[16.] Finite linear quotients of B3 of low dimension (with I. Tuba).
Submitted. (6/08)
[15.] Unitary braid representations with finite image (with M. J. Larsen).
Submitted. (5/08)
[14.] Two paradigms for topological quantum computation.
Submitted.
Abstract: We present two paradigms relating algebraic, topological and quantum
computational statistics for the topological model for quantum computation. In
particular we suggest correspondences between the computational power of
topological quantum computers, computational complexity of link invariants and
images of braid group representations. While at least parts of these paradigms
are well-known to experts, we provide supporting evidence for them in terms of recent
results. We give a fairly comprehensive list of known examples and formulate
two conjectures that would further support the paradigms.
Full paper (3/08) arXiv:math.QA
0803.1258
[13.] On classification of modular tensor categories (with Richard Stong and Zhenghan Wang)
Submitted.
Abstract: We classify all unitary modular tensor categories (UMTCs) of rank at most 4. There are a total of 70 UMTCs of rank at most 4. Each such UMTC can be obtained from 8 non-trivial prime UMTCs by direct product, and some symmetry operations. The 8 non-trivial prime UMTCs are the semion MTC, the Fibonacci MTC or (A1; 3) 1/2, the Z3 MTC, the Ising MTC, the even half of an SU(2) MTC at level 5 or (A1; 5) 1/2, the Z4 MTC, the toric code MTC, and the even half of an SU(2) MTC at level 7 or (A1; 7)1/2. Out of the 8 non-trivial prime UMTCs, 6 are quantum group categories for a simple Lie group: the semion=SU(2)1, the Fibonacci=(G2)1, the Z3=SU(3)1, Ising=complex conjugate of (E8)2, the Z4 = SU(4)1, and the toric code=SO(16)1. UMTCs encode topological properties of anyonic quantum systems and can be used to build fault-tolerant quantum computers. We conjecture that there are only finitely many equivalence classes of MTCs for any given rank, and a UMTC is universal for anyonic quantum computation if and only if its global quantum dimension D2 is not an integer.
Discovery of UMTCs in Nature with non-abelian anyons will be a landmark in condensed matter physics. The Z3 MTC is realized by fractional quantum Hall (FQH) liquids at filling fraction v = 1/3 . The Read-Rezayi conjecture predicts the realization of SU(2)k MTCs in FQH liquids at filling fractions v = 2+ k/(k+2). For v = 5/2 and SU(2)2, there is a numerical proof and experimental evidence for this conjecture. The Ising MTC is believed to be realized by chiral superconductors (strontium ruthenate). There are theoretical designs for the Fib X Fib MTC using optical lattice, and for the toric code MTC using Josephson junction.
Full paper (12/07) PDF
| arXiv: math.QA
0712.1377
[12.] On exotic modular tensor categories (with Seung-moon
Hong and Zhenghan Wang)
To appear in Commun. Contemp.
Math.
Abstract: It has been conjectured that every (2+1)-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair (G, l), where G is a compact Lie group, and LÎ H4(BG; Z) a cohomology class. We study two TQFTs constructed from Jones’ subfactor theory which are believed to be counterexamples to this conjecture: one is the quantum double of the even sectors of the E6 subfactor, and the other is the quantum double of the even sectors of the Haagerup subfactor. We cannot prove mathematically that the two TQFTs are indeed counterexamples because CSW TQFTs, while physically defined, are not yet mathematically constructed for every pair (G, l). The cases that are constructed mathematically include:
(1) G is a finite group—the Dijkgraaf-Witten TQFTs;
(2) G is torus Tn;
(3) G is a connected semi-simple Lie group—the Reshetikhin-Turaev TQFTs.
We prove that the two TQFTs are not among those mathematically constructed TQFTs or their direct products. Both TQFTs are of the Turaev-Viro type: quantum doubles of spherical tensor categories. We further prove that neither TQFT is a quantum double of a braided fusion category, and give evidence that neither is an orbifold or coset of TQFTs above. Moreover, representation of the braid groups from the half E6 TQFT can be used to build universal topological quantum computers, and the same is expected for the Haagerup case.
Full paper (10/07): arXiv: math.GT 0710.5761
[11.] Unitarizablity of premodular categories
J. Pure Appl. Algebra. 212
(2008), no. 8 1878-1887.
Abstract: We study the unitarizability of premodular categories constructed from representations of quantum group at roots of unity. We introduce Grothendieck unitarizability as a natural generalization of unitarizability to any class of premodular categories with a common Grothendieck semiring. We obtain new results for quantum groups of Lie types F4 and G2, and improve the known results for Lie types B and C.
Full paper (10/07): arXiv: math.QA 0710.1621
[10.] From Extra-Special Two-Groups to GHZ states (with Yong Zhang,
Yong-Shi Wu, Zhenghan Wang and Mo-Lin Ge)
Submitted.
Abstract: In this paper we explore natural connections among extraspecial 2-groups, almost-complex structures, unitary representations of the braid group and the Greenberger-Horne-Zeilinger (GHZ) states. We first present new representations of extraspecial 2-groups in terms of almost-complex structures and use them to derive new unitary braid representations as extensions of representations of the extraspecial 2-groups by the symmetric group. A few subtleties related to the correspondence between the unitary braid representations and the GHZ states (particularly those for an odd number of qubits) are clarified. We also discuss Yang–Baxterization of the new braid group representations and unitary evolution of the GHZ states. Our study suggests that the unitary braiding quantum gates may play an important role, through extraspecial 2-groups, in quantum error correction and topological quantum computing.
