Mølmer-Sørensen gate
| Identifier | Operator | Example statement |
|---|---|---|
| MS | \(MS\) | MS q[0], q[1] |
Description
The Mølmer-Sørensen, or MS, gate is a two-qubit gate. It is a maximally entangling gate, most commonly used in ion-trap quantum computers.
Properties
- Perfect Entangler (maximally entangles specific product states);
- Maximum Entangling Power over all uniformly random product states;
- Ising gate.
Representation
\[\begin{align}
MS &= \frac{1}{\sqrt{2}} \left(\begin{matrix}
1 & 0 & 0 & i \\
0 & 1 & i & 0 \\
0 & i & 1 & 0 \\
i & 0 & 0 & 1
\end{matrix}\right)
\end{align}\]
Operation examples
Standard basis
\[\begin{align}
MS\,|00\rangle &= \tfrac{1}{\sqrt{2}}|00\rangle + \tfrac{i}{\sqrt{2}}|11\rangle \\
\\
MS\,|01\rangle &= \tfrac{1}{\sqrt{2}}|01\rangle + \tfrac{i}{\sqrt{2}}|10\rangle \\
\\
MS\,|10\rangle &= \tfrac{i}{\sqrt{2}}|01\rangle + \tfrac{1}{\sqrt{2}}|10\rangle \\
\\
MS\,|11\rangle &= \tfrac{i}{\sqrt{2}}|00\rangle + \tfrac{1}{\sqrt{2}}|11\rangle \\
\end{align}\]
Qubit state ordering convention and matrix representation
Note that qubits in a ket are ordered with qubit indices decreasing from left to right, i.e.,
\[|\psi\rangle = \sum c_i~|q_nq_{n-1}~...q_1q_0\rangle_i\]
Note that for matrices a reversed basis ordering convention is adopted, as is done in most textbooks. For instance, in the case of a two-qubit control gate, the matrix is represented such that \(q_0\) is the control qubit and \(q_1\) is the target qubit; the state on which the matrix is applied should then effectively be written as \(|q_0\rangle \otimes |q_1\rangle\).