Session L30: Focus Session: Topologically Protected Qubits I

Non-Abelian Braiding of Lattice Bosons

We report on a numerical experiment in which we use time-dependent potentials to braid non-abelian quasiparticles. We consider lattice bosons in a uniform magnetic field within the fractional quantum Hall regime, where $\nu$, the ratio of particles to flux quanta, is near 1/2, 1 or 3/2. We introduce time-dependent potentials which move quasiparticle excitations around one another, explicitly simulating a braiding operation which could implement part of a gate in a quantum computation. We find that different braids do not commute for $\nu$ near $1$ and $3/2$, with Berry matrices respectively consistent with Ising and Fibonacci anyons. Near $\nu=1/2$, the braids commute.   

Parafermion braid statistics in quasi-one-dimensional networks

One dimensional systems with Majorana zero modes at phase boundaries may be thought of as physical realizations of a discrete quantum wire model first put forth by Kitaev [1]. Proposed methods for braiding such Majorana fermions in one-dimensional wire networks [2] have greatly expanded the set of plausible avenues toward topological quantum computation. Recently, a generalization of the Kitaev model to parafermion modes has been developed.[3] Here, we describe the transport of such parafermion modes along the chain by the adiabatic transformation of the Hamiltonian, analogous to the transport of Majorana fermion modes. We determine the (braid) transformations of the ground state sector allowed by the adiabatic exchange of the parafermion modes in wire networks. We show that, as with Majorana fermions, none of the parafermion braid sets are universal for quantum computation. Certain parafermion chain models, unlike Majorana fermion systems, become universal with the addition of measurement operations. We discuss possible physical realizations of the parafermion models. \\[4pt] [1] J Alicea et al., Nature Physics 7, 412-417 (2011) \\[0pt] [2] A. Kitaev, arXiv:cond-mat/0010440v2 \\[0pt] [3] P. Fendley, unpublished   

Ettingshausen effect due to Majorana modes

Due to the presence of Majorana fermions (zero mode) at the vortex core of topological superconductor, each vortex carries an extra entropy $s_0= k_B ln[2]/2$ that is independent of temperature. Utilizing this special property of Majorana fermions, one can show that the edge states appearing at the edge of a topological superconductor can be cooled (heated) due to the motion of the vortices. We will also discuss possible experimental setups to observe this cooling thermoelectric mechanism tied with the extra entropy carried by the vortex.   

Quantum information transfer between topological and spin qubit systems

In this talk I will introduce a method to coherently transfer quantum information, and to create entanglement, between topological qubits and conventional spin qubits. The transfer method uses gated control to transfer an electron (spin qubit) between a quantum dot and edge Majorana modes in adjacent topological superconductors. Because of the spin polarization of the Majorana modes, the electron transfer translates spin superposition states into superposition states of the Majorana system, and vice versa. Furthermore, I will discuss how a topological superconductor can be used to facilitate long-distance quantum information transfer and entanglement between spatially separated spin qubits. \\ References: \\ M. Leijnse, K. Flensberg, PRB 84, 140501(R) (2011) \\ M. Leijnse, K. Flensberg, PRL, in print, arXiv:1107.5703 \\   

Braiding anyons and communication between topological and non-topological systems

Quasi-particles with non-Abelian statistics are intriguing in both fundamental and applied physics. Here we propose a ``proof of principle'' experimental setup for braiding anyons and observing non-Abelian statistics using nearest-neighbor spin interactions inspired by the Kitaev honeycomb model. We also show an explicit method for teleportation between the topological and non-topological systems.   

