[Show abstract][Hide abstract] ABSTRACT: It is conceivable that for some strange anyon with quantum dimension $>1$
that the resulting representations of all $n$-strand braid groups $B_n$ are
overall phases, even though the ground state manifolds for $n$ such anyons are
in general Hilbert spaces of dimensions $>1$. We observe that this cannot occur
for any anyon with quantum dimension $>1$. Therefore, degeneracy implies
non-abelian statistics. It justifies the definition of a non-abelian anyon as
one with quantum dimension $>1$. Since non-abelian statistics presumes
degeneracy, degeneracy is more fundamental than non-abelian statistics.
[Show abstract][Hide abstract] ABSTRACT: The feasibility of a classification-by-rank program for modular categories
follows from the Rank-Finiteness Theorem. We develop arithmetic, representation
theoretic and algebraic methods for classifying modular categories by rank. As
an application, we determine all possible fusion rules for all rank=$5$ modular
categories and describe the corresponding monoidal equivalence classes.
[Show abstract][Hide abstract] ABSTRACT: We examine a class of operations for topological quantum computation based on
fusing and measuring topological charges for systems with SU$(2)_4$ or $k=4$
Jones-Kauffman anyons. We show that such operations augment the braiding
operations, which, by themselves, are not computationally universal. This
augmentation results in a computationally universal gate set through the
generation of an exact, topologically protected irrational phase gate and an
approximate, topologically protected controlled-$Z$ gate.
[Show abstract][Hide abstract] ABSTRACT: There are several possible theoretically allowed non-Abelian fractional
quantum Hall (FQH) states that could potentially be realized in one- and two-
component FQH systems at total filling fraction $\nu = n+ 2/3$, for integer
$n$. Some of these states even possess quasiparticles with non-Abelian
statistics that are powerful enough for universal topological quantum
computation, and are thus of particular interest. Here, we initiate a
systematic numerical study, using both exact diagonalization and variational
Monte Carlo, to investigate the phase diagram of FQH systems at total filling
fraction $\nu = n+2/3$, including in particular the possibility of the
non-Abelian $Z_4$ parafermion state. In $\nu = 2/3$ bilayers, we determine the
phase diagram as a function of interlayer tunneling and repulsion, finding only
three competing Abelian states, without the $Z_4$ state. On the other hand, in
single-component systems at $\nu = 8/3$, we find that the $Z_4$ parafermion
state has significantly higher overlap with the exact ground state than the
Laughlin state, together with a larger gap, suggesting that the experimentally
observed $\nu = 8/3$ state may be non-Abelian. Our results from the two
complementary numerical techniques agree well with each other qualitatively.
Physical Review B 02/2015; 92(3). DOI:10.1103/PhysRevB.92.035103 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Symmetry protected and symmetry enriched topological phases of matter are of
great interest in condensed matter physics due to new materials such as
topological insulators. The Levin-Wen model for spin/boson systems is an
important rigorously solvable model for studying $2D$ topological phases. The
input data for the Levin-Wen model is a unitary fusion category, but the same
model also works for unitary multi-fusion categories. In this paper, we provide
the details for this extension of the Levin-Wen model, and show that the
extended Levin-Wen model is a natural playground for the theoretical study of
symmetry protected and symmetry enriched topological phases of matter.
Journal of Physics A Mathematical and Theoretical 12/2014; 48(12). DOI:10.1088/1751-8113/48/12/12FT01 · 1.58 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We study representations of the loop braid group $LB_n$ from the perspective
of extending representations of the braid group $B_n$. We also pursue a
generalization of the braid/Hecke/Temperlely-Lieb paradigm---uniform finite
dimensional quotient algebras of the loop braid group algebras.
[Show abstract][Hide abstract] ABSTRACT: In this paper we classify all modular categories of dimension $4m$, where $m$
is an odd square-free integer, and all rank 6 and rank 7 weakly integral
modular categories. This completes the classification of weakly integral
modular categories through rank 7. In particular, our results imply that all
integral modular categories of rank at most $7$ are pointed (that is, every
simple object has dimension $1$). All the non-integral (but weakly integral)
modular categories of ranks $6$ and $7$ have dimension $4m$, with $m$ an odd
square free integer, so their classification is an application of our main
result.
