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.
The quantum CNOT gate acts on two qubits and can entangle them. It forms part of the standard universal set of gates,[2] meaning that any gate (unitary transformation) on the
N
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.[3]
The qubits on which the Cirac–Zoller gate operates are represented by two internal states, ground state and excited state (called in the following g and e) of trapped ions. An additional auxiliary excited state a is used to implement the gate. Due to their mutual Coulomb repulsion the ions line up in a linear chain. The ions are cooled to their collective ground state, so that the quantization of the motion of the chain becomes relevant. The proposal assumes that each ion can be individually addressed by laser pulses. Both the transitions "
g\toe
g\toa
The Cirac–Zoller gate between two qubits represented by ions A and B is then realized in a three-step process:
|gA,1\rangle\mapsto-i|eA,0\rangle
|eA,0\rangle\mapsto-i|gA,1\rangle
\pi
2\pi
g\toa
|gB,1\rangle\mapsto-|gB,1\rangle
\pi
In total, the three pulses realize the following transformation on the two-qubit subspace in the motional ground state:
\begin{matrix} &\xrightarrow{(1)}&&\xrightarrow{(2)}&&\xrightarrow{(3)}\\ |gg0\rangle&&|gg0\rangle&&|gg0\rangle&&|gg0\rangle\\ |ge0\rangle&&|ge0\rangle&&|ge0\rangle&&|ge0\rangle\\ |eg0\rangle&&-i|gg1\rangle&&i|gg1\rangle&&|eg0\rangle\\ |ee0\rangle&&-i|ge1\rangle&&-i|ge1\rangle&&-|ee0\rangle \end{matrix},
-1
UCZ
\sigmaz
UCZ
UCNOT=(1 ⊗ H)UCZ(1 ⊗ H).
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.[5] To achieve this, it is necessary that the light interacting with the ions can change their motional state. This requires Raman transitions. 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. In this regime, the coupling strength is reduced and leads to a relatively slow gate.
|gA,2\rangle