Electrical manipulation of Majorana Fermions in an interdigitated superconductor-ferromagnet device

We show that a topological phase supporting Majorana fermions can form in a 2DEG adjacent to an interdigitated superconductor-ferromagnet structure. An advantage of this setup is that the 2DEG can inherit the required Zeeman splitting and superconductivity from a single interface, allowing one to utilize a wide class of 2DEG's including the surface states of bulk InAs. We demonstrate that the interdigitated device supports a robust topological phase when the finger spacing $\lambda$ is smaller than the Fermi wavelength $\lambda_F$. In this regime the electrons effectively see a ``smeared" Zeeman splitting and pairing field despite the interdigitation. The topological phase survives even in the opposite limit $\lambda<\lambda_F$, though with a reduced bulk gap. We also describe how to electrically generate a vortex in this setup to trap a Majorana mode, which can be detected through edge tunneling spectroscopy.   

Interacting topological phases in multiband nanowires

We show that semiconductor nanowires coupled to an s-wave superconductor provide a playground to study effects of interactions between different topological superconducting phases supporting Majorana zero-energy modes. We consider quasi-one dimensional system where the topological phases emerge from different transverse subbands in the nanowire. In a certain parameter space, we show that there is a multi-critical point in the phase diagram where the low-energy theory is equivalent to the one describing two coupled Majorana chains. We study effect of interactions as well as symmetry-breaking perturbations on the topological phase diagram in the vicinity of this multi-critical point. Our results shed light on the stability of the topological phase around the multi-critical point and have important implications for the experiments on Majorana nanowires.   

Magnetic Control of Majorana Edge Modes in Topological Insulator-Ferromagnet-Superconductor Heterostructures

A surface of a strong 3D topological insulator (TI) doped with ferromagnetic atoms can be spin-polarized and similar to a 2D quantum anomalous Hall state. If an s-wave superconductivity can be induced by proximity effect on such a surface, a 2D topological superconducting phase is obtained. If we consider a TI-ferromagnet(FM)-superconductor(SC) heterostructure, a 2D time-reversal symmetry breaking topological superconducting (TSC) phases with Majorana edge mode(s) will be realized. We demonstrate that the existence of the edge modes critically depend on the combination of the directions and magnitudes of spin polarization on all surfaces, and that a model describing the states on only one surface is insufficient. We find that the number, the positions and the chirality of these edge modes corresponding to various TSC phases can be engineered by controlling the ferromagnetism on different surfaces. Our results are obtained by self-consistently solving for the edge modes in a 3D lattice model for topological insulator in contact with an s-wave BCS superconductor. We also offer an analysis to illustrate the underlying physics, using an effective 2D theory for the surface states.   

Self-consistent Study of Majorana Fermions on a Topological Insulator Surface $\pi$ Junction

It has been proposed that a Josephson $\pi$ junction that resides on the 2D surface of a 3D topological insulator (TI) is host to the Majorana fermion. We present a microscopic study of the $\pi$ junction TI surface through self-consistent calculations with the Bogoliubov-de Gennes equation. We calculate the order parameter, the singlet correlation function along with the energy spectrum, local density of states, and the spectral function. We also show the evolution of the energy dispersion of Majorana fermion as a function of the TI chemical potential.   

Unpaired Majorana fermions in a layered topological superconductor

We study the conditions for the existence of unpaired Majorana modes at the ends of vortex lines or the side edges of a layered topological superconductor. We show that the problem is mapped to that of a general Majorana chain and extend Kitaev's condition for the existence of its nontrivial phase by providing an additional condition when a supercurrent flows in the chain. Unpaired Majorana bound states may exist in a vortex line that threads the layers if the spin-orbit coupling has certain in-layer components but, interestingly, only if a nonzero supercurrent is maintained along the vortex. We discuss the exchange statistics of vortices in the presence of unpaired Majorana modes and comment on their experimental detection.   

Quantum point contact as a probe of a topological superconductor

We calculate the conductance of a ballistic point contact to a superconducting wire, produced by the s-wave proximity effect in a semiconductor with spin-orbit coupling in a parallel magnetic field. The conductance $G$ as a function of contact width or Fermi energy shows plateaus at half-integer multiples of $4e^2/h$ if the superconductor is in a topologically nontrivial phase, supporting Majorana fermions. In contrast, the plateaus are at the usual integer multiples in the topologically trivial phase without Majorana fermions. Disorder destroys all plateaus except the first, which remains precisely quantized in the case of a topological superconductor, consistent with previous results for a tunnel contact. The advantage of a ballistic contact over a tunnel contact as a probe of the topological phase is the strongly reduced sensitivity to finite voltage or temperature.   

