Mølmer–Sørensen gate
The Mølmer–Sørensen gate (or MS gate) is a two qubit gate used in trapped ion quantum computing. It was proposed by Klaus Mølmer and Anders Sørensen. 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 formula_1, where formula_2 is the energy splitting of the qubit states and formula_3 is a detuning close to the motional frequency of the ions. Depending on the interaction time, this produces the states formula_4 The above is equivalent to the Ising coupling gate  R(π/2); It can then be shown that this gate (along with arbitrary single-qubit rotation) produces a universal set of gates. An alternative definition of MS gate equates it to "R(π/2)", and is adopted as IonQ's native gate for two-qubit entanglement. In this definition, CNOT gate can be decomposed as 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. However, this thermal insensitivity is only valid in the Lamb–Dicke regime, so most  implementations first cool the ions to the motional ground state. 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. 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: formula_6 where formula_7 and formula_8 are the creation and annihilation operators of phonons  in the ions collective motional mode, formula_9 is the energy of those phonons, and formula_10 is the Pauli Z matrix on the formula_11"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: formula_12 where formula_13 is the Rabi frequency on the qubit carrier transition and formula_14 is the Lamb–Dicke parameter. In the Lamb–Dicke regime this Hamiltonian can be approximated and exactly integrated to yield the propagator formula_15: formula_16 formula_17 with formula_18, formula_19 and formula_20 is a displacement operator. So for formula_21 the displacement  operator vanishes and if formula_22 the gate will maximally entangle the ions.