Limits on Quantum Adiabatic Optimization Warning: Rough Manuscript!

• A quantum state Ψ is a linear superposition of basis states {|x〉}x and is described by the summation |Ψ〉 = ∑ x αx|x〉, with amplitudes αx ∈ C and |αx| the probability of observing the state x. • Formally, for an N dimensional system, the quantum state is represented by a "2 normalized vector in the N dimensional Hilbert with complex amplitudes. • Throughout this article we assume that our quantum system is finite dimensional (the dimension is typically denoted by N). • A change of the state Ψ is represented by a rotation of the vector |Ψ〉 in the Hilbert space. • How the state changes is determined by the Hamiltonian H , which describes the forces that act on the state Ψ. • The Hamiltonian, which can be time dependendent, acts as a linear operator on the state vector, and hence is described by a matrix ∈ CN×N . The time evolution is expressed by Schrödinger’s differential equation: