Cirac–Zoller controlled-NOT gate

The Cirac–Zoller controlled-NOT gate is an implementation of the controlled-NOT (CNOT) quantum logic gate using cold trapped ions that was proposed by Ignacio Cirac and Peter Zoller in 1995 and represents the central ingredient of the Cirac–Zoller proposal for a trapped-ion quantum computer.

[1] The key idea of the Cirac–Zoller proposal is to mediate the interaction between the two qubits through the joint motion of the complete chain of trapped ions.

-qubit Hilbert space can be approximated to arbitrary precision by a sequence of gates from the universal set.

The Cirac–Zoller gate was experimentally first realized in 2003 (in slightly modified form) at the University of Innsbruck, Austria by Ferdinand Schmidt-Kaler and coworkers in the group of Rainer Blatt using two calcium ions.

The ions are cooled to their collective ground state, so that the quantization of the motion of the chain becomes relevant.

The Cirac–Zoller gate between two qubits represented by ions A and B is then realized in a three-step process: In total, the three pulses realize the following transformation on the two-qubit subspace in the motional ground state: that is, the state ee acquires a phase

: The central theoretical realization, on which the above steps and much of the subsequent theoretical progress in trapped-ion quantum computation is based, is that the ion chain driven by red sideband pulses realizes the Jaynes–Cummings model for the two-level system formed by g and e and one of the normal modes of the chain.

To suppress transitions in which more than one quantum of motion is transferred, one has to work in the Lamb Dicke regime where the wavelength of the light used is large compared to the size of the wave packet of the trapped ion.