User:Gwen Leifer/sandbox

The Paul Trap
This saddle point $$\overrightarrow{p}=(x,y)$$is the point of minimized energy magnitude, $$|E(\overrightarrow{x})|$$, for the ions. The Paul trap is often described as a harmonic potential well that traps ions in the x and y directions and does not trap ions in the z direction. When multiple ions are at point $$\overrightarrow{p}$$ and the system is at equilibrium, the ions are only free to move in $$\widehat{z}$$. Therefore, the ions will repel each other and create a vertical configuration in $$\widehat{z}$$, 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. Furthermore, the additional vibrations of the added ions greatly complicate the quantum system, which makes initialization and computation more difficult.

Vibrational energy in the ion trap is quantized into phonons by the energy eigenstates of the ion strand, which are called the center of mass vibrational modes (a single phonon's energy is given by $$\hbar\omega_z$$). These quantum states occur when the trapped ions vibrate together and are completely isolated from the external environment. If the ions are not properly isolated, noise can result from ions interacting with external electromagnetic fields, which creates random movement and destroys the quantized energy states.

Once trapped, the ions should be cooled such that $$k_BT<<\hbar\omega_z$$. This can be achieved by a combination of Doppler cooling and Resolved sideband cooling.

Edit: components of a quantum computer
DiVincenzo outlined several of these criterion for quantum computing (see DiVincenzo's criteria).

Edit: 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. Such as trap, the quantum charge-coupled device (QCCD), has been designed by Kielpinski, Monroe, and Wineland. QCCD's 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.

Decoherence in Scalable Traps
When an ion is being transported between regions in an interconnected trap and is subjected to a nonuniform magnetic field, (see Zeeman effect), decoherence can occur in the form of the equation below. This is effectively changing the relative phase of the quantum state. The up and down arrows correspond to a general superposition qubit state, in this case the ground and excited states of the ion.

$$|\uparrow\rangle + |\downarrow\rangle\longrightarrow \exp(i\alpha)|\uparrow\rangle + |\downarrow\rangle$$

Another relative phase can be introduced by unintended electric fields affecting the positions of the ions within the trap. If the user could determine the parameter α, accounting for this decoherence would be relatively simple, as known quantum information processes exist for correcting a relative phase. However, since α is path-dependent, the problem is highly complex. Considering the multiple ways that decoherence of a relative phase can be introduced in an ion trap, reimagining the ion state in a new basis that minimizes decoherence could be a way to eliminate the issue.

One way to combat decoherence is to represent the quantum state in a new basis called the Decoherence Free Subspace, or DFS., with basis states $$|\uparrow\downarrow\rangle$$ and $$|\downarrow\uparrow\rangle$$. The DFS is actually the subspace of two ion states, such that if both ions acquire the same relative phase, the total quantum state in the DFS will be unaffected.

Arbitrary single-qubit rotation
(addition to section)

The operations $$R_x(\theta)$$and $$R_y(\theta)$$can be applied to individual ions by manipulating the frequency of an external electromagnetic field and exposing the ions to the field for specific amounts of time. These controls create a Hamiltonian of the form $$H_I^i=\hbar\Omega/2(S_+e(i\phi)+S_-e(-i\phi))$$.

CNOT Gate Implementation
The Controlled NOT gate is a crucial component for quantum computing, as any quantum gate can be created by a combination of CNOT gates and single-qubit rotations. It is therefore important that a trapped-ion quantum computer can preform this operation by meeting the following three requirements.

First, the trapped ion quantum computer must be able to perform arbitrary rotations on qubits, which are already discussed in the "arbitrary single-qubit rotation" section.

The next component of a CNOT gate is the controlled phase-flip gate, or the controlled-Z gate (see Quantum logic gate). In a trapped ion quantum computer, the state of the center of mass phonon functions as the control qubit, and the internal atomic spin state of the ion is the working qubit. The phase of the working qubit will therefore be flipped if the phonon qubit is in the state $$|1\rangle$$.

Lastly, a SWAP gate must be implemented to exchange the ion's spin state and the phonon state.

Two alternate schemes to represent the CNOT gates are presented in references