Topologization of electron liquids with Chern-Simons theory and quantum computation
We discuss a nexus among quantum topology, quantum physics and quantum computing.
cond-mat.mes-hall↗
arXiv subjects
Zhenghan Wang
Publications and source records attributed to Zhenghan Wang.
Universal manifold pairings and positivity
Gluing two manifolds M_1 and M_2 with a common boundary S yields a closed manifold M. Extending to formal linear combinations x=Sum_i(a_i M_i) yields a sesquilinear pairing p=<,> with values in (formal linear combinations of) closed manifolds. Topological quantum field theory (TQFT) represents this universal pairing p onto a finite dimensional quotient pairing q with values in C which in physically motivated cases is positive definite. To see if such a "unitary" TQFT can potentially detect any nontrivial x, we ask if is non-zero whenever x is non-zero. If this is the case, we call the pairing p positive. The question arises for each dimension d=0,1,2,.... We find p(d) positive for d=0,1, and 2 and not positive for d=4. We conjecture that p(3) is also positive. Similar questions may be phrased for (manifold, submanifold) pairs and manifolds with other additional structure. The results in dimension 4 imply that unitary TQFTs cannot distinguish homotopy equivalent simply connected 4-manifolds, nor can they distinguish smoothly s-cobordant 4-manifolds. This may illuminate the difficulties that have been met by several authors in their attempts to formulate unitary TQFTs for d=3+1. There is a further physical implication of this paper. Whereas 3-dimensional Chern-Simons theory appears to be well-encoded within 2-dimensional quantum physics, eg in the fractional quantum Hall effect, Donaldson-Seiberg-Witten theory cannot be captured by a 3-dimensional quantum system. The positivity of the physical Hilbert spaces means they cannot see null vectors of the universal pairing; such vectors must map to zero.
math.GT↗
The N-eigenvalue Problem and Two Applications
We consider the classification problem for compact Lie groups $G\subset U(n)$ 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.
math.RT↗
Extraspecial 2-groups and images of braid group representations
We investigate a family of (reducible) representations of Artin's 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.
math.RT↗
Quantum logic as motivated by quantum computing
We explore a connection between quantum logic and quantum computing.
math.LO↗
Density of the SO(3) TQFT representation of mapping class groups
We show that for an odd prime r > 3 and an integer g > 1, in the projective representation given by the SO(3) Witten-Chern-Simons theory at an rth root of unity, the image of the mapping class group of a surface of genus g is dense.
math.GT↗
A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
We describe a class of parity- and time-reversal-invariant topological states of matter which can arise in correlated electron systems in 2+1-dimensions. These states are characterized by particle-like excitations exhibiting exotic braiding statistics. $P$ and $T$ invariance are maintained by a `doubling' of the low-energy degrees of freedom which occurs naturally without doubling the underlying microscopic degrees of freedom. The simplest examples have been the subject of considerable interest as proposed mechanisms for high-$T_c$ superconductivity. One is the `doubled' version (i.e. two opposite-chirality copies) of the U(1) chiral spin liquid. The second example corresponds to $Z_2$ gauge theory, which describes a scenario for spin-charge separation. Our main concern, with an eye towards applications to quantum computation, are richer models which support non-Abelian statistics. All of these models, richer or poorer, lie in a tightly-organized discrete family. The physical inference is that a material manifesting the $Z_2$ gauge theory or a doubled chiral spin liquid might be easily altered to one capable of universal quantum computation. These phases of matter have a field-theoretic description in terms of gauge theories which, in their infrared limits, are topological field theories. We motivate these gauge theories using a parton model or slave-fermion construction and show how they can be solved exactly. The structure of the resulting Hilbert spaces can be understood in purely combinatorial terms. The highly-constrained nature of this combinatorial construction, phrased in the language of the topology of curves on surfaces, lays the groundwork for a strategy for constructing microscopic lattice models which give rise to these phases.
