The power of quantum systems on a line

We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one-dimensional MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum Church-Turing thesis and that quantum computers cannot efficiently solve QMA-complete problems) that there are one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses.Comment: 21 pages. v2 has numerous corrections and clarifications, and most importantly a new author, merged from arXiv:0705.4067. v3 is the published version, with additional clarifications, publisher's version available at http://www.springerlink.co

Non-equilibrium dynamics of quantum systems: order parameter evolution, defect generation, and qubit transfer

In this review, we study some aspects of the non-equilibrium dynamics of quantum systems. In particular, we consider the effect of varying a parameter in the Hamiltonian of a quantum system which takes it across a quantum critical point or line. We study both sudden and slow quenches in a variety of systems including one-dimensional ultracold atoms in an optical lattice, an infinite range ferromagnetic Ising model, and some exactly solvable spin models in one and two dimensions (such as the Kitaev model). We show that quenching leads to the formation of defects whose density has a power-law dependence on the quenching rate; the power depends on the dimensionalities of the system and of the critical surface and on some of the exponents associated with the critical point which is being crossed. We also study the effect of non-linear quenching; the power law of the defects then depends on the degree of non-linearity. Finally, we study some spin-1/2 models to discuss how a qubit can be transferred across a system.Comment: 36 pages, 14 figures; an updated version will be published in "Quantum Quenching, Annealing and Computation", Eds. A. Das, A. Chandra and B. K. Chakrabarti, Lect. Notes in Phys., Springer, Heidelberg (2009, to be published

Simulating 0+1 Dimensional Quantum Gravity on Quantum Computers: Mini-Superspace Quantum Cosmology and the World Line Approach in Quantum Field Theory

Quantum computers are a promising candidate to radically expand computational science through increased computing power and more effective algorithms. In particular quantum computing could have a tremendous impact in the field of quantum cosmology. The goal of quantum cosmology is to describe the evolution of the Universe through the Wheeler-DeWitt equation or path integral methods without having to first formulate a full theory of quantum gravity. The quantum computer provides an advantage in this endeavor because it can perform path integrals in Lorentzian space and does not require constructing contour integrations in Euclidean gravity. Also quantum computers can provide advantages in systems with fermions which are difficult to analyze on classical computers. In this study, we first employed classical computational methods to analyze a Friedmann-Robertson-Walker mini-superspace with a scalar field and visualize the calculated wave function of the Universe for a variety of different values of the spatial curvature and cosmological constant. We them used IBM's Quantum Information Science Kit Python library and the variational quantum eigensolver to study the same systems on a quantum computer. The framework can also be extended to the world line approach to quantum field theory.Comment: 5 pages, 4 figure

Line junction in a quantum Hall system with two filling fractions

We present a microscopic model for a line junction formed by counter or co-propagating single mode quantum Hall edges corresponding to different filling factors. The ends of the line junction can be described by two possible current splitting matrices which are dictated by the conditions of both lack of dissipation and the existence of a linear relation between the bosonic fields. Tunneling between the two edges of the line junction then leads to a microscopic understanding of a phenomenological description of line junctions introduced some time ago. The effect of density-density interactions between the two edges is considered, and renormalization group ideas are used to study how the tunneling parameter changes with the length scale. This leads to a power law variation of the conductance of the line junction with the temperature. Depending on the strength of the interactions the line junction can exhibit two quite different behaviors. Our results can be tested in bent quantum Hall systems fabricated recently.Comment: 9 pages including 4 figure

Critical Dynamics of Singlet Excitations in a Frustrated Spin System

We construct and analyze a two-dimensional frustrated quantum spin model with plaquette order, in which the low-energy dynamics is controlled by spin singlets. At a critical value of frustration the singlet spectrum becomes gapless, indicating a quantum transition to a phase with dimer order. This T=0 transition belongs to the 3D Ising universality class, while at finite temperature a 2D Ising critical line separates the plaquette and dimerized phases. The magnetic susceptibility has an activated form throughout the phase diagram, whereas the specific heat exhibits a rich structure and a power law dependence on temperature at the quantum critical point. We argue that the novel quantum critical behavior associated with singlet criticality discussed in this work can be relevant to a wide class of quantum spin systems, such as antiferromagnets on Kagome and pyrochlore lattices, where the low-energy excitations are known to be spin singlets, as well as to the CAVO lattice and several recently discovered strongly frustrated square-lattice antiferromagnets.Comment: 5 pages, 5 figures, additional discussion and figure added, to appear in Phys. Rev.

Abelian Conformal Field theories and Determinant Bundles

The present paper is the first in a series of papers, in which we shall construct modular functors and Topological Quantum Field Theories from the conformal field theory developed in [TUY]. The basic idea is that the covariant constant sections of the sheaf of vacua associated to a simple Lie algebra over Teichm\"uller space of an oriented pointed surface gives the vectorspace the modular functor associates to the oriented pointed surface. However the connection on the sheaf of vacua is only projectively flat, so we need to find a suitable line bundle with a connection, such that the tensor product of the two has a flat connection. We shall construct a line bundle with a connection on any family of pointed curves with formal coordinates. By computing the curvature of this line bundle, we conclude that we actually need a fractional power of this line bundle so as to obtain a flat connection after tensoring. In order to functorially extract this fractional power, we need to construct a preferred section of the line bundle. We shall construct the line bundle by the use of the so-called $bc$-ghost systems (Faddeev-Popov ghosts) first introduced in covariant quantization [FP]. We follow the ideas of [KNTY], but decribe it from the point of view of [TUY].Comment: A couple of typos correcte

Theory of defect production in nonlinear quench across a quantum critical point

We study defect production in a quantum system subjected to a nonlinear power law quench which takes it either through a quantum critical or multicritical point or along a quantum critical line. We elaborate on our earlier work [D. Sen, K. Sengupta, S. Mondal, \prl 101, 016806 (2008)] and present a detailed analysis of the scaling of the defect density $n$ with the quench rate $\tau$ and exponent \al for each of the above-mentioned cases. We also compute the correlation functions for defects generated in nonlinear quenches through a quantum critical point and discuss the dependence of the amplitudes of such correlation functions on the exponent \al. We discuss several experimental systems where these theoretical predictions can be tested.Comment: 9+ pages, v