Southwest Quantum Information and Technology

Quadratic forms in quantum Hall states

Jon Yard, Microsoft

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A promising approach to the construction of a fault-tolerant quantum computer aims to manipulate quantum information through the braiding of bulk quasiparticle excitations in strongly-correlated two-dimensional topologically ordered systems such as fractional quantum Hall states. The bulk excitations in such 2+1-dimensional topological phases are protected from noise by a finite energy gap, whereas the edges, which are actually what are probed in experiments, support gapless excitations. However, the same bulk topological phase can have multiple distinct edge phases. To make matters worse, a general classification of edge phases corresponding to a possible bulk phase is not currently known. In this talk, I will illustrate how the mathematical theory of integral quadratic forms yields a natural and complete mathematical classification of the bulk-boundary correspondence for abelian topological phases. For these states, edge phases correspond to integral lattices, while the bulk phase only depends on an invariant of the edge lattice called its genus by Gauss. I will also discuss ongoing research on a physical interpretation for an arithmetic refinement of the genus called the spinor genus. This is joint work with Cano, Cheng, Conrad, Mulligan, Nayak and Plamadeala.