Microscopic analogs of the Greenberger-Horne-Zeilinger experiment on an NMR quantum computer

A scheme is proposed to implement microscopic analogs of the Greenberger-Horne-Zeilinger experiment, as described by Lloyd, on a nuclear magnetic resonance (NMR) quantum computer. This scheme includes how to prepare the effective pure spin state |0000〉 (a state with all the spins pointing along the direction of the magnetic field) from a state at thermal equilibrium, how to transform it into states of the form, e.g., $(|000〉\ensuremath{-}|111〉)|0〉$ (which comprises the first three spins in an entangled state and the fourth register-spin), and how to measure the eigenvalue of the operator product like ${\ensuremath{\sigma}}_{x}^{1}{\ensuremath{\sigma}}_{y}^{2}{\ensuremath{\sigma}}_{y}^{3}$ for such three-spin entangled states, where ${\ensuremath{\sigma}}^{i}$ is the Pauli matrix for the $i\mathrm{th}$ spin, and store the result on the fourth spin. Also proposed is a general method to implement the controlled-not gate, an essential gate for any quantum algorithm, on n-qubit NMR quantum computers with $ng~4.$