Genetic braid optimization for topological quantum computation

In topologically-protected quantum computation quantum gates can be carried out by adiabatically braiding quasiparticles in two space dimensions, reminiscent of entangled world lines. Bonesteel {\em et al.}~[Phys.~Rev.~Lett.~{\bf 95}, 140503 (2005)] showed recently how to find braids that yield a universal set of quantum gates. Mathematically, the problem of executing a gate becomes that of finding a product of the matrices in that set that approximates the gate, up to an error. To date efficient methods to compute these gates only strive to optimize for accuracy. We explore the possibility of using evolutionary (genetic) algorithms to efficiently find optimal braids while allowing the user to optimize for the relative utilities of accuracy and length. Furthermore, when optimizing for error only, the method can efficiently produce braids of error $\sim 10^{-6}$ outperforming brute force approaches.   

Quantum memory on topological spin glass

We show that any topologically ordered local stabilizer model of spins in three dimensional lattices that lacks string logical operators can be used as a reliable quantum memory against thermal noise. It is shown that any local process creating a topologically charged particle separated from other particles by a distance $R$ must cross an energy barrier of height $c \log R$. This property makes the model glassy. We devise an efficient decoding algorithm that should be used at the final read-out, and prove a lower bound on the memory time until which the fidelity between the outcome of the decoder and the initial state is close to 1. The memory time increases as $L^{c \beta}$ where $L$ is the system size and $\beta$ the inverse temperature, as long as $L < L^\star \sim e^\beta$. Hence, the optimal memory time scales as $e^{c\beta^2}$. Our bound applies when the system interacts with thermal bath via a Markovian master equation. We give an example of a strictly local stabilizer code that satisfies all of our assumptions. We numerically verify for this example that our bound is tight up to constants.   

3D local qupit quantum code without string logical operator

Recently Haah introduced a new class of local quantum error correcting code embedded on a cubic lattice without any string logical operator. We present new codes with similar properties by relaxing the condition on the local particle dimension. The resulting code is well-defined when the local Hilbert space dimension is prime. These codes can be divided into two different classes: the local stabilizer generators are either symmetric or antisymmetric with respect to the inversion operation. We lower bound the number of encoded qudits by computing the commutation relation between the logical operators confined on a plane.   

Topological Decoding through Artificial Confinement

2D topological stabilizer codes have attracted a lot of attention in recent years for two main reasons. First, they provide exactly solvable models which exhibit topological order and anyonic excitations. Second, they naturally lead to quantum stabilizer error-correcting codes having macroscopic minimum distance. Although these codes are robust at zero temperature, quasi-particles appear and freely diffuse in the system at any finite temperature. If this diffusion is unchecked, errors will occur. Consequently, active error-correction is needed. We want to propose a cellular automaton that would perform this correction. It would ``manually'' confine the quasi-particles by simulating articifial attraction between them and moving them accordingly. We obtained encouraging preliminary results for error-correction and hope to generalize them to fault-tolerance.   

Andreev Bound states in One Dimensional Topological Superconductor with Broken Spatial Inversion Symmetry

We study the Andreev bound states (ABSs) at the Josephson junction of one dimensional topological superconductors (SC) when the spatial inversion symmetry (SIS) is broken. While in the absence of inversion symmetry, we show a hidden symmetry for the Bogoliubov de Gennes equations in the case of SC gap much smaller than Fermi energy in addition to the particle-hole symmetry, due to which the ABSs are predicted to carry irrational charge, with the charge value solely depending on the SIS breaking term, regardless of the details of the superconductor order parameter and whether the disorder scattering is present or not. We demonstrate that in the tunneling transport spectroscopy the irrationally charged ABSs are measured by the resonant differential tunneling conductance.