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A trapped ion quantum computer is one proposed approach to a large-scale quantum computer. Ions, or charged atomic particles, can be confined and suspended in free space using electromagnetic fields. Qubits are stored in stable electronic states of each ion, and quantum information can be transferred through the collective quantized motion of the ions in a shared trap (interacting through the Coulomb force). Lasers are applied to induce coupling between the qubit states (for single qubit operations) or coupling between the internal qubit states and the external motional states (for entanglement between qubits).

History of trapped ion quantum computing
The first implementation scheme for a controlled-NOT quantum gate was proposed by Ignacio Cirac and Peter Zoller in 1995, specifically for the trapped ion system. The same year, a key step in the controlled-NOT gate was experimentally realized at NIST Ion Storage Group, and research in quantum computing began to take off worldwide. Many traditional ion trapping research groups have made the transition to quantum computing research, while, more recently, many other new research groups have joined the effort. An enormous amount of progress in this field has been made in the past decade and trapped ions remain a leading candidate for quantum computation.

Qubit ( add a link to qubit wiki page)

Quantum Gates
In quantum computer, there're four basic types of gates: X, Y, Z gates which are corresponding to Pauli matrices. We also have a Hardmard gate which in matrix representation, is noted as:

$$H=(1/\sqrt{2})\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$$

Based on these gates, the three most basic logic gates: 'NOT' gate, 'FANOUT' gate, and 'AND' gate can be constructed. Theoretically any algorithm can be build based on these three gates.

So far scientists have physically realized 'controlled not gate'.

Other Two qubit entangling gates
Besides the controlled-NOT gate proposed by Cirac and Zoller in 1995, many equivalent, but more robust, schemes have been proposed and implemented experimentally since(citations probably need here). Recent theoretical work by Garcia-Ripoll, Cirac, and Zoller have shown that there are no fundamental limitations to the speed of entangling gates, but gates in this impulsive regime (faster than 1 microsecond) have not yet been demonstrated experimentally. The fidelity of these implementations has been greater than 99%.

All 'fidelity' link to wikipage 'fidelity of quantum system'

References