[Show abstract][Hide abstract] ABSTRACT: We examine the interplay of symmetry and topological order in $2+1$
dimensional topological phases of matter. We present a definition of the \it
topological symmetry \rm group, which characterizes the symmetry of the
emergent topological quantum numbers of a topological phase $\mathcal{C}$, and
describe its relation with the microscopic symmetry of the underlying physical
system. We derive a general framework to classify symmetry fractionalization in
topological phases, including non-Abelian phases and the possibility that the
symmetries permute the quasiparticle types. We develop a theory of extrinsic
defects (fluxes) associated with elements of the symmetry group, which provides
a general classification of symmetry-enriched topological phases derived from a
topological phase of matter $\mathcal{C}$ with (on-site) symmetry group $G$.
The algebraic theory of the defects, known as a $G$-crossed braided tensory
category $\mathcal{C}_{G}^{\times}$, allows one to compute many properties,
such as the number of topologically distinct types of defects associated with
each group element, their fusion rules, quantum dimensions, zero modes,
braiding exchange transformations, a generalized Verlinde formula for the
defects, and modular transformations of the $G$-crossed extensions of
topological phases. We also examine the promotion of the global symmetry to a
local gauge invariance, wherein the extrinsic $G$-defects are turned into
deconfined quasiparticle excitations, which results in a different topological
phase $\mathcal{C}/G$. A number of instructive and/or physically relevant
examples are studied in detail.
[Show abstract][Hide abstract] ABSTRACT: We generalize Ng's algebraic $0$-th knot contact homology for links in $S^3$
to links in $S^1 \times S^2$, and prove that the resulted link invariant is the
same as the cord ring of links. Our main tool is Lin's generalization of the
Markov theorem for braids in $S^3$ to braids in $S^1 \times S^2$. We conjecture
that our cord ring is always finitely generated for non-trivial links.
[Show abstract][Hide abstract] ABSTRACT: We show that braidings of the metaplectic anyons $X_\epsilon$ in
$\SO(3)_2=\SU(2)_4$ with their total charge equal to the metaplectic mode $Y$
supplemented with measurements of the total charge of two metaplectic anyons
are universal for quantum computation. We conjecture that similar universal
computing models can be constructed for all metaplectic anyon systems
$\SO(p)_2$ for any odd prime $p\geq 5$. In order to prove universality, we find
new conceptually appealing universal gate sets for qutrits and qupits.
[Show abstract][Hide abstract] ABSTRACT: Harnessing non-abelian statistics of anyons to perform quantum computational
tasks is getting closer to reality. While the existence of universal anyons by
braiding alone such as the Fibonacci anyon is theoretically a possibility,
accessible anyons with current technology all belong to a class that is called
weakly integral---anyons whose squared quantum dimensions are integers. We
analyze the computational power of the first non-abelian anyon system with only
integral quantum dimensions---$D(S_3)$, the quantum double of $S_3$. Since all
anyons in $D(S_3)$ have finite images of braid group representations, they
cannot be universal for quantum computation by braiding alone. Based on our
knowledge of the images of the braid group representations, we set up three
qutrit computational models. Supplementing braidings with some measurements and
ancillary states, we find a universal gate set for each model.
Quantum Information Processing 01/2014; 14(8). DOI:10.1007/s11128-015-1016-y · 1.92 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We prove a rank-finiteness conjecture for modular categories that there are
only finitely many modular categories of fixed rank $r$, up to equivalence. Our
main technical advance is a Cauchy theorem for modular categories: given a
modular category $\mathcal{C}$, the set of prime ideals of the global quantum
dimension $D^2$ of $\mathcal{C}$ in the cyclotomic number field $\mathcal{O}_N$
is identical to that of the Frobenius-Schur exponent $N=FSexp(\mathcal{C})$ of
$\mathcal{C}$. By combining the Galois symmetry of the modular $S,T$ matrices
with the knowledge of the modular representation of $SL(2,\mathbb{Z})$, we
determine all possible fusion rules for all rank=5 modular categories..
[Show abstract][Hide abstract] ABSTRACT: The second author previously discussed how classical complexity separation
conjectures, we call them "axioms", have implications in three manifold
topology: polynomial length stings of operations which preserve certain Jones
polynomial evaluations cannot produce exponential simplifications of link
diagrams. In this paper, we continue this theme, exploring now more subtle
separation axioms for quantum complexity classes. Surprisingly, we now find
that similar strings are unable to effect even linear simplifications of the
diagrams.
[Show abstract][Hide abstract] ABSTRACT: There are many interesting parallels between systems of interacting
non-Abelian anyons and quantum magnetism, occuring in ordinary SU(2) quantum
magnets. Here we consider theories of so-called su(2)_k anyons, well-known
deformations of SU(2), in which only the first k+1 angular momenta of SU(2)
occur. In this manuscript, we discuss in particular anyonic generalizations of
ordinary SU(2) spin chains with an emphasis on anyonic spin S=1 chains. We find
that the overall phase diagrams for these anyonic spin-1 chains closely mirror
the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including
anyonic generalizations of the Haldane phase, the AKLT construction, and
supersymmetric quantum critical points. A novel feature of the anyonic spin-1
chains is an additional topological symmetry that protects the gapless phases.
