arXiv subjects

Jiannis K. Pachos

Publications and source records attributed to Jiannis K. Pachos.

At least 73 records · Page 4Linked to original sources

arXiv2014

Focus on topological quantum computation

Topological quantum computation started as a niche area of research aimed at employing particles with exotic statistics, called anyons, for performing quantum computation. Soon it evolved to include a wide variety of disciplines. Advances in the understanding of anyon properties inspired new quantum algorithms and helped in the characterisation of topological phases of matter and their experimental realisation. The conceptual appeal of topological systems as well as their promise for building fault-tolerant quantum technologies fuelled the fascination in this field. This `focus on' brings together several of the latest developments in the field and facilitates the synergy between different approaches.

quant-ph↗

arXiv2013

Entropic Barriers for Two-Dimensional Quantum Memories

Comprehensive no-go theorems show that information encoded over local two-dimensional topologically ordered systems cannot support macroscopic energy barriers, and hence will not maintain stable quantum information at finite temperatures for macroscopic timescales. However, it is still well motivated to study low-dimensional quantum memories due to their experimental amenability. Here we introduce a grid of defect lines to Kitaev's quantum double model where different anyonic excitations carry different masses. This setting produces a complex energy landscape which entropically suppresses the diffusion of excitations that cause logical errors. We show numerically that entropically suppressed errors give rise to super-exponential inverse temperature scaling and polynomial system size scaling for small system sizes over a low-temperature regime. Curiously, these entropic effects are not present below a certain low temperature. We show that we can vary the system to modify this bound and potentially extend the described effects to zero temperature.

quant-ph↗

arXiv2013

Seeing Majorana fermions in time-of-flight images of spinless fermions coupled by s-wave pairing

The Chern number, nu, as a topological invariant that identifies the winding of the ground state in the particle-hole space, is a definitive theoretical signature that determines whether a given superconducting system can support Majorana zero modes. Here we show that such a winding can be faithfully identified for any superconducting system (p-wave or s-wave with spin-orbit coupling) through a set of time-of-flight measurements, making it a diagnostic tool also in actual cold atom experiments. As an application, we specialize the measurement scheme for a chiral topological model of spinless fermions. The proposed model only requires the experimentally accessible s-wave pairing and staggered tunnelling that mimics spin-orbit coupling. By adiabatically connecting this model to Kitaev's honeycomb lattice model, we show that it gives rise to nu = \pm 1 phases, where vortices bind Majorana fermions, and nu=\pm 2 phases that emerge as the unique collective state of such vortices. Hence, the preparation of these phases and the detection of their Chern numbers provide an unambiguous signature for the presence of Majorana modes. Finally, we demonstrate that our detection procedure is resilient against most inaccuracies in experimental control parameters as well as finite temperature.

cond-mat.supr-con↗

arXiv2013

Abelian Chern-Simons-Maxwell theory from a tight binding model of spinless fermions

Abelian Chern-Simons-Maxwell theory can emerge from the bosonisation of the 2+1-dimensional Thirring model that describes interacting Dirac fermions. Here we show how the Thirring model manifests itself in the low energy limit of a two-dimensional tight-binding model of spinless fermions. To establish that we employ a modification of Haldane's model, where the "doubling" of fermions is rectified by adiabatic elimination. Subsequently, fermionic interactions are introduced that lead to the analytically tractable Thirring model. By local density measurements of the lattice fermions we can establish that for specific values of the couplings the model exhibits the confining 2+1-dimensional QED phase or a topological ordered phase that corresponds to the Chern-Simons theory. The implementation of the model as well as the measurement protocol are accessible with current technology of cold atoms in optical lattices.

cond-mat.str-el↗

arXiv2013

Simulating Dirac fermions with Abelian and non-Abelian gauge fields in optical lattices

In this work we present an optical lattice setup to realize a full Dirac Hamiltonian in 2+1 dimensions. We show how all possible external potentials coupled to the Dirac field can arise from perturbations of the existing couplings of the honeycomb lattice model, without the need of additional laser fields. This greatly simplifies the proposed implementations, requiring only spatial modulations of the intensity of the laser beams. We finally suggest several experiments to observe the properties of the Dirac field in the setup.