Full paper (6/07): arXiv: quant-ph/0706.1761v2
[9]. Braid representations
from twisted quantum doubles of finite groups (with Pavel
Etingof and Sarah Witherspoon)
Abstract: We investigate the braid group representations
arising from categories of representations of twisted quantum doubles of finite
groups. For these categories, we show that the resulting braid group
representations always factor through finite groups, in contrast to the
categories associated with quantum groups at roots of unity. We also show that
in the case of p-groups, the corresponding pure braid group representations
factor through a finite p-group, which answers a question asked by V. Drinfeld.
Full paper (3/07): arXiv: math.QA/0703274
[8].
An algebra-level version of a link-polynomial identity of Lickorish (with Michael Larsen)
Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 3, 623-638.
Abstract: We establish isomorphisms
between certain specializations of Birman-Murakami-Wenzl algebras and the symmetric squares of Temperley-Lieb algebras. These isomorphisms
imply a link-polynomial identity due to W. B. R. Lickorish.
As an application, we compute the closed images of the irreducible braid group
representations factoring over these specialized BMW algebras.
Full paper
(5/06): arXiv: math.QA/0605455
[7]. Generating functions for ranks of pre-modular
categories.
Subsumed in [4] below.
Abstract: We derive generating functions for the
ranks of pre-modular categories associated with quantum groups at roots of
unity.
Full paper (8/05): arXiv math.QA/0509457
[6]. The N-eigenvalue problem and two applications (with
Michael Larsen and Zhenghan Wang)
Int.
Math. Res. Not. 2005 no. 64 (2005)
3987-4018.
Abstract: We consider the classification problem for
compact Lie subgroups of U(n) which are generated by a
single conjugacy class with a fixed number N of
distinct eigenvalues. We give an explicit
classification when N=3 and apply this to extract information about Galois
representations and braid group representations.
Full paper (11/05) PDF | arXiv
math.RT/0506025
[5]. Extraspecial 2-groups and images
of braid group representations (with Jennifer Franko
and Zhenghan Wang)
J. Knot Theory
Ramifications 15 no. 4 (2006) 413-428.
Abstract: We investigate a family of (reducible) representations
of the braid groups corresponding to a specific solution to the Yang-Baxter
equation. The images of the braid groups
under these representations are
finite groups, and we identify them precisely as extensions of extra-special
2-groups. The decompositions of the representations into their
irreducible constituents are determined, which allows us to relate them to the
well-known Jones representations of the braid groups factoring over Temperley-Lieb algebras and the corresponding link invariants.
Full paper (3/05): PDF | arXiv math.RT/0503435
[4]. From quantum
groups to unitary modular tensor categories
Abstract:. Modular tensor
categories are generalizations of the representation theory of quantum groups
at roots of unity axiomatizing the properties
necessary to produce 3-dimensional TQFT and invariants of 3-manifolds. Although
other constructions have since been found, quantum groups remain the most
prolific source. Recently proposed applications to quantum computing have
provided an impetus to understand and describe these examples as explicitly as
possible, especially those that are "physically feasible." We survey
the current status of the problem of producing unitary modular tensor
categories from quantum groups.
Full paper (1/06): PDF | arXiv
math.QA/0503226
[3]. A note on tensor categories of Lie type E9
Journal of
Algebra 284 no. 1 (2005), 296-309.
Abstract:. We consider the
problem of decomposing tensor powers of the fundamental level 1 highest weight
representation V of the affine Kac-Moody algebra
associated to the Dynkin diagram E9. We describe an elementary
algorithm for determining the decomposition of the submodule
of the nth tensor power of V whose irreducible direct summands have highest
weights which are maximal with respect to the null-root. This decomposition is
based on Littelmann's path algorithm and conforms with the uniform combinatorial behavior recently discovered
by H. Wenzl for the series EN for N different from 9.
Full paper (6/7/04): PDF | arXiv:math.RT/0406122
[2]. On a family on non-unitarizable
ribbon categories
Mathematische
Zeitschrift 250 no. 4 (2005), 745-774.
Abstract:. We consider two
families of categories. The first is the family of semisimple
quotients of H. Andersen's tilting module categories for quantum groups of Lie
type B specialized at odd roots of unity. The second consists of categories
constructed from a particular family of finite-dimensional quotients of the
group algebra of Artin's braid group known as
BMW-algebras of type BC. Our main result is to show that these families
coincide as braided tensor categories using a recent theorem of Tuba and Wenzl. The morphism spaces in
these categories can be equipped with a Hermitian
form, and we are able to show that these categories are never unitary, and no
braided tensor category sharing the Grothendieck semiring common to these families is unitarizable.
Full paper (3/12/04): PDF | arXiv: math.QA/0403217
[1]. On tensor categories arising from quantum
groups and BMW-algebras at odd roots of unity.
PhD Thesis,
Abstract: Most of the results in my thesis are published in [2].
Full paper (
Last updated after 11/02/07