Distinctions further arise in the form of an even/odd effect in the deformation
parameter k when considering su(2)_k anyonic theories with k>4, as well as for
the special case of the su(2)_4 theory for which the spin-1 representation
plays a special role. We also address anyonic generalizations of spin-1/2
chains with a focus on the topological protection provided for their gapless
ground states. Finally, we put our results into context of earlier
generalizations of SU(2) quantum spin chains, in particular so-called (fused)
Temperley-Lieb chains.
[Show abstract][Hide abstract] ABSTRACT: For non-abelian simple objects in a unitary modular category, the density of
their braid group representations, the #P-hard evaluation of their associated
link invariants, and the BQP-completeness of their anyonic quantum computing
models are closely related. We systematically study such properties of the
non-abelian simple objects in metaplectic modular categories, which are unitary
modular categories with fusion rules of SO(m)_2 for an odd integer m \geq 3.
The simple objects with quantum dimensions \sqrt{m} have finite image braid
group representations, and their link invariants are classically efficient to
evaluate. We also provide classically efficient simulation of their braid group
representations. These simulations of the braid group representations can be
regarded as qudit generalizations of the Knill-Gottesmann theorem for the qubit
case. The simple objects of dimension 2 give us a surprising result: while
their braid group representations have finite images and are efficiently
simulable classically after a generalized localization, their link invariants
are #P-hard to evaluate exactly. We sharpen the #P-hardness by showing that any
sufficiently accurate approximation of their associated link invariants is
already #P-hard.
[Show abstract][Hide abstract] ABSTRACT: We begin by discussing the history of quantum logic, dividing it into three
eras or lives. The first life has to do with Birkhoff and von Neumann's
algebraic approach in the 1930's. The second life has to do with attempt to
understand quantum logic as logic that began in the late 1950's and blossomed
in the 1970's. And the third life has to do with recent developments in quantum
logic coming from its connections to quantum computation. We discuss our own
work connecting quantum logic to quantum computation (viewing quantum logic as
the logic of quantum registers storing qubits), and make some speculations
about mathematics based on quantum principles.
Journal of Philosophical Logic 02/2013; 42(3). DOI:10.1007/s10992-013-9273-7
[Show abstract][Hide abstract] ABSTRACT: There is compelling theoretical evidence that quantum physics will change the
face of information science. Exciting progress has been made during the last
two decades towards the building of a large scale quantum computer. A quantum
group approach stands out as a promising route to this holy grail, and provides
hope that we may have quantum computers in our future.
[Show abstract][Hide abstract] ABSTRACT: The Blind Source Separation problem consists in estimating a set of unknown source signals from their measured combinations. It was only investigated in a non-quantum framework up to now. We propose its first quantum extensions. We thus introduce the ...
Quantum Information Processing 12/2012; 11(6). DOI:10.1007/s11128-012-0488-2 · 1.92 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We introduce and study a class of anyon models that are a natural
generalization of Ising anyons and Majorana fermion zero modes. These models
combine an Ising anyon sector with a sector associated with $SO(m)_2$
Chern-Simons theory. We show how they can arise in a simple scenario for
electron fractionalization and give a complete account of their quasiparticles
types, fusion rules, and braiding. We show that the image of the braid group is
finite for a collection of $2n$ fundamental quasiparticles and is a proper
subgroup of the metaplectic representation of $Sp(2n-2,\mathbb{F}_m)\ltimes
H(2n-2,\mathbb{F}_m)$, where $Sp(2n-2,\mathbb{F}_m)$ is the symplectic group
over the finite field $\mathbb{F}_m$ and $H(2n-2,\mathbb{F}_m)$ is the extra
special group (also called the $(2n-1)$-dimensional Heisenberg group) over
$\mathbb{F}_m$. Moreover, the braiding of fundamental quasiparticles can be
efficiently simulated classically. However, computing the result of braiding a
certain type of composite quasiparticle is $# P$-hard, although it is not
universal for quantum computation because it has a finite braid group image.
This a rare example of a topological phase that is not universal for quantum
computation through braiding but nevertheless has $# P$-hard link invariants.
We argue that our models are closely related to recent analyses finding
non-Abelian anyonic properties for defects in quantum Hall systems,
generalizing Majorana zero modes in quasi-1D systems.