arXiv subjects

Michael Freedman

Publications and source records attributed to Michael Freedman.

64 records · Page 4Linked to original sources

arXiv2005

Towards Universal Topological Quantum Computation in the $ν=5/2$ Fractional Quantum Hall State

The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction $ν= 5/2$, can support topologically-protected qubits with extremely low error rates. Braiding operations also allow perfect implementation of certain unitary transformations of these qubits. However, in the case of the Pfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is not dense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian state supports universal quantum computation, albeit with some operations which require error correction. On the other hand, if certain topology-changing operations can be implemented, then fully topologically-protected universal quantum computation is possible. In order to accomplish this, it is necessary to measure the interference between quasiparticle trajectories which encircle other moving trajectories in a time-dependent Hall droplet geometry.

cond-mat.mes-hall↗

arXiv2004

Topologically-Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State

The Pfaffian state is an attractive candidate for the observed quantized Hall plateau at Landau level filling fraction $ν=5/2$. This is particularly intriguing because this state has unusual topological properties, including quasiparticle excitations with non-Abelian braiding statistics. In order to determine the nature of the $ν=5/2$ state, one must measure the quasiparticle braiding statistics. Here, we propose an experiment which can simultaneously determine the braiding statistics of quasiparticle excitations and, if they prove to be non-Abelian, produce a topologically-protected qubit on which a logical NOT operation is performed by quasiparticle braiding. Using the measured excitation gap at $ν=5/2$, we estimate the error rate to be $10^{-30}$ or lower.

cond-mat.mes-hall↗

arXiv2004

An extended Hubbard model with ring exchange: a route to a non-Abelian topological phase

We propose an extended Hubbard model on a 2D Kagome lattice with an additional ring-exchange term. The particles can be either bosons or spinless fermions . At a special filling fraction of 1/6 the model is analyzed in the lowest non-vanishing order of perturbation theory. Such ``undoped'' model is closely related to the Quantum Dimer Model. We show how to arrive at an exactly soluble point whose ground state manifold is the extensively degenerate ``d-isotopy space'', a necessary precondition for for a certain type of non-Abelian topological order. Near the ``special'' values, $d = 2 \cos π/(k+2)$, this space is expected to collapse to a stable topological phase with anyonic excitations closely related to SU(2) Chern-Simons theory at level k.

cond-mat.str-el↗

arXiv2004

On the asymptotics of quantum SU(2) representations of mapping class groups

We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid groups and hyperelliptic mapping class groups do not have Kazhdan's property (T). On the other hand, the representations of the mapping class group of the torus do not have almost invariant vectors, in fact they converge to the metaplectic representation of SL(2,Z) on L^2(R). As a consequence we obtain a curious analytic fact about the Fourier transform on the real line which may not have been previously observed.

math.QA↗

arXiv2004

A Line of Critical Points in 2+1 Dimensions: Quantum Critical Loop Gases and Non-Abelian Gauge Theory

We (1) construct a one-parameter family of lattice models of interacting spins; (2) obtain their exact ground states; (3) derive a statistical-mechanical analogy which relates their ground states to O(n) loop gases; (4) show that the models are critical for $d\leq \sqrt{2}$, where $d$ parametrizes the models; (5) note that for the special values $d=2\cos(π/(k+2))$, they are related to doubled level-$k$ SU(2) Chern-Simons theory; (6) conjecture that they are in the universality class of a non-relativistic SU(2) gauge theory; and (7) show that its one-loop $β$-function vanishes for all values of the coupling constant, implying that it is also on a critical line.

cond-mat.str-el↗

arXiv2003

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↗

arXiv2003

Non-Abelian topological phases in an extended Hubbard model

We describe four closely related Hubbard-like models (models A, B, C and D) of particles that can hop on a 2D Kagome lattice interacting via Coulomb repulsion. The particles can be either bosons (models A and B) or (spinless) fermions (models C and D). Models A and C also include a ring exchange term. In all four cases we solve equations in the model parameters to arrive at an exactly soluble point whose ground state manifold is the extensively degenerate ``d-isotopy space'' $\bar{V}_d$, 0<d<2. Near the ``special'' values, $d = 2 \cos π/k+2$, $\bar{V}_d$ should collapse to a stable topological phase with anyonic excitations closely related to SU(2) Chern-Simons theory at level k. We mention simplified models $A^-$ and $C^-$ which may also lead to these topological phases.

cond-mat.str-el↗

arXiv2002

Diameters of Homogeneous Spaces

Let G be a compact connected Lie group with trivial center. Using the action of G on its Lie algebra, we define an operator norm | |_{G} which induces a bi-invariant metric d_G(x,y)=|Ad(yx^{-1})|_{G} on G. We prove the existence of a constant β\approx .12 (independent of G) such that for any closed subgroup H \subsetneq G, the diameter of the quotient G/H (in the induced metric) is \geq β.

quant-ph↗

arXiv2000

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↗

arXiv1997

Extension of incompressible surfaces on the boundary of 3-manifolds

An incompressible surface $F$ on the boundary of a compact orientable 3-manifold $M$ is arc-extendible if there is an arc $γ$ on $\partial M - $ Int $F$ such that $F \cup N(γ)$ is incompressible, where $N(γ)$ is a regular neighborhood of $γ$ in $\partial M$. Suppose for simplicity that $M$ is irreducible, and $F$ has no disk components. If $M$ is a product $F\times I$, or if $\partial M - F$ is a set of annuli, then clearly $F$ is not arc-extendible. The main theorem of this paper shows that these are the only obstructions for $F$ to be arc-extendible.

math.GT↗