cond-mat.quant-gas↗

arXiv2012

Braiding Interactions in Anyonic Quantum Walks

The anyonic quantum walk is a dynamical model describing a single anyon propagating along a chain of stationary anyons and interacting via mutual braiding statistics. We review the recent results on the effects of braiding statistics in anyonic quantum walks in quasi-one dimensional ladder geometries. For anyons which correspond to spin-1/2 irreps of the quantum groups $SU(2)_k$, the non-Abelian species $(1<k<\infty)$ gives rise to entanglement between the walker and topological degrees of freedom which is quantified by quantum link invariants over the trajectories of the walk. The decoherence is strong enough to reduce the walk on the infinite ladder to classical like behaviour. We also present numerical results on mixing times of $SU(2)_2$ or Ising model anyon walks on cyclic graphs. Finally, the possible experimental simulation of the anyonic quantum walk in Fractional Quantum Hall systems is discussed.

quant-ph↗

arXiv2012

Lifetime of topological quantum memories in thermal environment

Here we investigate the effect lattice geometry has on the lifetime of two-dimensional topological quantum memories. Initially, we introduce various lattice patterns and show how the error-tolerance against bit-flips and phase-flips depends on the structure of the underlying lattice. Subsequently, we investigate the dependence of the lifetime of the quantum memory on the structure of the underlying lattice when it is subject to a finite temperature. Importantly, we provide a simple effective formula for the lifetime of the memory in terms of the average degree of the lattice. Finally, we propose optimal geometries for the Josephson junction implementation of topological quantum memories.

quant-ph↗

arXiv2012

Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model

We provide a comprehensive microscopic understanding of the nucleation of topological quantum liquids, a general mechanism where interactions between non-Abelian anyons cause a transition to another topological phase, which we study in the context of Kitaev's honeycomb lattice model. For non-Abelian vortex excitations arranged on superlattices, we observe the nucleation of several distinct Abelian topological phases whose character is found to depend on microscopic parameters such as the superlattice spacing or the spin exchange couplings. By reformulating the interacting vortex superlattice in terms of an effective model of Majorana fermion zero modes, we show that the nature of the collective many-anyon state can be fully traced back to the microscopic pairwise vortex interactions. Due to RKKY-type sign oscillations in the interactions, we find that longer-range interactions beyond nearest neighbor can influence the collective state and thus need to be included for a comprehensive picture. The omnipresence of such interactions implies that corresponding results should hold for vortices forming an Abrikosov lattice in a p-wave superconductor, quasiholes forming a Wigner crystal in non-Abelian quantum Hall states or topological nanowires arranged in regular arrays.

cond-mat.mes-hall↗

arXiv2011

Incoherent dynamics in the toric code subject to disorder

We numerically study the effects of two forms of quenched disorder on the anyons of the toric code. Firstly, a new class of codes based on random lattices of stabilizer operators is presented, and shown to be superior to the standard square lattice toric code for certain forms of biased noise. It is further argued that these codes are close to optimal, in that they tightly reach the upper bound of error thresholds beyond which no correctable CSS codes can exist. Additionally, we study the classical motion of anyons in toric codes with randomly distributed onsite potentials. In the presence of repulsive long-range interaction between the anyons, a surprising increase with disorder strength of the lifetime of encoded states is reported and explained by an entirely incoherent mechanism. Finally, the coherent transport of the anyons in the presence of both forms of disorder is investigated, and a significant suppression of the anyon motion is found.

quant-ph↗

arXiv2011

Conformal flatness, non-Abelian Kaluza-Klein and the Quaternions

Maximally symmetric manifolds with holonomy in the unitary quaternionic group Sp(d/4) emerge from the non-Abelian Kaluza-Klein reduction of conformally flat spaces. Thus, all special manifolds with constant properly `holonomy-related' sectional curvature, are in natural correspondence with conformally flat, possibly non-Abelian, Kaluza-Klein spaces.

math-ph↗

arXiv2011

Engineering complex topological memories from simple Abelian models

In three spatial dimensions, particles are limited to either bosonic or fermionic statistics. Two-dimensional systems, on the other hand, can support anyonic quasiparticles exhibiting richer statistical behaviours. An exciting proposal for quantum computation is to employ anyonic statistics to manipulate information. Since such statistical evolutions depend only on topological characteristics, the resulting computation is intrinsically resilient to errors. So-called non-Abelian anyons are most promising for quantum computation, but their physical realization may prove to be complex. Abelian anyons, however, are easier to understand theoretically and realize experimentally. Here we show that complex topological memories inspired by non-Abelian anyons can be engineered in Abelian models. We explicitly demonstrate the control procedures for the encoding and manipulation of quantum information in specific lattice models that can be implemented in the laboratory. This bridges the gap between requirements for anyonic quantum computation and the potential of state-of-the-art technology.

quant-ph↗

arXiv2011

Bringing order through disorder: Localization of errors in topological quantum memories

