Abstract. 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−αℓ, where ℓ is a length scale, and α is some positive constant. In contrast, the “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. Quantum computation is a catch-all for several models of computation based on a theoretical ability to manufacture, manipulate and measure quantum states. In this context, there are three areas where remarkable algorithms have been found: searching a data base [15], abelian groups (factoring and discrete logarithm) [19],

Abstract: 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 field 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. 1.

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After a brief introduction to the principles and promise of quantum information processing, the requirements for the physical implementation of quantum computation are discussed. These five requirements, plus two relating to the communication of quantum information, are extensively explored and related to the many schemes in atomic physics, quantum optics, nuclear and electron magnetic resonance spectroscopy, superconducting electronics, and quantum-dot physics, for achieving quantum computing. 1.

Kevin Walker, and Zhenghan Wang. Their work has been the inspiration for this lecture. The mathematical problem of localizing modular functors to neighborhoods of points is shown to be closely related to the physical problem of engineering a local Hamiltonian for a computationally universal quantum medium. For genus = 0 surfaces, such a local Hamiltonian is mathematically defined. Braiding defects of this medium implements a representation associated to the Jones polynomial and this representation is known to be universal for quantum computation. 1 The Picture Principle Reality has the habit of intruding on the prodigies of purest thought and encumbering them with unpleasant embellishments. So it is astonishing when the chthonian hammer of the engineer resonates precisely to the gossamer fluttering of theory. Such a moment may soon be at hand in the practice and theory of quantum computation. The most compelling theoretical question, “localization, ” is yielding an answer which points the way to a solution of Based on lectures prepared for the joint Microsoft/University of Washington celebration

Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing. ” The section on soap bubbles even includes some “experimental ” results. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1

Merely by existing, all physical systems register information. And by evolving dynamically in time, they transform and process that information. The laws of physics determine the amount of information that a physical system can register (number of bits) and the number of elementary logic operations that a system can perform (number of ops). The universe is a physical system. This paper quantifies the amount of information that the universe can register and the number of elementary operations that it can have performed over its history. The universe can have performed no more than 10 120 ops on 10 90 bits. ‘Information is physical ’ 1. This statement of Landauer has two complementary interpretations. First, information is registered and processed by physical systems. Second, all physical systems register and process information. The description of physical systems in terms of information and information processing is complementary to the conventional description of physical system in terms of the laws of physics. A recent paper by the author2 put bounds on the amount of information processing that can be performed by physical systems. The first limit is on speed. The Margolus/Levitin theorem3 implies that the total

A rather elementary family of local Hamiltonians H◦,ℓ,ℓ = 1,2,3,..., is described for a 2−dimensional quantum mechanical system of spin = 1 2 particles. On the torus, the ground state space G◦,ℓ is essentially infinite dimensional but may collapse under “perturbation ” to an anyonic system with a complete mathematical description: the quantum double of the SO(3)−Chern-Simons modular functor at q = e 2πi/ℓ+2 which we call DEℓ. The Hamiltonian H◦,ℓ defines a quantum loop gas. We argue that for ℓ = 1 and 2, G◦,ℓ is unstable and the collapse to Gǫ,ℓ ∼ = DEℓ can occur truly by perturbation. For ℓ ≥ 3 G◦,ℓ is stable and in this case finding Gǫ,ℓ ∼ = DEℓ must require either ǫ> ǫℓ> 0, help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes “ ”. A hypothetical phase diagram is included in the introduction. The effect of perturbation is studied algebraically: the ground state G◦,ℓ of H◦,ℓ is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state Gǫ,ℓ described by a quotient algebra. By classification, this implies Gǫ,ℓ ∼ = DEℓ. The fundamental point is that nonlinear structures

We analyze relationships between the Jones polynomial and quantum computation. First, we present two polynomial-time quantum algorithms which give additive approximations of the Jones polynomial, in the sense of Bordewich, Freedman, Lovász and Welsh, of any link obtained from a certain general family of closures of braids, evaluated at any primitive root of unity. This family encompasses the well-known plat and trace closures, generalizing results recently obtained by Aharonov, Jones and Landau. We base our algorithms on a local qubit implementation of the unitary Jones-Wenzl representations of the braid group which makes the underlying representation theory apparent, allowing us to provide an algorithm for approximating the HOMFLYPT two-variable polynomial of the trace closure of a braid at certain pairs of values as well. Next, we provide a self-contained proof that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. This theorem was originally proved by Freedman, Larsen and Wang in the context of topological quantum computation, and the necessary notion of approximation was later provided by Bordewich et al. Our proof is simpler as it uses a more natural encoding of two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. Finally, we conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, without taking the usual route through the Tutte polynomial and graph coloring. 1

A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and non-Abelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.