Publications (54)162.43 Total impact
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ABSTRACT: Fibonacci anyons are nonAbelian particles for which braiding is universal for quantum computation. Reichardt has shown how to systematically generate nontrivial braids for three Fibonacci anyons which yield unitary operations with offdiagonal matrix elements that can be made arbitrarily small in a particular natural basis through a simple and efficient iterative procedure. This procedure does not require brute force search, the SolovayKitaev method, or any other numerical technique, but the phases of the resulting diagonal matrix elements cannot be directly controlled. We show that despite this lack of control the resulting braids can be used to systematically construct entangling gates for two qubits encoded by Fibonacci anyons.  [Show abstract] [Hide abstract]
ABSTRACT: The Majorana code is an example of a stabilizer code where the quantum information is stored in a system supporting wellseparated Majorana bound states (MBSs). We focus on onedimensional realizations of the Majorana code, as well as networks of such structures, and investigate their lifetime when coupled to a paritypreserving thermal environment. We apply the Davies prescription, a standard method that describes the basic aspects of a thermal environment, and derive a master equation in the BornMarkov limit. We first focus on a single wire with immobile MBSs and perform error correction to annihilate thermal excitations. In the hightemperature limit, we show both analytically and numerically that the lifetime of the Majorana qubit grows logarithmically with the size of the wire. We then study a trijunction with four MBSs when braiding is executed. We study the occurrence of dangerous error processes that prevent the lifetime of the Majorana code from growing with the size of the trijunction. The origin of the dangerous processes is the braiding itself, which separates pairs of excitations and renders the noise nonlocal; these processes arise from the basic constraints of moving MBSs in onedimensional (1D) structures. We confirm our predictions with Monte Carlo simulations in the lowtemperature regime, i.e., the regime of practical relevance. Our results put a restriction on the degree of selfcorrection of this particular 1D topological quantum computing architecture.  [Show abstract] [Hide abstract]
ABSTRACT: We give an analytic construction of a class of twoqubit gate pulse sequences that act on five of the six spin$\frac12$ particles used to encode a pair of exchangeonly threespin qubits. Within this class, the problem of gate construction reduces to that of finding a smaller sequence that acts on four spins and is subject to a simple constraint. The optimal sequence satisfying this constraint yields a twoqubit gate sequence equivalent to that found numerically by Fong and Wandzura. Our construction is sufficiently simple that it can be carried out entirely with pen, paper, and knowledge of a few basic facts about quantum spin. We thereby analytically derive the FongWandzura sequence that has so far escaped intuitive explanation.  [Show abstract] [Hide abstract]
ABSTRACT: The Majorana code is an example of a stabilizer code where the quantum information is stored in a system supporting wellseparated Majorana Bound States (MBSs). We focus on onedimensional realizations of the Majorana code, as well as networks of such structures, and investigate their lifetime when coupled to a paritypreserving thermal environment. We apply the Davies prescription, a standard method that describes the basic aspects of a thermal environment, and derive a master equation in the BornMarkov limit. We first focus on a single wire with immobile MBSs and perform error correction to annihilate thermal excitations. In the hightemperature limit, we show both analytically and numerically that the lifetime of the Majorana qubit grows logarithmically with the size of the wire. We then study a trijunction with four MBSs when braiding is executed. We study the occurrence of dangerous error processes that prevent the lifetime of the Majorana code from growing with the size of the trijunction. The origin of the dangerous processes is the braiding itself, which separates pairs of excitations and renders the noise nonlocal; these processes arise from the basic constraints of moving MBSs in 1D structures. We confirm our predictions with Monte Carlo simulations in the lowtemperature regime, i.e. the regime of practical relevance. Our results put a restriction on the degree of selfcorrection of this particular 1D topological quantum computing architecture.  [Show abstract] [Hide abstract]
