[Show abstract][Hide abstract]ABSTRACT: Fibonacci anyons are non-Abelian 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 off-diagonal 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 Solovay-Kitaev 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 well-separated Majorana bound states (MBSs). We focus on one-dimensional realizations of the Majorana code, as well as networks of such structures, and investigate their lifetime when coupled to a parity-preserving 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 Born-Markov limit. We first focus on a single wire with immobile MBSs and perform error correction to annihilate thermal excitations. In the high-temperature 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 one-dimensional (1D) structures. We confirm our predictions with Monte Carlo simulations in the low-temperature regime, i.e., the regime of practical relevance. Our results put a restriction on the degree of self-correction of this particular 1D topological quantum computing architecture.
[Show abstract][Hide abstract]ABSTRACT: We give an analytic construction of a class of two-qubit gate pulse sequences
that act on five of the six spin-$\frac12$ particles used to encode a pair of
exchange-only three-spin 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 two-qubit 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 Fong-Wandzura
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 well-separated Majorana Bound
States (MBSs). We focus on one-dimensional realizations of the Majorana code,
as well as networks of such structures, and investigate their lifetime when
coupled to a parity-preserving 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 Born-Markov limit. We first
focus on a single wire with immobile MBSs and perform error correction to
annihilate thermal excitations. In the high-temperature 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
low-temperature regime, i.e. the regime of practical relevance. Our results put
a restriction on the degree of self-correction of this particular 1D
topological quantum computing architecture.
[Show abstract][Hide abstract]ABSTRACT: We present pulse sequences for two-qubit gates acting on encoded qubits for
exchange-only 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 three-dimensional rotations
of effective spin-1/2 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 d-1 Cartesian axes has leading entanglement entropy (N^{d-1}/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.
Full-text Article · Nov 2013 · Physical Review Letters
[Show abstract][Hide abstract]ABSTRACT: We study the effect of the Chern-Simons 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 layer-dependent Aharonov-Bohm 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 random-phase 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 out-of-phase 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 non-Abelian anyons in 2+1 dimensional space-time.
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 off-diagonal 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 Solovay-Kitaev
method. There is, however, a downside---the phases of the diagonal
matrix elements cannot be directly controlled. Despite this, we show
that the resulting braids can be used to construct leakage-free
entangling two-qubit gates for qubits encoded using four Fibonacci
anyons each. We give two explicit constructions---one 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 spin-1/2
particles, for example electron spins in quantum dots. We present a
procedure for analytically constructing sequences of exchange pulses for
carrying out leakage free two-qubit gates on logical three-spin qubits.
At each stage of our construction we reduce the problem to that of
finding a sequence of rotations for an effective two-level system. The
resulting pulse sequences are 39 pulses long, longer than the original
19-pulse sequence of DiVincenzo et al. [1] and the more recent 18-pulse
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 leakage-free sequence of
exchange pulses must act on at least five of the six spins to produce an
entangling two-qubit 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 Levin-Wen model for
Fibonacci anyons as a non-Abelian surface code for fault-tolerant
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
Levin-Wen 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 Levin-Wen model for doubled Fibonacci
anyons. Such measurements can be viewed as syndrome measurements for the
quantum error-correcting 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 n-qubit 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 self-consistency conditions, including the pentagon
equation, are satisfied by the Fibonacci code.
Article · Jun 2012 · Physical review. B, Condensed matter
[Show abstract][Hide abstract]ABSTRACT: There is compelling theoretical evidence1 that the ν = 5/2 fractional quantum Hall state is a Moore-Read state2 - a state which can be viewed as a spin-polarized p-wave superconductor of composite fermions. The question remains, how can one test this hypothesis experimentally? To address this we have developed a semi-phenomenological description of this state in which the Halperin-Lee-Read3 theory of the half-filled Landau level is modified by adding a p-wave pairing interaction between composite fermions by hand. The electromagnetic response functions for the resulting mean-field p-wave superconducting state are then calculated and used in an RPA calculation of the physical electronic response. In particular, we predict the wave-vector and frequency dependence of the longitudinal conductivity σxx(q,ω) which can be measured in surface-acoustic-wave propagation experiments.4
Article · Jan 2012 · International Journal of Modern Physics B
[Show abstract][Hide abstract]ABSTRACT: The ground state of the uniform antiferromagnetic spin-1/2 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 long-length 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 valence-bond 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 long-length 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 spin-1/2 Heisenberg chain, the uniform and disordered critical 1D transverse-field Ising model and chains of interacting non-Abelian anyons.
[Show abstract][Hide abstract]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 well-defined 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 layer-dependent fluctuations in the Aharonov-Bohm 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 valence-bond Monte Carlo method is used to study ground state properties of the 1 + 1 dimensional quantum Q-state Potts models. For appropriate values of Q these models can be used to describe interacting chains of non-Abelian 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 non-topological 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: Read-Rezayi fractional quantum Hall states are among the prime candidates for realizing non-Abelian 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 Read-Rezayi 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 Read-Rezayi state.
[Show abstract][Hide abstract]ABSTRACT: We demonstrate numerically that non-Abelian 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.
Full-text Article · Aug 2009 · Physical Review Letters
[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 one-dimensional chains of interacting quasiparticles in nonabelian states is also reviewed.
Full-text Article · May 2009 · International Journal of Modern Physics B
[Show abstract][Hide abstract]ABSTRACT: In valence-bond Monte Carlo (VBMC) ootnotetextA. Sandvik, PRL 95, 207203 (2005). the ground state of a quantum spin system is sampled directly from the valence-bond (VB) basis --- a useful basis for visualizing the properties of singlet ground states. For example, the ground state of the uniform AFM spin-12 Heisenberg chain is characterized by strongly fluctuating bonds with power-law length distribution, while in the random-singlet phase (RSP) of a random Heisenberg chain these bonds, while still having a power-law 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 =