UC Santa Barbara

We provide a mathematical definition of a low energy scaling limit of a sequence of general non-relativistic quantum theories in any dimension, and apply our formalism to anyonic chains. We formulate Conjecture 2.5.3 on conditions when a unitary rational (1+1)-d conformal field theory would arise as such a limit and verify the conjecture for the Ising minimal model M(4,3) using su(2)_2 anyonic chains. Our work is based on the existence of a Fourier transform relation between Temperley-Lieb generators {e_i} and some finite stage operators of the Virasoro generators {L_m+L_{-m}} and {i(L_m-L_{-m})} for unitary minimal models M(k+2,k+1) (proven for k=2). An earlier attempt [1], called the Koo-Saleur formula, has slower convergence with characteristics that hinder the convergence of algebras of observables, an important contribution of this work. Assuming Conjecture 2.5.3, most of our main results for M(4,3) hold for higher (k >= 3) unitary minimal models M(k+2,k+1) as well. Our approach is supported by extensive numerical simulation and physical proofs in the physics literature. It is also inspired by an eventual application to an efficient simulation of conformal field theories by quantum computers. We approach the definition of the unitary evolution and correlator simulation problems in the same spirit of topological quantum field theory simulation as established by M. Freedman et. al. [2]. Under certain conditions, we present complexity theoretic hardness results on the simulation problems by using the framework of fermionic quantum computation by Bravyi and Kitaev [3].