-Carlo calculations.Comment: 15 pages, 8 fugures, submitted to JSTA