Lachezar S. Georgiev

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

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

n\ensuremath{\ge}3

A universal conformal field theory approach to the chiral persistent currents in the mesoscopic fractional quantum Hall states

qubits we prove that the image of the braid group is a nontrivial 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.

Coulomb blockade in hierarchical quantum Hall droplets

Abstract 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.

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

Abstract We propose a general and compact scheme for the computation of the periods and amplitudes of the chiral persistent currents, magnetizations and magnetic susceptibilities in mesoscopic fractional quantum Hall disk samples threaded by Aharonov–Bohm magnetic field. This universal approach uses the effective conformal field theory for the edge states in the quantum Hall effect to derive explicit formulas for the corresponding partition functions in presence of flux. We point out the crucial role of a special invariance condition for the partition function, following from the Bloch–Byers–Yang theorem, which represents the Laughlin spectral flow. As an example we apply this procedure to the Z k parafermion Hall states and show that they have universal non-Fermi liquid behavior without anomalous oscillations. For the analysis of the high-temperature asymptotics of the persistent currents in the parafermion states we derive the modular S-matrices constructed from the S matrices for the u ( 1 ) sector and that for the neutral parafermion sector which is realized as a diagonal affine coset.

Thermoelectric properties of Coulomb-blockaded fractional quantum Hall islands

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 ± 1), and find a non-trivial pattern of conductance peaks. In particular, each one of them occurs with a characteristic multiplicity, which 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.

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

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 numbers 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 in terms of those for , as well as explicit formulae for almost all braid matrices for exchanges of Ising anyons. Finally we prove that the braid-group representations obtained from the multi-anyon Pfaffian wavefunctions 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 σ do indeed realize one of the two inequivalent spinor representations of the rotation group SO(2n) as conjectured by Nayak and Wilczek.

Thermopower in the Coulomb Blockade Regime for Laughlin Quantum Dots

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 sharing the same electric properties. We also propose a procedure for measuring the ratio r=vn/vc of the Fermi velocities of the neutral and charged edge modes for filling factor νH=5/2 from the power-factor data in the low-temperature limit.

Hilbert Space Decomposition for Coulomb Blockade in Fabry–Pérot Interferometers

We demonstrate that the two inequivalent spinor representations of the braid group , describing the exchanges of 2n+2 non-Abelian Ising anyons in the Pfaffian topological quantum computer, are equivalent from the 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.

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

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.