ABSTRACT: We present pulse sequences for twoqubit gates acting on encoded qubits for exchangeonly quantum computation. Previous work finding such sequences has always required numerical methods due to the large search space of unitary operators acting on the space of the encoded qubits. By contrast, our construction can be understood entirely in terms of threedimensional rotations of effective spin1/2 pseudospins which allows us to use geometric intuition to determine the required sequence of operations analytically. The price we pay for this simplification is that, at 39 pulses, our sequences are significantly longer than the best numerically obtained sequences.  [Show abstract] [Hide abstract]
ABSTRACT: We show the violation of the entanglement area law for bosonic systems with Bose surfaces. For bosonic systems with gapless factorized energy dispersions on an N^{d} Cartesian lattice in d dimensions, e.g., the exciton Bose liquid in two dimensions, we explicitly show that a belt subsystem with width L preserving translational symmetry along d1 Cartesian axes has leading entanglement entropy (N^{d1}/3)lnL. Using this result, the strong subadditivity inequality, and lattice symmetries, we bound the entanglement entropy of a rectangular subsystem from below and above showing a logarithmic violation of the area law. For subsystems with a single flat boundary, we also bound the entanglement entropy from below showing a logarithmic violation, and argue that the entanglement entropy of subsystems with arbitrary smooth boundaries are similarly bounded.  [Show abstract] [Hide abstract]
ABSTRACT: We study the effect of the ChernSimons gauge fields on the possible transition from two decoupled composite fermion metals to the interlayer coherent composite fermion state proposed by Alicea et al. [Phys. Rev. Lett. 103, 256403 (2009)] in a symmetrically doped quantum Hall bilayer with total Landau level filling fraction $\nu_{tot} = 1$. In this transition, interlayer Coulomb repulsion leads to excitonic condensation of composite fermions which are then free to tunnel coherently between layers. We find that this coherent tunneling is strongly suppressed by the layerdependent AharonovBohm phases experienced by composite fermions as they propagate through the fluctuating gauge fields in the system. This suppression is analyzed by treating these gauge fluctuations within the randomphase approximation and calculating their contribution to the energy cost for forming an exciton condensate of composite fermions. This energy cost leads to (1) an increase in the critical interlayer repulsion needed to drive the transition; and (2) a discontinuous jump in the energy gaps to outofphase excitations (i.e., excitations involving currents with opposite signs in the two layers) at the transition.  [Show abstract] [Hide abstract]
ABSTRACT: In topological quantum computation, quantum gates are carried out by braiding worldlines of nonAbelian anyons in 2+1 dimensional spacetime. The simplest such anyons for which braiding is universal for quantum computation are Fibonacci anyons. Reichardt [1] has shown how to construct nontrivial braids for three Fibonacci anyons which yield 2 x2 unitary operations whose offdiagonal matrix elements (in the appropriate basis) can be made arbitrarily small through a simple and efficient iterative procedure. A great advantage of this construction is that it does not require either brute force search or the SolovayKitaev method. There is, however, a downsidethe phases of the diagonal matrix elements cannot be directly controlled. Despite this, we show that the resulting braids can be used to construct leakagefree entangling twoqubit gates for qubits encoded using four Fibonacci anyons each. We give two explicit constructionsone based on the ``functional braid" approach of Hu and Wan [2], and another based on the ``effective qubit" approach of Hormozi et al. [3]. [1] B.W. Reichardt, Quant. Inf. and Comp. 12, 876 (2012). [2] H. Xu and X. Wan, PRA 78, 042325 (2008). [3] L. Hormozi et al., PRL 103, 160501 (2009).  [Show abstract] [Hide abstract]
ABSTRACT: Exchange pulses are local unitary operations obtained by turning on and off the isotropic exchange interaction between pairs of spin1/2 particles, for example electron spins in quantum dots. We present a procedure for analytically constructing sequences of exchange pulses for carrying out leakage free twoqubit gates on logical threespin qubits. At each stage of our construction we reduce the problem to that of finding a sequence of rotations for an effective twolevel system. The resulting pulse sequences are 39 pulses long, longer than the original 19pulse sequence of DiVincenzo et al. [1] and the more recent 18pulse sequence of Fong and Wandzura [2], both of which were obtained numerically. Like the latter sequence, our sequences work regardless of the total spin of the six spins used to encode two qubits. After introducing our method, we prove that any leakagefree sequence of exchange pulses must act on at least five of the six spins to produce an entangling twoqubit gate.[4pt] [1] D.P. DiVincenzo et al., Nature 408, 339 (2000). [2] B.H. Fong & S.M. Wandzura, Quantum Info. Comput., 11, 1003 (2011).  [Show abstract] [Hide abstract]
