thumb|upright=1.1|Chip ion trap for quantum computing from 2011 at NIST
A trapped-ion quantum computer (TIQC) 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
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 accelerate worldwide.
[[File:Simplified scale model of the quantum computing demonstrator housed in two 19-inch racks with major components labeled.png|thumb|upright=1.6|Simplified scale model
Paul trap
thumb|upright=2|Classical linear Paul trap in Innsbruck for a string of calcium ions
The electrodynamic quadrupole ion trap now used in trapped-ion quantum computing research was invented in the 1950s by Wolfgang Paul (who received the Nobel Prize for his work in 1989). Charged particles cannot be trapped in 3D by only electrostatic forces because of Earnshaw's theorem. Instead, an electric field oscillating at radio frequency (RF) is applied, forming a potential with the shape of a saddle spinning at the RF frequency. If the RF field has the right parameters (oscillation frequency and field strength), the charged particle becomes effectively trapped at the saddle point by a restoring force, with the motion described by a set of Mathieu equations. The Paul trap is often described as a harmonic potential well that traps ions in two dimensions (assume <math>\hat{x
}</math> and <math>\widehat{y}</math> without loss of generality) and does not trap ions in the <math>\widehat{z}</math> direction. When multiple ions are at the saddle point and the system is at equilibrium, the ions are only free to move in <math>\widehat{z}</math>. Therefore, the ions will repel each other and create a vertical configuration in <math>\widehat{z}</math>, the simplest case being a linear strand of only a few ions. Coulomb interactions of increasing complexity will create a more intricate ion configuration if many ions are initialized in the same trap.
The system's initial state for quantum computation can therefore be described by the ions in their hyperfine and motional ground states, resulting in an initial center of mass phonon state of <math>|0\rangle</math> (zero phonons).
Arbitrary single qubit rotation
One of the requirements of universal quantum computing is to coherently change the state of a single qubit. For example, this can transform a qubit starting out in 0 into any arbitrary superposition of 0 and 1 defined by the user. In a trapped-ion system, this is often done using magnetic dipole transitions or stimulated Raman transitions for hyperfine qubits and electric quadrupole transitions for optical qubits. The term "rotation" alludes to the Bloch sphere representation of a qubit pure state. Gate fidelity can be greater than 99%.
The rotation operators <math>R_x(\theta)</math> and <math>R_y(\theta)</math> can be applied to individual ions by manipulating the frequency of an external electromagnetic field from and exposing the ions to the field for specific amounts of time. These controls create a Hamiltonian of the form <math>H_I^i=\hbar\Omega/2(S_+\exp(i\phi)+S_-\exp(-i\phi))</math><!--Please Define all variables -->. Here, <math>S_+</math> and <math>S_-</math> are the raising and lowering operators of spin (see Ladder operator). These rotations are the universal building blocks for single-qubit gates in quantum computing.
Scalable trap designs
Quantum computers must be capable of initializing, storing, and manipulating many qubits at once in order to solve difficult computational problems. However, as previously discussed, a finite number of qubits can be stored in each trap while still maintaining their computational abilities. It is therefore necessary to design interconnected ion traps that are capable of transferring information from one trap to another. Ions can be separated from the same interaction region to individual storage regions and brought back together without losing the quantum information stored in their internal states. Ions can also be made to turn corners at a "T" junction, allowing a two dimensional trap array design. Semiconductor fabrication techniques have also been employed to manufacture the new generation of traps, making the 'ion trap on a chip' a reality. An example is the quantum charge-coupled device (QCCD) designed by D. Kielpinski, Christopher Monroe and David J. Wineland. QCCDs resemble mazes of electrodes with designated areas for storing and manipulating qubits.
The variable electric potential created by the electrodes can both trap ions in specific regions and move them through the transport channels, which negates the necessity of containing all ions in a single trap. Ions in the QCCD's memory region are isolated from any operations and therefore the information contained in their states is kept for later use. Gates, including those that entangle two ion states, are applied to qubits in the interaction region by the method already described in this article.