Review of the linear algebra of quantum mechanics
Kristopher A. Innanen
The continuing (though gradual), development of applied geophysical algorithms and ideas in the area of quantum computing at CREWES, as well as our growing interest in characterizing inversion uncertainty with quasi-dynamical systems (inspired by the so-called Hamiltonian Monte Carlo methods), both motivate some review work on the linear algebra of quantum mechanics. The description of simple systems, in isolation and in combination, is developed, with the latter allowing simple entangled states to be discussed, which is relevant in quantum information and computing. Laddering operators in the description of harmonic oscillators are also given special focus: since in the above mentioned Hamiltonian MC methods, energy and data misfit are associated, laddering operators may be useful descriptors of convergence. Steps needed to increase the dimensionality of systems, and steps needed to increase the number of particles/oscillators in systems are also given attention, since both will be needed in evolving applications in optimization and geophysical quantum computing.