arXiv subjects

Lachezar S. Georgiev

Publications and source records attributed to Lachezar S. Georgiev.

At least 19 recordsLinked to original sources

arXiv2024

Braiding Fibonacci anyons

Fibonacci anyons provide the simplest possible model of non-Abelian fusion rules: [1] x [1] = [0] + [1]. We propose a conformal field theory construction of topological quantum registers based on Fibonacci anyons realized as quasiparticle excitations in the Z_3 parafermion fractional quantum Hall state. To this end, the results of Ardonne and Schoutens for the correlation function of n = 4 Fibonacci fields are extended to the case of arbitrary n (and 3 r electrons). Special attention is paid to the braiding properties of the obtained correlators. We explain in details the construction of a monodromy representation of the Artin braid group acting on n-point conformal blocks of Fibonacci anyons. For low n (up to n = 8), the matrices of braid group generators are displayed explicitly. A simple recursion formula makes it possible to extend without efforts the construction to any n. Finally, we construct N qubit computational spaces in terms of conformal blocks of 2N + 2 Fibonacci anyons.

hep-th↗

arXiv2024

Diagonal Coset Approach to Topological Quantum Computation with Fibonacci Anyons

We investigate a promising conformal field theory realization scheme for topological quantum computation based on the Fibonacci anyons, which are believed to be realized as quasiparticle excitations in the $\mathbb{Z}_3$ parafermion fractional quantum Hall state in the second Landau level with filling factor $ν=12/5$. These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the $n$-qubit topological quantum register in terms of $2n+2$ Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in $\mathbb{Z}_3$ parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.

quant-ph↗

arXiv2019

Exact modular $S$ matrix for the ${\mathbb Z}_k$ parafermion quantum Hall islands and measurement of non-Abelian anyons

Using the decomposition of rational conformal filed theory characters for the ${\mathbb Z}_k$ parafermion quantum Hall droplets for general $k=2,3, \ldots$, we derive analytically the full modular $S$ matrix for these states, including the $\widehat{u(1)}$ parts corresponding to the charged sector of the full conformal field theory and the neutral parafermion contributions corresponding to the diagonal affine coset models. This precise neutral-part parafermion $S$ matrix is derived from the explicit relations between the coset matrix and those for the numerator and denominator of the coset and the latter is expressed in compact form due to the level--rank duality between the affine Lie algebras $\widehat{su(k)_2}$ and $\widehat{su(2)_k}$. The exact results obtained for the $S$ matrix elements are expected to play an important role for identifying interference patterns of fractional quantum Hall states in Fabry-Pérot interferometers which can be used to distinguish between Abelian and non-Abelian statistics of quasiparticles localized in the bulk of fractional quantum Hall droplets as well as for nondestructive interference measurement of Fibonacci anyons which can be used for universal topological quantum computation.

math-ph↗

arXiv2016

Topological Quantum Computation with Non-Abelian Anyons in Fractional Quantum Hall States

We review the general strategy of topologically protected quantum information processing based on non-Abelian anyons, in which quantum information is encoded into the fusion channels of pairs of anyons and in fusion paths for multi-anyon states, realized in two-dimensional fractional quantum Hall systems. The quantum gates which are needed for the quantum information processing in these multi-qubit registers are implemented by exchange or braiding of the non-Abelian anyons that are at fixed positions in two-dimensional coordinate space. As an example we consider the Pfaffian topological quantum computer based on the fractional quantum Hall state with filling factor $ν_H=5/2$. The elementary qubits are constructed by localizing Ising anyons on fractional quantum Hall antidots and various quantum gates, such as the Hadamard gate, phase gates and CNOT, are explicitly realized by braiding. We also discuss the appropriate experimental signatures which could eventually be used to detect non-Abelian anyons in Coulomb blockaded quantum Hall islands.

cond-mat.str-el↗

arXiv2016

Thermoelectric characteristics of ${\mathbb Z}_k$ parafermion Coulomb islands

Using the explicit rational conformal field theory partition functions for the ${\mathbb Z}_k$ parafermion quantum Hall states on a disk we compute numerically the thermoelectric power factor for Coulomb-blockaded islands at finite temperature. We demonstrate that the power factor is rather sensitive to the neutral degrees of freedom and could eventually be used to distinguish experimentally between different quantum Hall states having identical electric properties. This might help us to confirm whether non-Abelian quasiparticles, such as the Fibonacci anyons, are indeed present in the experimentally observed quantum Hall states.

cond-mat.str-el↗

arXiv2015

Thermopower and thermoelectric power factor of ${\mathbb Z}_k$ parafermion quantum dots