cond-mat.str-el↗
Quantum SU(2) faithfully detects mapping class groups modulo center
The Jones-Witten theory gives rise to representations of the (extended) mapping class group of any closed surface Y indexed by a semi-simple Lie group G and a level k. In the case G=SU(2) these representations (denoted V_A(Y)) have a particularly simple description in terms of the Kauffman skein modules with parameter A a primitive 4r-th root of unity (r=k+2). In each of these representations (as well as the general G case), Dehn twists act as transformations of finite order, so none represents the mapping class group M(Y) faithfully. However, taken together, the quantum SU(2) representations are faithful on non-central elements of M(Y). (Note that M(Y) has non-trivial center only if Y is a sphere with 0, 1, or 2 punctures, a torus with 0, 1, or 2 punctures, or the closed surface of genus = 2.) Specifically, for a non-central h in M(Y) there is an r_0(h) such that if r>= r_0(h) and A is a primitive 4r-th root of unity then h acts projectively nontrivially on V_A(Y). Jones' [J] original representation rho_n of the braid groups B_n, sometimes called the generic q-analog-SU(2)-representation, is not known to be faithful. However, we show that any braid h not= id in B_n admits a cabling c = c_1,...,c_n so that rho_N (c(h)) not= id, N=c_1 + ... + c_n.
math.GT↗
On Freedman's lattice models for topological phases
Freedman proposes a family of Hamiltonians $H_{0,l}$ which define quantum loop gas models on any celluated compact surface. We study the simplest nontrivial cases: celluations of the torus. Our numerical data support Freedman's conjecture, but the conjectured space of ground states does not come out in full.
math-ph↗
Topological Quantum Computation
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like $e^{-\ał}$, where $ł$ is a length scale, and $α$ is some positive constant. In contrast, the $\q$presumptive" qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about $10^{-4}$) before computation can be stabilized.
quant-ph↗
The two-eigenvalue problem and density of Jones representation of braid groups
Introduction 1. The two-eigenvalue problem 2. Hecke algebra representations of braid groups 3. Duality of Jones-Wenzl representations 4. Closed images of Jones-Wenzl sectors 5. Distribution of evaluations of Jones polynomials 6. Fibonacci representations
math.GT↗
Simulation of topological field theories by quantum computers
Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned topological models having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H = 0. These are called topological quantum filed theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model BQP. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.
quant-ph↗
A modular functor which is universal for quantum computation
We show that the topological modular functor from Witten-Chern-Simons theory is universal for quantum computation in the sense a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern-Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation, have topological implications which will be considered elsewhere.
quant-ph↗
Fermat limit and congruence of Ohtsuki invariants
By calculating the Fermat limit of certain q-Fermat functions, we get explicit surgery formulae for the second and third Ohtsuki invariants for homology 3-spheres. The surgery formula of the second Ohtsuki invariant λ_2, combined with an argument using the general theory of finite type invariants of homology 3-spheres, leads to a congruence relation between the first and second Ohtsuki invariants: λ_1=2λ_2 mod 24.
math.GT↗
Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function
A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a 1-string link whose closure is the knot in question. The colored Jones polynomial then counts with weights families of ``self-avoiding'' cycles of random walk on the cabling of the 1-string link. As a consequence of such interpretations of the Alexander and colored Jones polynomials, the computation of the limit of the renormalized colored Jones polynomial when the coloring (or cabling) parameter tends to infinity whereas the weight parameter tends to 1 leads immediately to a new proof of the Melvin-Morton conjecture, which was first established by Rozansky and by Bar-Natan and Garoufalidis.
math.GT↗
Burau representation and random walk on string links
Using a probabilistic interpretation of the Burau representation of the braid group offered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string links. This representation is determined by a linear system, and is dominated by finite type string link invariants. For positive string links, the representation matrix can be interpreted as the transition matrix of a Markov process. For positive non-separable links, we show that all states are persistent. (This is an extensively revised version of the first author's original paper titled "Burau representation and random walk on knots".)
q-alg↗
On Ohtsuki's invariants of integral homology 3-spheres, I
An attempt is made to conceptualize the derivation as well as to facilitate the computation of Ohtsuki's rational invariants $λ_n$ of integral homology 3-spheres extracted from Reshetikhin-Turaev SU(2) quantum invariants. Several interesting consequences will follow from our computation of $λ_2$. One of them says that $λ_2$ is always an integer divisible by 3. It seems interesting to compare this result with the fact shown by Murakami that $λ_{1}$ is 6 times the Casson invariant. Other consequences include some general criteria for distinguishing homology 3-spheres obtained from surgery on knots by using the Jones polynomial.
q-alg↗
Integral geometry of plane curves and knot invariants
We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related with invariants of generic plane curves defined by Arnold recently with deep motivations in symplectic and contact geometry. Quadratic bounds on these plane curve invariants are derived using their relationship with the knot invariant.
dg-ga↗