ABSTRACT: It may be possible to use the ground states of the LevinWen model for Fibonacci anyons as a nonAbelian surface code for faulttolerant quantum computation [1]. To do this, it will be necessary to repeatedly measure the vertex and plaquette operators of the model to check for errors. Recently, two of us have constructed quantum circuits for performing such measurements [2]. Here we present an alternate measurement scheme based on simulating an interference experiment. This ``experiment'' can be thought of, roughly, as first inserting a pair of Fibonacci anyons with trivial total topological charge onto one edge of a plaquette, ``braiding'' one anyon all the way around the plaquette while the other remains fixed, and then either measuring the total topological charge of the two anyons or manipulating their state in a specific way. We construct explicit quantum circuits which can be used to simulate these processes and show how they can be used to measure the LevinWen plaquette operator on a quantum computer.[4pt] [1] R. Koenig, G. Kuperberg, and B.W. Reichardt, Ann. Phys. 325, 2707 (2010).[0pt] [2] N.E. Bonesteel and D.P. DiVincenzo, Phys. Rev. B 86, 165113 (2012).  [Show abstract] [Hide abstract]
ABSTRACT: We construct quantum circuits for measuring the commuting set of vertex and plaquette operators that appear in the LevinWen model for doubled Fibonacci anyons. Such measurements can be viewed as syndrome measurements for the quantum errorcorrecting code defined by the ground states of this model (the Fibonacci code). We quantify the complexity of these circuits with gate counts using different universal gate sets and find these measurements become significantly easier to perform if nqubit Toffoli gates with n = 3,4 and 5 can be carried out directly. In addition to measurement circuits, we construct simplified quantum circuits requiring only a few qubits that can be used to verify that certain selfconsistency conditions, including the pentagon equation, are satisfied by the Fibonacci code.  [Show abstract] [Hide abstract]
ABSTRACT: There is compelling theoretical evidence1 that the ν = 5/2 fractional quantum Hall state is a MooreRead state2  a state which can be viewed as a spinpolarized pwave superconductor of composite fermions. The question remains, how can one test this hypothesis experimentally? To address this we have developed a semiphenomenological description of this state in which the HalperinLeeRead3 theory of the halffilled Landau level is modified by adding a pwave pairing interaction between composite fermions by hand. The electromagnetic response functions for the resulting meanfield pwave superconducting state are then calculated and used in an RPA calculation of the physical electronic response. In particular, we predict the wavevector and frequency dependence of the longitudinal conductivity σxx(q,ω) which can be measured in surfaceacousticwave propagation experiments.4  [Show abstract] [Hide abstract]
ABSTRACT: The ground state of the uniform antiferromagnetic spin1/2 Heisenberg chain can be viewed as a strongly fluctuating liquid of valence bonds, while in disordered chains these bonds lock into random singlet states on longlength scales. We show that this phenomenon can be studied numerically, even in the case of weak disorder, by calculating the mean value of the number of valence bonds leaving a block of L contiguous spins (the valencebond entanglement entropy) as well as the fluctuations in this number. These fluctuations show a clear crossover from a small L regime, in which they behave similar to those of the uniform model, to a large L regime, in which they saturate in a way consistent with the formation of a random singlet state on longlength scales. A scaling analysis of these fluctuations is used to study the dependence on disorder strength of the length scale characterizing the crossover between these two regimes. Results are obtained for a class of models that include, in addition to the spin1/2 Heisenberg chain, the uniform and disordered critical 1D transversefield Ising model and chains of interacting nonAbelian anyons. 