Using the conformal field theory approach to the thermoelectric characteristics of fractional quantum Hall states, previously developed in Nucl. Phys. B 894 (2015) 284, we show that the thermoelectric power factor of Coulomb-blockaded islands, realized by point contacts in Fabry--Pérot interferometers in the $\mathbb{Z}_k$ parafermion Hall states, could give reliable signatures for distinguishing the topological orders of different quantum Hall states having identical electric properties. For example, while the conductance peak patterns in the Coulomb blockade regime for such states are practically indistinguishable for $v_n \ll v_c$ even at finite temperature, where $v_n$ and $v_c$ are the Fermi velocities of the neutral and charged modes respectively, the power factors $\mathcal{P}_T$ of the corresponding states are much more sensitive to the neutral modes. In particular, the smaller $r=v_n/v_c$ the bigger the asymmetries in the power factor which combined with the thermal broadening of the conductance peaks due to the neutral modes' multiplicities could give us the ultimate tool to figure out which of the competing quantum Hall universality classes are indeed realized in the experiments. We give a complete description of the power factor profiles in the $\mathbb{Z}_3$ and $\mathbb{Z}_4$ parafermion states with arbitrary number of quasiparticles localized in the bulk which could be useful for comparison with the experiments.

cond-mat.mes-hall↗

arXiv2015

Thermoelectric properties of Coulomb-blockaded fractional quantum Hall islands

We show that it is possible and rather efficient to compute at non-zero temperature the thermoelectric characteristics of Coulomb blockaded fractional quantum Hall islands, formed by two quantum point contacts inside of a Fabry-Perot interferometer, using the conformal field theory partition functions for the chiral edge excitations. The oscillations of the thermopower with the variation of the gate voltage as well as the corresponding figure-of-merit and power factors, provide finer spectroscopic tools which are sensitive to the neutral multiplicities in the partition functions and could be used to distinguish experimentally between different universality classes. We also propose a procedure for measuring the ratio r=v_n/v_c of the Fermi velocities of the neutral and charged edge modes for filling factor ν=5/2 from the power-factor data in the low-temperature limit.

cond-mat.str-el↗

arXiv2014

Thermopower in the Coulomb blockade regime for Laughlin quantum dots

Using the conformal field theory partition function of a Coulomb-blockaded quantum dot, constructed by two quantum point contacts in a Laughlin quantum Hall bar, we derive the finite-temperature thermodynamic expression for the thermopower in the linear-response regime. The low-temperature results for the thermopower are compared to those for the conductance and their capability to reveal the structure of the single-electron spectrum in the quantum dot is analyzed.

cond-mat.str-el↗

arXiv2011

Hilbert space decomposition for Coulomb blockade in Fabry--Pérot interferometers

We show how to construct the thermodynamic grand potential of a droplet of incompressible fractional quantum Hall liquid, formed inside of an electronic Fabry--Pérot interferometer, in terms of the conformal field theory disk partition function for the edge states in presence of Aharonov-Bohm flux. To this end we analyze in detail the algebraic structure of the edge states' Hilbert space and identify the effect of the variation of the flux. This allows us to compute, in the linear response approximation, all thermodynamic properties of the conductance in the regime when the Coulomb blockade is softly lifted by the change of the magnetic flux due to the weak coupling between the droplet and the two quantum point contacts.

math-ph↗

arXiv2010

Thermal broadening of the Coulomb blockade peaks in quantum Hall interferometers

We demonstrate that the differential magnetic susceptibility of a fractional quantum Hall disk, representing a Coulomb island in a Fabry--Perot interferometer, is exactly proportional to the island's conductance and its paramagnetic peaks are the equilibrium counterparts of the Coulomb blockade conductance peaks. Using as a thermodynamic potential the partition functions of the edge states' effective conformal field theory we find the positions of the Coulomb blockade peaks, when the area of the island is varied, the modulations of the distance between them as well as the thermal decay and broadening of the peaks when temperature is increased. The finite-temperature estimates of the peak's heights and widths could give important information about the experimental observability of the Coulomb blockade. In addition, the predicted peak asymmetry and displacement at finite temperature due to neutral multiplicities could serve to distinguish different fractional quantum Hall states with similar zero-temperature Coulomb blockade patterns.

cond-mat.mes-hall↗

arXiv2009

Computational equivalence of the two inequivalent spinor representations of the braid group in the Ising topological quantum computer

We demonstrate that the two inequivalent spinor representations of the braid group \B_{2n+2}, describing the exchanges of 2n+2 non-Abelian Ising anyons in the Pfaffian topological quantum computer, are equivalent from computational point of view, i.e., the sets of topologically protected quantum gates that could be implemented in both cases by braiding exactly coincide. We give the explicit matrices generating almost all braidings in the spinor representations of the 2n+2 Ising anyons, as well as important recurrence relations. Our detailed analysis allows us to understand better the physical difference between the two inequivalent representations and to propose a process that could determine the type of representation for any concrete physical realization of the Pfaffian quantum computer.

cond-mat.mes-hall↗

arXiv2009

Monodromy analysis of the computational power of the Ising topological quantum computer

We show that all quantum gates which could be implemented by braiding of Ising anyons in the Ising topological quantum computer preserve the n-qubit Pauli group. Analyzing the structure of the Pauli group's centralizer, also known as the Clifford group, for n\geq 3 qubits, we prove that the image of the braid group is a non-trivial subgroup of the Clifford group and therefore not all Clifford gates could be implemented by braiding. We show explicitly the Clifford gates which cannot be realized by braiding estimating in this way the ultimate computational power of the Ising topological quantum computer.

