Braiding and Entanglement in Spin Networks: a combinatorial approach to topological phases

Zoltán Kádár, Annalisa Marzuoli, Mario Rasetti

The spin network quantum simulator relies on the su(2) representation ring (or its q-deformed counterpart at q = root of unity) and its basic features naturally include (multipartite) entanglement and braiding. In particular, q-deformed spin network automata are able to perform efficiently approximate calculations of topological invarians of knots and 3-manifolds. The same algebraic background is shared by 2D lattice models supporting topological phases of matter that have recently gained much interest in condensed matter physics. These developments are motivated by the possibility to store quantum information fault-tolerantly in a physical system supporting fractional statistics since a part of the associated Hilbert space is insensitive to local perturbations. Most of currently addressed approaches are framed within a "double" quantum Chern–Simons field theory, whose quantum amplitudes represent evolution histories of local lattice degrees of freedom. We propose here a novel combinatorial approach based on "state sum" models of the Turaev–Viro type associated with SU(2)q-colored triangulations of the ambient 3-manifolds. We argue that boundary 2D lattice models (as well as observables in the form of colored graphs satisfying braiding relations) could be consistently addressed. This is supported by the proof that the Hamiltonian of the Levin–Wen condensed string net model in a surface Σ coincides with the corresponding Turaev–Viro amplitude on Σ × [0,1] presented in the last section.

Michael Mariën

Silvano Garnerone, Annalisa Marzuoli, Mario Rasetti

The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFT). More precisely, when working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum automata which in turn represent discrete versions of topological quantum computation models. Such a quantum combinatorial scheme, which essentially encodes SU (2) Racah-Wigner algebra and its braided counterpart, is particularly suitable to address problems in topology and group theory and we discuss here a finite states-quantum automaton able to accept the language of braid group in view of applications to the problem of estimating link polynomials in Chern-Simons field theory.

Bogdan Andrei Bernevig

Mauro Carfora B, Annalisa Marzuoli

Garnerone, S., Marzuoli, A., Mario Rasetti +1 more

The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFT). More precisely, when working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum automata which in turn represent discrete versions of topological quantum computation models. Such a quantum combinatorial scheme, which essentially encodes SU(2) Racah–Wigner algebra and its braided counterpart, is particularly suitable to address problems in topology and group theory and we discuss here a finite states– quantum automaton able to accept the language of braid group in view of applications to the problem of estimating link polynomials in Chern–Simons field theory. 1 1