A Modular Functor Which is Universal¶for Quantum Computation

By Michael H. Freedman, Michael Larsen, Zhenghan Wang. 2000

578 citations recorded by the source corpus.

Abstract:We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that 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 has topological implications which will be considered elsewhere.

Subfields: Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b1737d644119955bf3849cd4b3d960003dbea95e

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Citations in the corpus

Quantum memories at finite temperature (2014, background): “Models that achieve universal fault-tolerant quantum computation by anyon braiding are well known (Brennen and Pachos, 2007; Freedman et al., 2002; Kitaev, 2003).”
A Polynomial Quantum Algorithm for Approximating the Jones Polynomial (2005, methodology): “A model of quantum computation based on Topological Quantum Field Theory (TQFT) and Chern-Simons theory was defined in [10, 11], and Kitaev, Larsen, Freedman and Wang showed that this model is polynomially equivalent in computational power to the standard quantum computation model in [12, 13 ].”
Quantum algorithms: an overview (2015, unknown): “The inspiration for other quantum algorithms, not discussed here, includes topological quantum (cid:12)eld theory [43]; connections between quantum circuits and spin models [32]; the Elitzur-Vaidman quantum bomb tester [63]; and directly solving the semide(cid:12)nite programming problem…”
Non-Abelian Anyons and Interferometry (2007, unknown): “[79, 80, 81, 82, 83, 84, 85].”
Quantum algorithm for linear systems of equations. (2008, background): “[11] M.”
Universal topological quantum computation from a superconductor/Abelian quantum Hall heterostructure (2013, background): “Fibonacci anyon from the vacuum, is sufficient to perform any desired quantum computation in a completely faulttolerant manner [148,149].”