Anderson localization emerges in quantum systems when randomised parameters cause the exponential suppression of motion. Here we consider this phenomenon in topological models and establish its usefulness for protecting topologically encoded quantum information. For concreteness we employ the toric code. It is known that in the absence of a magnetic field this can tolerate a finite initial density of anyonic errors, but in the presence of a field anyonic quantum walks are induced and the tolerable density becomes zero. However, if the disorder inherent in the code is taken into account, we demonstrate that the induced localization allows the topological quantum memory to regain a finite critical anyon density, and the memory to remain stable for arbitrarily long times. We anticipate that disorder inherent in any physical realisation of topological systems will help to strengthen the fault-tolerance of quantum memories.

quant-ph↗

arXiv2010

Fractional quantum Hall effect in a U(1)xSU(2) gauge field

We consider the bosonic fractional quantum Hall effect in the presence of a non-Abelian gauge field in addition to the usual Abelian magnetic field. The non-Abelian field breaks the twofold internal state degeneracy, but preserves the Landau level degeneracy. Using exact diagonalization, we find that for moderate non-Abelian field strengths the system's behaviour resembles a single internal state quantum Hall system, while for stronger fields there is a phase transition to either two internal state behaviour or the complete absence of fractional quantum Hall plateaus. Usually the energy gap is reduced by the presence of a non-Abelian field, but some non-Abelian fields appear to slightly increase the gap of the $ν=1$ and $ν=3/2$ Read-Rezayi states.

cond-mat.mes-hall↗

arXiv2010

Energy gaps of Hamiltonians from graph Laplacians

The Cheeger inequalities give an upper and lower bound on the spectral gap of discrete Laplacians defined on a graph in terms of the geometric characteristics of the graph. We generalise this approach and we employ it to determine if a given discrete Hamiltonian with non-positive elements is gapped or not in the thermodynamic limit. First, we define the graph that corresponds to such a generic Hamiltonian. Then we present a suitable generalisation of the Cheeger inequalities that overcomes scaling deficiencies of the original version. By employing simple examples we illustrate how the generalised Cheeger inequalities can successfully identify gapped or gapless phases and we comment on the computational complexity of this approach.

quant-ph↗

arXiv2010

Cold atom simulation of interacting relativistic quantum field theories

We demonstrate that Dirac fermions self-interacting or coupled to dynamic scalar fields can emerge in the low energy sector of designed bosonic and fermionic cold atom systems. We illustrate this with two examples defined in two spacetime dimensions. The first one is the self-interacting Thirring model. The second one is a model of Dirac fermions coupled to a dynamic scalar field that gives rise to the Gross-Neveu model. The proposed cold atom experiments can be used to probe spectral or correlation properties of interacting quantum field theories thereby presenting an alternative to lattice gauge theory simulations.

cond-mat.str-el↗

arXiv2010

Universal quantum computation with a non-Abelian topological memory

An explicit lattice realization of a non-Abelian topological memory is presented. The correspondence between logical and physical states is seen directly by use of the stabilizer formalism. The resilience of the encoded states against errors is studied and compared to that of other memories. A set of non-topological operations are proposed to manipulate the encoded states, resulting in universal quantum computation. This work provides insight into the non-local encoding non-Abelian anyons provide at the microscopical level, with an operational characterization of the memories they provide.

quant-ph↗

arXiv2010

Statistical dynamics of a non-Abelian anyonic quantum walk

We study the single particle dynamics of a mobile non-Abelian anyon hopping around many pinned anyons on a surface. The dynamics is modelled by a discrete time quantum walk and the spatial degree of freedom of the mobile anyon becomes entangled with the fusion degrees of freedom of the collective system. Each quantum trajectory makes a closed braid on the world lines of the particles establishing a direct connection between statistical dynamics and quantum link invariants. We find that asymptotically a mobile Ising anyon becomes so entangled with its environment that its statistical dynamics reduces to a classical random walk with linear dispersion in contrast to particles with Abelian statistics which have quadratic dispersion.

quant-ph↗

arXiv2010

Topological phase transitions driven by gauge fields in an exactly solvable model

We demonstrate the existence of a new topologically ordered phase in Kitaev's honeycomb lattice model. This new phase appears due to the presence of a vortex lattice and it supports chiral Abelian anyons. We characterize the phase by its low-energy behavior that is described by a distinct number of Dirac fermions. We identify two physically distinct types of topological phase transitions and obtain analytically the critical behavior of the extended phase space. The Fermi surface evolution associated with the transitions is shown to be due to the Dirac fermions coupling to chiral gauge fields. Finally, we describe how the new phase can be understood in terms of interactions between the anyonic vortices.

cond-mat.str-el↗