Topological quantum computation

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<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript minus alpha script l"><mml:semantics><mml:msup><mml:mi>e</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mi>α</mml:mi><mml:mi>ℓ</mml:mi></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">e^{-\alpha \ell }</mml:annotation></mml:semantics></mml:math></inline-formula>, where<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"><mml:semantics><mml:mi>ℓ</mml:mi><mml:annotation encoding="application/x-tex">\ell</mml:annotation></mml:semantics></mml:math></inline-formula>is a length scale, and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"><mml:semantics><mml:mi>α</mml:mi><mml:annotation encoding="application/x-tex">\alpha</mml:annotation></mml:semantics></mml:math></inline-formula>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<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="10 Superscript negative 4"><mml:semantics><mml:msup><mml:mn>10</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">10^{-4}</mml:annotation></mml:semantics></mml:math></inline-formula>) before computation can be stabilized.