Mølmer–Sørensen gate
This procedure is based on the original proposition by Klaus Mølmer and Anders Sørensen in 1999–2000.
[4] In an MS gate, entangled states are prepared by illuminating ions with a bichromatic light field.
Mølmer and Sørensen identified two regimes in which this is possible: In both regimes, a red and blue sideband interaction are applied simultaneously to each ion, with the red and blue tones symmetrically detuned by
When an MS gate is applied globally to all ions in a chain, multipartite entanglement is created, with the form of the gate being a sum of local XX (or YY, or XY depending on experimental parameters) interactions applied to all qubit pairs.
Trapped ions were identified by Ignacio Cirac and Peter Zoller at the University of Innsbruck, Austria in 1995, as the first realistic system with which to implement a quantum computer, in a proposal which included a procedure for implementing a CNOT gate by coupling ions through their collective motion.
[4] A major drawback of Cirac and Zoller's scheme was that it required the trapped ion system to be restricted to its joint motional ground state, which is difficult to achieve experimentally.
The Cirac-Zoller CNOT gate was not experimentally demonstrated with two ions until 8 years later, in 2003, with a fidelity of 70-80%.
[5] Around 1998, there was a collective effort to develop two-qubit gates independent of the motional state of individual ions,[6][1][7] one of which was the scheme proposed by Klaus Mølmer and Anders Sørensen in Aarhus University, Denmark.
In 1999, Mølmer and Sørensen proposed a native multi-qubit trapped ion gate as an alternative to Cirac and Zoller's scheme, insensitive to the vibrational state of the system and robust against changes in the vibrational number during gate operation.
[1][2] Mølmer and Sørensen's scheme requires only that the ions be in the Lamb-Dicke regime, and it produces an Ising-like interaction Hamiltonian using a bichromatic laser field.
Following Mølmer and Sørensen's 1999 papers, Gerard J. Milburn proposed a 2-qubit gate that makes use of a stroboscopic Hamiltonian in order to couple internal state operators to different quadrature components.
[8] Soon after, in 2000, Mølmer and Sørensen published a third article[3] illustrating that their 1999 scheme was already a realization of Milburn's, just with a harmonic rather than stroboscopic application of the Hamiltonian coupling terms.
Mølmer and Sørensen's 2000 article also takes a more general approach to the gate scheme compared to the 1999 proposal.
In the 1999 papers, only the "slow gate" regime is considered, in which a large detuning from resonance is required to avoid off-resonant coupling to unwanted phonon modes.
[9] In 2003, Wineland's group produced better results by using a geometric phase gate,[10] which is a specific case of the more general formalism put forward by Mølmer, Sørensen, Milburn, and Xiaoguang Wang.
Today, the MS gate is widely used and accepted as the standard by trapped ion groups (and companies),[11][12] and optimizing and generalizing MS gates is currently an active field in the trapped ion community.
[17] To implement the scheme, two ions are irradiated with a bichromatic laser field with frequencies
[19] In this definition, CNOT gate can be decomposed as The Mølmer–Sørensen gate implementation 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.
[20] However, this thermal insensitivity is only valid in the Lamb–Dicke regime, so most implementations first cool the ions to the motional ground state.
Lee, and C. Monroe where this gate was used to produce all four Bell states and to implement Grover's algorithm successfully.
are the creation and annihilation operators of phonons in the ions' collective motional mode,
parameterizes the size of the ground state wavepacket compared to radiation wavelength
By making a second rotating wave approximation to neglect oscillation terms, each piece can be examined independently.
The MS Hamiltonian is the application of simultaneous, symmetrically detuned red and blue sideband tones over
We also assume that the tones are detuned near a motional mode which is far from the carrier such that the RWA is invoked to drop
, this term can be neglected, as the phase space trajectory consists of very small, fast loops about the origin.
In the strong field regime, ions are coherently excited and the motional state is highly entangled with the internal state until all undesirable excitations are deterministically removed toward the end of the interaction.
Mølmer and Sørensen's original proposition considers operations in the limit
In this 'weak-field regime', there is insensitivity to vibrational state and robustness against changes in vibrational motion throughout the entire gate operation, due to exploiting two important effects of quantum mechanics: If we consider two ions, each illuminated by lasers with detunings
can be neglected, as the phase space trajectory consists of very small, fast loops about the origin.