math-ph↗

arXiv2009

Coulomb Blockade in Hierarchical Quantum Hall Droplets

The degeneracy of energy levels in a quantum dot of Hall fluid, leading to conductance peaks, can be readily derived from the partition functions of conformal field theory. Their complete expressions can be found for Hall states with both Abelian and non-Abelian statistics, upon adapting known results for the annulus geometry. We analyze the Abelian states with hierarchical filling fractions, ν=m/(mp \pm 1), and find a non trivial pattern of conductance peaks. In particular, each one of them occurs with a characteristic multiplicity, that is due to the extended symmetry of the m-folded edge. Experimental tests of the multiplicity can shed more light on the dynamics of this composite edge.

cond-mat.mes-hall↗

arXiv2009

Ultimate braid-group generators for coordinate exchanges of Ising anyons from the multi-anyon Pfaffian wave functions

We give a rigorous and self-consistent derivation of the elementary braid matrices representing the exchanges of adjacent Ising anyons in the two inequivalent representations of the Pfaffian quantum Hall states with even and odd number of Majorana fermions. To this end we use the distinct operator product expansions of the chiral spin fields in the Neveu-Schwarz and Ramond sectors of the two-dimensional Ising conformal field theory. We find recursive relations for the generators of the irreducible representations of the braid group B_{2n+2} in terms of those for B_{2n}, as well as explicit formulas for almost all braid matrices for exchanges of Ising anyons. Finally we prove that the braid-group representations obtained from the multi-anyon Pfaffian wave functions are completely equivalent to the spinor representations of SO(2n+2) and give the equivalence matrices explicitly. This actually proves that the correlation functions of 2n chiral Ising spin fields sigma do indeed realize one of the two inequivalent spinor representations of the rotation group SO(2n) as conjectured by Nayak and Wilczek.

math-ph↗

arXiv2008

Topological Quantum Computation with the universal R matrix for Ising anyons

We show that the braid-group extension of the monodromy-based topological quantum computation scheme of Das Sarma et al. can be understood in terms of the universal R matrix for the Ising model giving similar results to those obtained by direct analytic continuation of multi-anyon Pfaffian wave functions. It is necessary, however, to take into account the projection on spinor states with definite total parity which is responsible for the topological entanglement in the Pfaffian topological quantum computer.

math-ph↗

arXiv2008

Implementation of Clifford gates in the Ising-anyon topological quantum computer

We give a general proof for the existence and realizability of Clifford gates in the Ising topological quantum computer. We show that all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates. We find that the braiding gates for two qubits exhaust the entire two-qubit Clifford group. Analyzing the structure of the Clifford group for n \geq 3 qubits we prove that the the image of the braid group is a non-trivial subgroup of the Clifford group so that not all Clifford gates could be implemented by braiding in the Ising topological quantum computation scheme. We also point out which Clifford gates cannot in general be realized by braiding.

quant-ph↗

arXiv2007

Towards a universal set of topologically protected gates for quantum computation with Pfaffian qubits

We review the topological quantum computation scheme of Das Sarma et al. from the perspective of the conformal field theory for the two-dimensional critical Ising model. This scheme originally used the monodromy properties of the non-Abelian excitations in the Pfaffian quantum Hall state to construct elementary qubits and execute logical NOT on them. We extend the scheme of Das Sarma et al. by exploiting the explicit braiding transformations for the Pfaffian wave functions containing 4 and 6 quasiholes to implement, for the first time in this context, the single-qubit Hadamard and phase gates and the two-qubit Controlled-NOT gate over Pfaffian qubits in a topologically protected way. In more detail, we explicitly construct the unitary representations of the braid groups B_4, B_6 and B_8 and use the elementary braid matrices to implement one-, two- and three-qubit gates. We also propose to construct a topologically protected Toffoli gate, in terms of a braid-group based Controlled-Controlled-Z gate precursor. Finally we discuss some difficulties arising in the embedding of the Clifford gates and address several important questions about topological quantum computation in general.

hep-th↗

arXiv2006

Topologically protected quantum gates for computation with non-Abelian anyons in the Pfaffian quantum Hall state

We extend the topological quantum computation scheme using the Pfaffian quantum Hall state, which has been recently proposed by Das Sarma et al., in a way that might potentially allow for the topologically protected construction of a universal set of quantum gates. We construct, for the first time, a topologically protected Controlled-NOT gate which is entirely based on quasihole braidings of Pfaffian qubits. All single-qubit gates, except for the pi/8 gate, are also explicitly implemented by quasihole braidings. Instead of the pi/8 gate we try to construct a topologically protected Toffoli gate, in terms of the Controlled-phase gate and CNOT or by a braid-group based Controlled-Controlled-Z precursor. We also give a topologically protected realization of the Bravyi-Kitaev two-qubit gate g_3.

cond-mat.mes-hall↗