Article: Suppression of Interlayer Phase Coherence by Gauge Fluctuations in Bilayer Composite Fermi Liquids
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ABSTRACT: The nu =1/2+1/2 bilayer quantum Hall system exhibits at least two phases as a function of layer spacing, d. For d/l 1, (l is magnetic length), the system decouples into two nu= 1/2 composite fermion (CF) liquids. For d/l sufficiently small, the system enters an incompressible bilayer quantum Hall state. Recently, Alicea et al. [1] have proposed a state which might exist for intermediate layer spacing (d ˜l). In this "interlayer phase coherent" state, CFs tunnel coherently between layers forming welldefined bonding and antibonding Fermi seas, though there is no actual tunneling of physical electrons. Here we show that scattering from gauge fields in the CF liquids leads to strong layerdependent fluctuations in the AharonovBohm phases seen by CFs which suppress interlayer phase coherence. This suppression appears as a singular contribution to the correlation energy which inhibits any T=0 phase transition into an interlayer phase coherent state, and drives any such transition first order. Work supported by US DOE.[4pt] [1] J. Alicea, O.I. Motrunich, G. Refael, M.P.A. Fisher, PRL 103, 256403 (2009).  [Show abstract] [Hide abstract]
ABSTRACT: a b s t r a c t A generalized version of the valencebond Monte Carlo method is used to study ground state properties of the 1 + 1 dimensional quantum Qstate Potts models. For appropriate values of Q these models can be used to describe interacting chains of nonAbelian anyons — quasiparticle excitations of certain exotic fractional quantum Hall states. Published by Elsevier B.V.  [Show abstract] [Hide abstract]
ABSTRACT: We consider a hypothetical topological quantum computer where the qubits are comprised of either Ising or Fibonacci anyons. For each case, we calculate the time and number of qubits (space) necessary to execute the most computationally expensive step of Shor's algorithm, modular exponentiation. For Ising anyons, we apply Bravyi's distillation method [S. Bravyi, Phys. Rev. A 73, 042313 (2006)] which combines topological and nontopological operations to allow for universal quantum computation. With reasonable restrictions on the physical parameters we find that factoring a 128 bit number requires approximately 10^3 Fibonacci anyons versus at least 3 x 10^9 Ising anyons. Other distillation algorithms could reduce the resources for Ising anyons substantially.  [Show abstract] [Hide abstract]
ABSTRACT: ReadRezayi fractional quantum Hall states are among the prime candidates for realizing nonAbelian anyons which, in principle, can be used for topological quantum computation. We present a prescription for efficiently finding braids which can be used to carry out a universal set of quantum gates on encoded qubits based on anyons of the ReadRezayi states with k>2, k not equal 4. This work extends previous results which only applied to the case k=3 (Fibonacci) and clarifies why, in that case, gate constructions are simpler than for a generic ReadRezayi state.  [Show abstract] [Hide abstract]
ABSTRACT: We demonstrate numerically that nonAbelian quasihole (qh) excitations of the nu=5/2 fractional quantum Hall state have some of the key properties necessary to support quantum computation. We find that as the qh spacing is increased, the unitary transformation which describes winding two qh's around each other converges exponentially to its asymptotic limit and that the two orthogonal wave functions describing a system with four qh's become exponentially degenerate. We calculate the length scales for these two decays to be xi(U) approximately 2.7l(0) and xi(E) approximately 2.3l(0), respectively. Additionally, we determine which fusion channel is lower in energy when two qh's are brought close together.  [Show abstract] [Hide abstract]
ABSTRACT: Certain fractional quantum Hall states, including the experimentally observed ν = 5/2 state, and, possibly, the ν = 12/5 state, may have a sufficiently rich form of topological order (i.e. they may be nonabelian) to be useful for quantum information processing. For example, in some cases they may be used for topological quantum computation, an intrinsically fault tolerant form of quantum computation which is carried out by braiding the world lines of quasiparticle excitations in 2+1 dimensional space time. Here we briefly review the relevant properties of nonabelian quantum Hall states and discuss some of the methods we have found for finding specific braiding patterns which can be used to carry out universal quantum computation using them. Recent work on onedimensional chains of interacting quasiparticles in nonabelian states is also reviewed.  [Show abstract] [Hide abstract]
ABSTRACT: In valencebond Monte Carlo (VBMC) ootnotetextA. Sandvik, PRL 95, 207203 (2005). the ground state of a quantum spin system is sampled directly from the valencebond (VB) basis  a useful basis for visualizing the properties of singlet ground states. For example, the ground state of the uniform AFM spin12 Heisenberg chain is characterized by strongly fluctuating bonds with powerlaw length distribution, while in the randomsinglet phase (RSP) of a random Heisenberg chain these bonds, while still having a powerlaw length distribution, lock into a particular VB state on long length scales. ootnotetextD. S. Fisher, PRB 50, 3799 (1994). We use VBMC to directly probe the formation of a RSP by calculating both the average number of bonds nL leaving a block of L spins (the VB entanglement entropy ootnotetextF. Alet, et al., PRL 99, 117204 (2007).), and its fluctuations, sigmanL^2 =
Publication Stats
834  Citations  
162.43  Total Impact Points  
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19932015

Florida State University
 Department of Physics
Tallahassee, Florida, United States


19982001

National High Magnetic Field Laboratory
Tallahassee, Florida, United States
