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〉-|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 σ_x¹σ_y²σ_y³ for such three-spin entangled states, where σⁱ is the Pauli matrix for the ith 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 n>~4.