In ergodic theory, a branch of mathematics, the spectral gap conjecture of Alexander Lubotzky, Ralph S. Phillips, and Peter Sarnak is a statement on the spectral gaps of certain actions of a free group on the sphere S 2 {\displaystyle S^{2}} .
Statement Any matrix U ∈ S U ( 2 ) {\displaystyle U\in SU(2)} defines an isometry of the sphere S 2 {\displaystyle S^{2}} , which in turn defines an operator ϕ U {\displaystyle \phi _{U}} on the Hilbert space L 2 ( S U ( 2 ) ) {\displaystyle L^{2}(SU(2))} . The spectral gap conjecture states that for any integer n > 2 {\displaystyle n>2} , if n {\displaystyle n} isometries U 1 , … , U n {\displaystyle U_{1},\dots ,U_{n}} are chosen uniformly at random, then the operator ϕ U 1 + ϕ U 1 − 1 + ⋯ + ϕ U n + ϕ U n − 1 {\displaystyle \phi _{U_{1}}+\phi _{U_{1}}^{-1}+\cdots +\phi _{U_{n}}+\phi _{U_{n}}^{-1}} has a nontrivial spectral gap with probability 1.
Progress In 2007, Jean Bourgain and Alex Gamburd proved that when the matrices U i {\displaystyle U_{i}} have entries which are all algebraic numbers up to simultaneous conjugation, the resulting operator has a spectral gap. This result was later generalized to the case of S U ( d ) {\displaystyle SU(d)} . It is known that either there is a nontrivial spectral gap with probability 1 or that the spectral gap is trivial with probability 1. If true, the statement would have applications to quantum computing and the design of universal quantum gate sets.
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