Topological Quantum Computing with Only One Mobile Quasiparticle

In a topological quantum computer, universal quantum computation is performed by dragging quasiparticle excitations of certain two dimensional systems around each other to form braids of their world lines in 2+1 dimensional space-time. In this paper we show that any such quantum computation that can be done by braiding $n$ identical quasiparticles can also be done by moving a single quasiparticle around n-1 other identical quasiparticles whose positions remain fixed.Comment: 4 pages, 5 figure

Braid Topologies for Quantum Computation

In topological quantum computation, quantum information is stored in states which are intrinsically protected from decoherence, and quantum gates are carried out by dragging particle-like excitations (quasiparticles) around one another in two space dimensions. The resulting quasiparticle trajectories define world-lines in three dimensional space-time, and the corresponding quantum gates depend only on the topology of the braids formed by these world-lines. We show how to find braids that yield a universal set of quantum gates for qubits encoded using a specific kind of quasiparticle which is particularly promising for experimental realization.Comment: 4 pages, 4 figures, minor revision

Parafermionic edge zero modes in Z_n-invariant spin chains

A sign of topological order in a gapped one-dimensional quantum chain is the existence of edge zero modes. These occur in the Z_2-invariant Ising/Majorana chain, where they can be understood using free-fermion techniques. Here I discuss their presence in spin chains with Z_n symmetry, and prove that for appropriate coupling they are exact, even in this strongly interacting system. These modes are naturally expressed in terms of parafermions, generalizations of fermions to the Z_n case. I show that parafermionic edge zero modes do not occur in the usual ferromagnetic and antiferromagnetic cases, but rather only when the interactions are chiral, so that spatial-parity and time-reversal symmetries are broken.Comment: 22 pages. v2: small changes, added reference

Fluctuating Cu-O-Cu Bond model of high temperature superconductivity in cuprates

Twenty years of extensive research has yet to produce a general consensus on the origin of high temperature superconductivity (HTS). However, several generic characteristics of the cuprate superconductors have emerged as the essential ingredients of and/or constraints on any viable microscopic model of HTS. Besides a Tc of order 100K, the most prominent on the list include a d-wave superconducting gap with Fermi liquid nodal excitations, a d-wave pseudogap with the characteristic temperature scale T*, an anomalous doping-dependent oxygen isotope shift, nanometer-scale gap inhomogeneity, etc.. The key role of planar oxygen vibrations implied by the isotope shift and other evidence, in the context of CuO2 plane symmetry and charge constraints from the strong intra-3d Coulomb repulsion U, enforces an anharmonic mechanism in which the oxygen vibrational amplitude modulates the strength of the in-plane Cu-Cu bond. We show, within a Fermi liquid framework, that this mechanism can lead to strong d-wave pairing and to a natural explanation of the salient features of HTS

Hard loss of stability in Painlev\'e-2 equation

A special asymptotic solution of the Painlev\'e-2 equation with small parameter is studied. This solution has a critical point $t_*$ corresponding to a bifurcation phenomenon. When $t<t_*$ the constructed solution varies slowly and when $t>t_*$ the solution oscillates very fast. We investigate the transitional layer in detail and obtain a smooth asymptotic solution, using a sequence of scaling and matching procedures

Hamiltonian Formulation of Quantum Error Correction and Correlated Noise: The Effects Of Syndrome Extraction in the Long Time Limit

We analyze the long time behavior of a quantum computer running a quantum error correction (QEC) code in the presence of a correlated environment. Starting from a Hamiltonian formulation of realistic noise models, and assuming that QEC is indeed possible, we find formal expressions for the probability of a faulty path and the residual decoherence encoded in the reduced density matrix. Systems with non-zero gate times (``long gates'') are included in our analysis by using an upper bound on the noise. In order to introduce the local error probability for a qubit, we assume that propagation of signals through the environment is slower than the QEC period (hypercube assumption). This allows an explicit calculation in the case of a generalized spin-boson model and a quantum frustration model. The key result is a dimensional criterion: If the correlations decay sufficiently fast, the system evolves toward a stochastic error model for which the threshold theorem of fault-tolerant quantum computation has been proven. On the other hand, if the correlations decay slowly, the traditional proof of this threshold theorem does not hold. This dimensional criterion bears many similarities to criteria that occur in the theory of quantum phase transitions.Comment: 19 pages, 5 figures. Includes response to arXiv:quant-ph/0702050. New title and an additional exampl

Quantum Measurements and Gates by Code Deformation

The usual scenario in fault tolerant quantum computation involves certain amount of qubits encoded in each code block, transversal operations between them and destructive measurements of ancillary code blocks. We introduce a new approach in which a single code layer is used for the entire computation, in particular a surface code. Qubits can be created, manipulated and non-destructively measured by code deformations that amount to `cut and paste' operations in the surface. All the interactions between qubits remain purely local in a two-dimensional setting.Comment: Revtex4, 6 figure

A quantum search for zeros of polynomials

A quantum mechanical search procedure to determine the real zeros of a polynomial is introduced. It is based on the construction of a spin observable whose eigenvalues coincide with the zeros of the polynomial. Subsequent quantum mechanical measurements of the observable output directly the numerical values of the zeros. Performing the measurements is the only computational resource involved

Strictly contractive quantum channels and physically realizable quantum computers

We study the robustness of quantum computers under the influence of errors modelled by strictly contractive channels. A channel $T$ is defined to be strictly contractive if, for any pair of density operators $\rho,\sigma$ in its domain, $\| T\rho - T\sigma \|_1 \le k \| \rho-\sigma \|_1$ for some $0 \le k < 1$ (here $\| \cdot \|_1$ denotes the trace norm). In other words, strictly contractive channels render the states of the computer less distinguishable in the sense of quantum detection theory. Starting from the premise that all experimental procedures can be carried out with finite precision, we argue that there exists a physically meaningful connection between strictly contractive channels and errors in physically realizable quantum computers. We show that, in the absence of error correction, sensitivity of quantum memories and computers to strictly contractive errors grows exponentially with storage time and computation time respectively, and depends only on the constant $k$ and the measurement precision. We prove that strict contractivity rules out the possibility of perfect error correction, and give an argument that approximate error correction, which covers previous work on fault-tolerant quantum computation as a special case, is possible.Comment: 14 pages; revtex, amsfonts, amssymb; made some changes (recommended by Phys. Rev. A), updated the reference