User:ScienceCy/Mølmer–Sørensen gate

The Mølmer-Sørensen interaction is a two-qubit gate (or more generally a multi-qubit entangling gate) used in trapped-ion quantum computing, named for Klaus Mølmer and Anders Sørensen, who originally proposed the process in 1999 as a quantum entangling interaction. It uses a bichromatic laser field tuned to motional sidebands of the ion chain to drive the ions into different electronic (qubit) states.

The ion trap
Given that tuning, the ion-trap system can be described by the Hamiltonian

$$H = H_0 + H_{int}$$

$$H_0 = \hbar \nu (\hat{a}^\dagger \hat{a} + 1/2) + \hbar \omega_{eg} \sum_i \sigma_{zi}/2$$

$$H_{int} = \sum_i \frac{\hbar \Omega_i}{2} (\sigma_{+i} e^{\eta_i(\hat{a}+\hat{a}^\dagger) - \omega_i t} + h.c.),$$

where the motion of the ions are treated as a single oscillatory mode with frequency $$\nu$$ and ladder operators $$\hat{a}$$ and $$\hat{a}^\dagger$$. Each qubit (with Pauli matrices $$\sigma_{i}$$) consists of two electronic energy levels of an ion, separated by $$\hbar\omega_{eg}$$, and controlled via a laser with frequency $$\omega_i$$ and Rabi frequency $$\Omega_i$$.

The Lamb-Dicke approximation
The gate is designed to operate in the Lamb Dicke regime, where the Lamb Dicke parameters $$\eta_i$$ are such that $$\eta_i \sqrt{n+1} << 1$$. This allows the use of second-order perturbation theory to describe the gate interaction. Assuming the ions all have the same Rabi frequency and Lamb-Dicke parameter, the only transitions allowed by conservation of energy are between $$\mid 0,0,n \rangle$$ and $$\mid 1,1,n \rangle$$, where the first two numbers indicate the qubit states and $$n$$ is the state of the motional mode.

The Rabi frequency $$\tilde{\Omega}$$ for that transition is given perturbatively as a sum over intermediate states $$\mid m \rangle$$,

$$ \left(\frac{\tilde{\Omega}}{2}\right)^2 = \frac{1}{\hbar^2} \left|\sum_m \frac{\langle 1,1,n \mid H_{int} \mid m \rangle \langle m \mid H_{int} \mid 0,0,n \rangle}{E_{0,0,n} + \hbar \omega_i - E_m} \right|^2, $$

where $$\hbar\omega_i$$ is the laser energy.

If the beam addressing the first ion is detuned to the upper sideband and the second ion to the lower sideband, then in the Lamb-Dicke approximation only two such intermediate states need be considered, which are $$\mid 1,0,n+1\rangle$$ and $$\mid 0,1,n-1 \rangle$$. If the laser is detuned by $$\delta$$, then the sum reduces to

$$ \tilde{\Omega} = -\frac{(\Omega\eta)^2}{2(\nu - \delta)}. $$



Interference between the two possible paths results in the Rabi frequency being independent of the motional quantum number $$n$$, which means that the Mølmer-Sørensen gate can be performed from any motional state, even a mixed state.

The result is thus a simple sinusoidal oscillation between $$\mid 0,0,n \rangle$$ and $$\mid 1,1,n \rangle$$; depending on how long the interaction is run for, $$\mid 0,0,n \rangle$$ can be excited to $$\mid 1,1,n \rangle$$ or any superposition of the two.

Bichromatic fields
If the system starts in a state $$\mid 0,1,n \rangle$$ or $$\mid 1,0,n \rangle$$, which have no resonant transitions for the detunings described above, it will remain in that state. However, the energies of those states are shifted by an amount dependent on $$n$$, meaning that unless the motional state is a pure eigenstate, the system will decohere.

This decoherence can be eliminated by applying beams of both detunings to both ions. For the oscillations between $$\mid 0,0,n \rangle$$ and $$\mid 1,1,n \rangle$$, this adds two other possible intermediate states: $$\mid 1,0,n-1\rangle$$ and $$\mid 0,1,n+1 \rangle$$. Due to the symmetry of the system, these two paths contribute the same value to the sum as the other two, which simply doubles the Rabi frequency. However, the two bichromatic fields creates the possibility of a transition between $$\mid 0,1,n \rangle$$ and $$\mid 1,0,n \rangle$$, with the opposite Rabi frequency. Thus a Mølmer-Sørensen gate with duration $$T$$ is given by the matrix

$$ MS = \begin{bmatrix} \cos \frac{\tilde{\Omega} T}{2} & 0 & 0 & i \sin \frac{\tilde{\Omega} T}{2} \\ 0 & \cos \frac{\tilde{\Omega} T}{2} & - i \sin \frac{\tilde{\Omega} T}{2} & 0 \\ 0 & - i \sin \frac{\tilde{\Omega} T}{2} & \cos \frac{\tilde{\Omega} T}{2} & 0 \\ i \sin \frac{\tilde{\Omega} T}{2} & 0 & 0 & \cos \frac{\tilde{\Omega} T}{2} \end{bmatrix}. $$

Multi-particle entanglement
If there are more than two ions in the trap and every such ion is illuminated by the bichromatic beam, the Mølmer-Sørensen interaction can be used to create a multi-particle entangled state. This can be seen by writing the state in the collective spin representation $$\mid J,M \rangle$$, where $$J = N/2$$ is the total spin of the system, and $$M \in \{-J, ..., J\}$$ is the collective Z spin of the system such that if $$N_e$$ of the ions are in the excited state, $$M = N_e - N/2$$. Applying the Mølmer-Sørensen bichromatic beams results in the system evolving under the Hamiltonian

$$ H = \frac{\chi}{\hbar}(J_+^2 + J_-^2 + J_+ J_- + J_- J_+). $$

If the ion trap begins in a state

$$ \mid \Psi \rangle = \mid N/2, -N/2 \rangle = \mid 0 \rangle ^{\otimes N} $$

then applying $$H$$ for a time $$t = \frac{\pi}{8\chi}$$ will create the fully entangled state

$$ \frac{e^{-i\pi / 4}}{\sqrt{2}} \mid N/2, -N/2 \rangle + \frac{e^{+i\pi/4 + iN\pi/2}}{\sqrt{2}} \mid N/2, N/2 \rangle. $$

Universal quantum computation
Trapped-ion quantum computers also natively support single-qubit rotations, which in combination with the two-qubit Mølmer-Sørensen gate can be used to perform universal quantum computation. For example, the controlled not gate can be implemented by a $$\pi/2$$ rotation around the Y axis on the control qubit, a Mølmer-Sørensen gate with $$\tilde{\Omega} T/2 = \pi/4$$, a $$\pi/2$$ rotation around the X axis on each qubit, and finally a $$-\pi/2$$ rotation around Y on the control qubit.

Experimental realization
The Mølmer-Sørensen gate was first implemented in 2000 by Sackett et al. at NIST to create two-ion and four-ion entangled states. The implementation used beryllium ions in a radio frequency trap, driven by 313 nm lasers with an 80 GHz detuning. While the accuracy of this initial experiment was limited, other groups (Haljan et al. at University of Michigan, Benhelm et al. at Universität Innsbruck , etc.) have since achieved higher fidelities with cadmium and calcium ions.