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 −α , where 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], [27], and simulating physical systems [5], [21]. To this list we may add a fourth class of algorithms which yield approximate, but rapid, evaluations of many quantum invariants of three dimensional manifolds, e.g., the absolute value of the Jones polynomial of a link L at certain roots of unity: |V L (e 2πi 5 )|. This seeming curiosity is actually the tip of an iceberg which links quantum computation both to low dimensional topology and the theory of anyons; the motion of anyons in a two dimensional system defines a braid in 2 + 1 dimension. This iceberg is a model of quantum computation based on topological, rather than local, degrees of freedom.The class of functions, BQP (functions computable with bounded error, given quantum resources, in polynomial time), has been defined in three distinct but equivalent ways: via quantum Turing machines [2], quantum circuits [3], [6], and modular functors [7], [8]. The last is the subject of this article. We may now propose a "thesis" in the spirit of Alonzo Church: all "reasonable" computational models which add the resources of quantum mechanics (or quantum field theory) to classical computation yield (efficiently) inter-simulable classes: there is one quantum theory of computation. (But alas, we are not so sure of our thesis at Planck scale energies. Who is to say that all the observables there must even be computable functions in the sense of Turing?)The case for quantum computation rests on three pillars: inevitability-Moore's law suggests we will soon be doing it whether we want to or not, desirability-the above mentioned algorithms, and finally feasibility-which in the past has been
Topological phases are unique states of matter incorporating long-range quantum entanglement, hosting exotic excitations with fractional quantum statistics. We report a practical method to identify topological phases in arbitrary realistic models by accurately calculating the Topological Entanglement Entropy (TEE) using the Density Matrix Renormalization Group (DMRG). We argue that the DMRG algorithm naturally produces a minimally entangled state, from amongst the quasi-degenerate ground states in a topological phase.This proposal both explains the success of this method, and the absence of ground state degeneracy found in prior DMRG sightings of topological phases.We demonstrate the effectiveness of the calculational procedure by obtaining the TEE for several microscopic models, with an accuracy of order 10 −3 when the circumference of the cylinder is around ten times the correlation length.As an example, we definitively show the ground state of the quantum S = 1/2 antiferromagnet on the kagomé lattice is a topological spin liquid, and strongly constrain the full identification of this phase of matter. 1 arXiv:1205.4289v1 [cond-mat.str-el]
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.
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