Synthesis of arbitrary unitary transformations of collective states of trapped ions by quantum Householder reflections
Peter A. Ivanov, Nikolay V. Vitanov · Physical Review A · 2008
We propose an efficient technique for creation and manipulation of entangled states and for construction of arbitrary many-particle unitary transformations in a linear ion trap. We show that, under suitable conditions on the laser pulse areas and detunings, the propagator within the single-excitation manifold has the form of a Householder reflection (HR). We demonstrate that a single HR is sufficient to create an entangled $W$-type state or to transfer one entangled state into another. An arbitrary $N$-dimensional unitary transformation can be constructed by at most $N\ensuremath{-}1$ HRs, i.e., significantly faster than $O({N}^{2})$ steps in previous proposals, which makes this technique less sensitive to heating and other types of decoherence. We present examples of synthesis of quantum Fourier transforms for four, five, and six ions, which are constructed using only two, three, and four HRs, respectively.