Zhenghan Wang

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

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Publications (52)100.62 Total impact

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    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.
    11/2014;
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    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.
    11/2014;
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    Shawn X. Cui, Zhenghan Wang
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    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.
    07/2014;
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    Shawn X. Cui, Zhenghan Wang
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    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.
    05/2014;
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    Shawn X. Cui, Seung-Moon Hong, Zhenghan Wang
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    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.
    01/2014;
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    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..
    10/2013;
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    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.
    CoRR. 05/2013; abs/0810.0033.
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    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.
    Physical review. B, Condensed matter 03/2013; 87(23). · 3.77 Impact Factor
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    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.
    Communications in Mathematical Physics 03/2013; · 1.97 Impact Factor
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    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).
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    Zhenghan Wang
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    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.
    01/2013;
  • Zhenghan Wang
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    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). · 1.75 Impact Factor
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    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.
    Physical review. B, Condensed matter 10/2012; 87(16). · 3.77 Impact Factor
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    ABSTRACT: Galois conjugation relates unitary conformal field theories (CFTs) and topological quantum field theories (TQFTs) to their non-unitary counterparts. Here we investigate Galois conjugates of quantum double models, such as the Levin-Wen model. While these Galois conjugated Hamiltonians are typically non-Hermitian, we find that their ground state wave functions still obey a generalized version of the usual code property (local operators do not act on the ground state manifold) and hence enjoy a generalized topological protection. The key question addressed in this paper is whether such non-unitary topological phases can also appear as the ground states of Hermitian Hamiltonians. Specific attempts at constructing Hermitian Hamiltonians with these ground states lead to a loss of the code property and topological protection of the degenerate ground states. Beyond this we rigorously prove that no local change of basis can transform the ground states of the Galois conjugated doubled Fibonacci theory into the ground states of a topological model whose Hermitian Hamiltonian satisfies Lieb-Robinson bounds. These include all gapped local or quasi-local Hamiltonians. A similar statement holds for many other non-unitary TQFTs. One consequence is that the "Gaffnian" wave function cannot be the ground state of a gapped fractional quantum Hall state.
    Physical review. B, Condensed matter 06/2011; 85(4). · 3.77 Impact Factor
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    Liang Chang, Zhenghan Wang
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    ABSTRACT: M. Hennings and G. Kuperberg defined quantum invariants Z_{Henn} and Z_{Kup} of closed oriented 3-manifolds 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.
    06/2011;
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    Kevin Walker, Zhenghan Wang
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    ABSTRACT: Levin-Wen models are microscopic spin models for topological phases of matter in (2+1)-dimension. We introduce a generalization of such models to (3+1)-dimension based on unitary braided fusion categories, also known as unitary premodular categories. We discuss the ground state degeneracy on 3-manifolds and statistics of excitations which include both points and defect loops. Potential connections with recently proposed fractional topological insulators and projective ribbon permutation statistics are described.
    Frontiers of Physics 04/2011; 7(2). · 1.59 Impact Factor
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    Eric C. Rowell, Zhenghan Wang
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    ABSTRACT: Governed by locality, we explore a connection between unitary braid group representations associated to a unitary $R$-matrix and to a simple object in a unitary braided fusion category. Unitary $R$-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q=exp(\pi i/6) specialization of the unitary Jones representation of the braid groups can be localized by a unitary $9\times 9$ R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N),2) for an odd prime N>1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science. Comment: 27 pages; version 2: clarified definitions, expanded section 6
    Communications in Mathematical Physics 09/2010; · 1.97 Impact Factor
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    ABSTRACT: In a recent paper, Teo and Kane proposed a 3D model in which the defects support Majorana fermion zero modes. They argued that exchanging and twisting these defects would implement a set R of unitary transformations on the zero mode Hilbert space which is a 'ghostly' recollection of the action of the braid group on Ising anyons in 2D. In this paper, we find the group T_{2n} which governs the statistics of these defects by analyzing the topology of the space K_{2n} of configurations of 2n defects in a slowly spatially-varying gapped free fermion Hamiltonian: T_{2n}\equiv {\pi_1}(K_{2n})$. We find that the group T_{2n}= Z \times T^r_{2n}, where the 'ribbon permutation group' T^r_{2n} is a mild enhancement of the permutation group S_{2n}: T^r_{2n} \equiv \Z_2 \times E((Z_2)^{2n}\rtimes S_{2n}). Here, E((Z_2)^{2n}\rtimes S_{2n}) is the 'even part' of (Z_2)^{2n} \rtimes S_{2n}, namely those elements for which the total parity of the element in (Z_2)^{2n} added to the parity of the permutation is even. Surprisingly, R is only a projective representation of T_{2n}, a possibility proposed by Wilczek. Thus, Teo and Kane's defects realize `Projective Ribbon Permutation Statistics', which we show to be consistent with locality. We extend this phenomenon to other dimensions, co-dimensions, and symmetry classes. Since it is an essential input for our calculation, we review the topological classification of gapped free fermion systems and its relation to Bott periodicity.
    Physical review. B, Condensed matter 05/2010; 83. · 3.77 Impact Factor
  • Physical Review Letters 08/2009; 103(9). · 7.73 Impact Factor
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    ABSTRACT: Quantum mechanical systems, whose degrees of freedom are so-called su(2)k anyons, form a bridge between ordinary SU(2) quantum magnets (of arbitrary spin-S) and systems of interacting non-Abelian anyons. Anyonic spin-1/2 chains exhibit a topological protection mechanism that stabilizes their gapless ground states and which vanishes only in the limit (k-->infinity) of the ordinary spin-1/2 Heisenberg chain. For anyonic spin-1 chains the phase diagram closely mirrors the one of the biquadratic SU(2) spin-1 chain. Our results describe, at the same time, nucleation of different 2D topological quantum fluids within a "parent" non-Abelian quantum Hall state, arising from a macroscopic occupation with localized, interacting anyons. The edge states between the "nucleated" and the parent liquids are neutral, and correspond precisely to the gapless modes of the anyonic chains.
    Physical Review Letters 08/2009; 103(7):070401. · 7.73 Impact Factor

Publication Stats

1k Citations
100.62 Total Impact Points

Institutions

  • 2006–2013
    • University of California, Santa Barbara
      • • Kavli Institute for Theoretical Physics
      • • Department of Physics
      Santa Barbara, California, United States
  • 2009
    • ETH Zurich
      • Institute for Theoretical Physics
      Zürich, ZH, Switzerland
  • 2004
    • Indiana University East
      Indiana, United States
  • 2000–2004
    • Indiana University Bloomington
      • Department of Mathematics
      Bloomington, Indiana, United States