Publications (68)114.74 Total impact
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ABSTRACT: We obtain a classification of metaplectic modular categories: every metaplectic modular category is a gauging of the particlehole symmetry of a cyclic modular category. Our classification suggests a conjecture that every weaklyintegral modular category can be obtained by gauging a symmetry of a pointed modular category.  [Show abstract] [Hide abstract]
ABSTRACT: Topological order of a topological phase of matter in two spacial dimensions is encoded by a unitary modular (tensor) category (UMC). A group symmetry of the topological phase induces a group symmetry of its corresponding UMC. Gauging is a wellknown theoretical tool to promote a global symmetry to a local gauge symmetry. We give a mathematical formulation of gauging in terms of higher category formalism. Roughly, given a UMC with a symmetry group $G$, gauging is a $2$step process: first extend the UMC to a $G$crossed braided fusion category and then take the equivariantization of the resulting category. Gauging can tell whether or not two enriched topological phases of matter are different, and also provides a way to construct new UMCs out of old ones. We derive a formula for the $H^4$obstruction, prove some properties of gauging, and carry out gauging for two concrete examples.  [Show abstract] [Hide abstract]
ABSTRACT: We generalize Ng's twovariable algebraic/combinatorial zeroth framed knot contact homology for framed oriented knots in S3 to knots in S1 × S2, and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Actually, our cord algebra has an extra variable, which potentially corresponds to the third variable in Ng's threevariable knot contact homology. Our main tool is Lin's generalization of the Markov theorem for braids in S3 to braids in S1 × S2. We conjecture that our framed cord algebras are always finitely generated for nonlocal knots.  [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 nonabelian statistics. It justifies the definition of a nonabelian anyon as one with quantum dimension $>1$. Since nonabelian statistics presumes degeneracy, degeneracy is more fundamental than nonabelian statistics.  [Show abstract] [Hide abstract]
ABSTRACT: The feasibility of a classificationbyrank program for modular categories follows from the RankFiniteness 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 prove a rankfiniteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category $\mathcal{C}$ with $N=ord(T)$, the order of the modular $T$matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension $D^2$ in the Dedekind domain $\mathbb{Z}[e^{\frac{2\pi i}{N}}]$ is identical to that of $N$.  [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 $k=4$ 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.  [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 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.  [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.  [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. 
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.  [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.  [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.
Publication Stats
2k  Citations  
114.74  Total Impact Points  
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Institutions

20072015

University of California, Santa Barbara
 Department of Physics
Santa Barbara, California, United States


19982006

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


1992

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