Draft:Spectral gap conjecture

Any matrix ${\displaystyle U\in SU(2)}$ defines an isometry of the sphere ${\displaystyle S^{2}}$, which in turn defines an operator ${\displaystyle \phi _{U}}$ on the Hilbert space ${\displaystyle L^{2}(SU(2))}$. The spectral gap conjecture states that for any integer ${\displaystyle n>2}$, if ${\displaystyle n}$ isometries ${\displaystyle U_{1},\dots ,U_{n}}$ are chosen uniformly at random, then the operator ${\displaystyle \phi _{U_{1}}+\phi _{U_{1}}^{-1}+\cdots +\phi _{U_{n}}+\phi _{U_{n}}^{-1}}$ has a nontrivial spectral gap with probability 1.^[1]

In 2007, Jean Bourgain and Alex Gamburd proved that when the matrices ${\displaystyle U_{i}}$ have entries which are all algebraic numbers up to simultaneous conjugation, the resulting operator has a spectral gap.^[2] This result was later generalized to the case of ${\displaystyle SU(d)}$.^[3] It is known that either there is a nontrivial spectral gap with probability 1 or that the spectral gap is trivial with probability 1.^[4] If true, the statement would have applications to quantum computing and the design of universal quantum gate sets.^[5][6]