Mølmer–Sørensen gate

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v t

The qubit occupation over four gate cycles.

The Mølmer–Sørensen gate is a two qubit gate used in trapped ion quantum computing. It was proposed by Klaus Mølmer and Anders Sørensen.^[1] Their proposal also extends to gates on more than two qubits.

Implementation[]

To implement the gate, two ions are irradiated with a bichromatic laser field with frequencies ${\displaystyle \omega _{eg}\pm \delta }$, where ${\displaystyle \hbar \omega _{eg}}$ is the energy splitting of the qubit states and ${\displaystyle \delta }$ is a detuning close to the motional frequency of the ions. Depending on the interaction time, this produces the states^[2]

${\displaystyle {\begin{aligned}\mid ee\rangle \rightarrow (|ee\rangle +i|gg\rangle )/{\sqrt {2}}\\\mid eg\rangle \rightarrow (|eg\rangle -i|ge\rangle )/{\sqrt {2}}\\\mid ge\rangle \rightarrow (|ge\rangle -i|eg\rangle )/{\sqrt {2}}\\\mid gg\rangle \rightarrow (|gg\rangle +i|ee\rangle )/{\sqrt {2}}\end{aligned}}}$

The above can then be shown to produce a universal set of gates. The Mølmer–Sørensen gate has the advantage that it does not fail if the ions were not cooled completely to the ground state, and it does not require the ions to be individually addressed.^[3] However, this thermal insensitivity is only valid in the Lamb–Dicke regime, so most implementations first cool the ions to the motional ground state.^[4] An experiment was done by P.C. Haljan, K. A. Brickman, L. Deslauriers, P.J. Lee, and C. Monroe where this gate was used to produce all four Bell states and to implement Grover's algorithm successfully.^[5]

Interaction Hamiltonian and Evolution[]

Given that the Mølmer-Sørenson gate only uses a single motional mode of the trapped ions and two of the electronic states to form the qubit, the Hamiltonian of the system for two ions can be expressed as:^[1]

${\displaystyle H_{0}=\hbar \nu (a^{\dagger }a+1/2)+{\frac {\hbar \omega _{eg}}{2}}\left(\sigma _{z}^{(1)}+\sigma _{z}^{(2)}\right)}$

The time evolution of the state of a two qubit implementation of the Mølmer-Sørensen gate over two gate times from the ions starting in the ground state. States are labelled first by the qubit of each ion and then by the phonon occupancy of the motional mode. The grey circle represents a probability of 1/2 of that state.

where ${\displaystyle a^{\dagger }}$ and ${\displaystyle a}$ are the creation and annihilation operators of phonons in the ions collective motional mode, ${\displaystyle \hbar \nu }$ is the energy of those phonons, and ${\displaystyle \sigma _{z}^{(i)}}$ is the Pauli Z matrix on the ${\displaystyle i}$th qubit. The Hamiltonian of the system in the interaction picture and with the biochromatic light, after applying the rotating wave approximation, is given by:^[4]

${\displaystyle H(t)=\hbar \Omega (e^{-i\delta t}+e^{i\delta t})e^{i\eta (ae^{-i\nu t}+a^{\dagger }e^{i\nu t})}(\sigma _{+}^{(1)}+\sigma _{+}^{(2)})+h.c.}$

where ${\displaystyle \Omega }$ is the Rabi frequency on the qubit carrier transition and ${\displaystyle \eta }$ is the Lamb–Dicke parameter. In the Lamb–Dicke regime this Hamiltonian can be approximated and exactly integrated to yield the propagator ${\displaystyle U(t)}$:^[4]

${\displaystyle H(t)\approx -\hbar \eta \Omega (a^{\dagger }e^{i\epsilon t}+ae^{-i\epsilon t})S_{y}}$

${\displaystyle U(t)={\hat {D}}\left({\frac {\eta \Omega }{\epsilon }}(e^{i\epsilon t}-1)S_{y}\right)exp\left[i{\frac {\eta ^{2}\Omega ^{2}}{\epsilon }}\left(t-{\frac {\sin {\epsilon t}}{\epsilon }}\right)S_{y}^{2}\right]}$

with ${\displaystyle \epsilon =\nu -\delta }$, ${\displaystyle S_{y}=\sigma _{1y}+\sigma _{2y}}$ and ${\displaystyle {\hat {D}}(\alpha )=e^{\alpha a^{\dagger }-\alpha ^{*}a}}$ is a displacement operator. So for ${\displaystyle t_{gate}=2\pi /\epsilon }$ the displacement operator vanishes and if ${\displaystyle \Omega =\epsilon /(4\eta )}$ the gate will maximally entangle the ions.

References[]