Publications (61)109.53 Total impact
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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 $k=4$ JonesKauffman 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 nonAbelian 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 nonAbelian 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 nonAbelian $Z_4$ 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 singlecomponent 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 nonAbelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.  Journal of the American Mathematical Society 01/2015; DOI:10.1090/jams/842 · 3.06 Impact Factor
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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 LevinWen model for spin/boson systems is an important rigorously solvable model for studying $2D$ topological phases. The input data for the LevinWen model is a unitary fusion category, but the same model also works for unitary multifusion categories. In this paper, we provide the details for this extension of the LevinWen model, and show that the extended LevinWen 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:10.1088/17518113/48/12/12FT01 · 1.69 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/TemperlelyLieb paradigmuniform 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 squarefree 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 nonintegral (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 nonAbelian 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 symmetryenriched topological phases derived from a topological phase of matter $\mathcal{C}$ with (onsite) 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 nontrivial 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.Journal of Mathematical Physics 05/2014; 56(3). DOI:10.1063/1.4914941 · 1.18 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Harnessing nonabelian 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 integralanyons whose squared quantum dimensions are integers. We analyze the computational power of the first nonabelian 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; DOI:10.1007/s111280151016y · 2.96 Impact Factor 
Article: On Modular Categories
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ABSTRACT: We prove a rankfiniteness 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 FrobeniusSchur 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 nonAbelian anyons and quantum magnetism, occuring in ordinary SU(2) quantum magnets. Here we consider theories of socalled su(2)_k anyons, wellknown 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 spin1 chains closely mirror the phase diagram of the ordinary bilinearbiquadratic spin1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin1 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 spin1 representation plays a special role. We also address anyonic generalizations of spin1/2 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 socalled (fused) TemperleyLieb chains.Physical review. B, Condensed matter 03/2013; 87(23). DOI:10.1103/PhysRevB.87.235120 · 3.66 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: For nonabelian simple objects in a unitary modular category, the density of their braid group representations, the #Phard evaluation of their associated link invariants, and the BQPcompleteness of their anyonic quantum computing models are closely related. We systematically study such properties of the nonabelian 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 KnillGottesmann 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 #Phard to evaluate exactly. We sharpen the #Phardness by showing that any sufficiently accurate approximation of their associated link invariants is already #Phard.Communications in Mathematical Physics 03/2013; 330(1). DOI:10.1007/s0022001420447 · 1.90 Impact Factor  [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/s1099201392737  [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 nonquantum framework up to now. We propose its first quantum extensions. We thus introduce the ...Quantum Information Processing 12/2012; 11(6). DOI:10.1007/s1112801204882 · 2.96 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 $SO(m)_2$ ChernSimons 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 $Sp(2n2,\mathbb{F}_m)\ltimes H(2n2,\mathbb{F}_m)$, where $Sp(2n2,\mathbb{F}_m)$ is the symplectic group over the finite field $\mathbb{F}_m$ and $H(2n2,\mathbb{F}_m)$ is the extra special group (also called the $(2n1)$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 nonAbelian anyonic properties for defects in quantum Hall systems, generalizing Majorana zero modes in quasi1D systems.Physical review. B, Condensed matter 10/2012; 87(16). DOI:10.1103/PhysRevB.87.165421 · 3.66 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Inspired by quantum information theory, we look for representations of the braid groups $B_n$ on $V^{\otimes (n+m2)}$ for some fixed vector space $V$ such that each braid generator $\sigma_i, i=1,...,n1,$ acts on $m$ consecutive tensor factors from $i$ through $i+m1$. The braid relation for $m=2$ is essentially the YangBaxter equation, and the cases for $m>2$ are called generalized YangBaxter equations. We observe that certain objects in ribbon fusion categories naturally give rise to such representations for the case $m=3$. Examples are given from the Ising theory (or the closely related $SU(2)_2$), $SO(N)_2$ for $N$ odd, and $SU(3)_3$. The solution from the JonesKauffman theory at a $6^{th}$ root of unity, which is closely related to $SO(3)_2$ or $SU(2)_4$, is explicitly described in the end.Geometry and Topology Monographs 03/2012; DOI:10.2140/gtm.2012.18.191  [Show abstract] [Hide abstract]
ABSTRACT: M. Hennings and G. Kuperberg defined quantum invariants Z_{Henn} and Z_{Kup} of closed oriented 3manifolds based on certain Hopf algebras, respectively. We prove that Z_{Kup}=Z_{Henn}^2 for lens spaces when both invariants are based on factorizable finite dimensional ribbon Hopf algebras.
Publication Stats
2k  Citations  
109.53  Total Impact Points  
Top Journals
Institutions

2007–2013

University of California, Santa Barbara
 Kavli Institute for Theoretical Physics
Santa Barbara, California, United States


2000–2006

Indiana University Bloomington
 Department of Mathematics
Bloomington, Indiana, United States


1992

Princeton University
 Department of Mathematics
Princeton, New Jersey, United States
