Mølmer-Sørensen Gate

Figure

Description

The Mølmer-Sørensen (MS) gate is the standard high-fidelity entangling gate in trapped-ion quantum computing. It uses a bichromatic laser field — two tones placed symmetrically around the qubit carrier frequency, near the red and blue motional sidebands — to generate an effective spin-spin (XX) interaction between two or more ions sharing a common motional mode.

The key physics is that each drive tone couples the ion’s internal state to the shared motional bus: the blue-detuned tone drives $|0,n⟩\to |1,n+1⟩$ transitions while the red-detuned tone drives $|0,n⟩\to |1,n-1⟩$. When both tones act simultaneously on all addressed ions, the motional excitation is virtual — the motional state returns to its initial value after the gate duration — while a geometric phase accumulates that depends on the product of the ions’ spin states. This produces the desired entangling interaction.

A crucial practical advantage is that the MS gate does not require ground-state cooling of the motional mode. Because the interaction is mediated through virtual phonon excitation, the gate works for thermal motional states as long as the Lamb-Dicke parameter $\eta \sqrt{\bar{n}+1}\ll 1$ is satisfied.

Hamiltonian

In the interaction picture, the MS gate Hamiltonian for $N$ ions is:

$H_{\text{MS}}=\hslash \Omega {\sum }_{i=1}^N{\eta }_i{\sigma }_{\varphi }^{(i)}\left(a{e}^{-i\delta t}+a^{†}{e}^{i\delta t}\right)$

where $\Omega$ is the Rabi frequency, ${\eta }_i$ is the Lamb-Dicke parameter for ion $i$, $\delta$ is the detuning from the motional sideband, and $a$, $a^{†}$ are the motional mode operators. After adiabatic elimination of the motional degrees of freedom, this reduces to an effective spin-spin interaction:

$H_{\text{eff}}=\hslash {\sum }_{i<j}{J}_{ij}{\sigma }_{\varphi }^{(i)}{\sigma }_{\varphi }^{(j)},J_{ij}\propto \frac{{\eta }_i{\eta }_j{\Omega }^2}{\delta }$

For a two-ion gate, evolution for time $\tau =\pi /(4J_{12})$ produces the maximally entangling unitary:

$U_{\text{MS}}=\exp \left(-i\frac{\pi }{4}{\sigma }_{\varphi }^{(1)}{\sigma }_{\varphi }^{(2)}\right)$

Motivation

• The MS gate became the dominant trapped-ion entangling gate because it is experimentally robust to motional-state preparation errors — unlike early Cirac–Zoller gates, it does not require ground-state cooling.
• Maps naturally onto global-beam plus local-addressing hardware used in modern ion-trap systems.
• Scalable to multi-ion entangling operations via amplitude/frequency pulse shaping.
• Achieves the highest two-qubit gate fidelities reported for any qubit platform (99.9%+).

Experimental Status

Original proposal — Mølmer and Sørensen (1999):

• Proposed bichromatic driving scheme for entangling hot trapped ions without ground-state cooling.
• Two papers: Mølmer and Sørensen, PRL 82, 1835 (1999) and Sørensen and Mølmer, PRL 82, 1971 (1999).

First multi-particle entanglement — Sackett et al. (2000):

• Demonstrated entanglement of up to 4 ions using the MS interaction at NIST.

High-fidelity demonstration — Benhelm et al. (2008):

• Achieved 99.3% Bell-state fidelity with ${}^{40}$Ca${}^{+}$ ions using an MS gate.
• First demonstration approaching fault-tolerant thresholds.

State-of-the-art — Ballance et al. (2016):

• Achieved 99.9% two-qubit gate fidelity with ${}^{43}$Ca${}^{+}$ hyperfine qubits.
• Used careful pulse shaping and error suppression techniques.

Key Metrics

Metric	Value	Notes	Fidelity reference
2Q gate fidelity	99.9%+	State-of-the-art trapped-ion MS gate	Ballance et al. 2016
Gate time	10–200 μs	Detuning/fidelity tradeoff	—
Cooling requirement	No ground-state cooling required	Major practical advantage over Cirac–Zoller	—
Gate type	XX (or YY) entangling	Geometric phase gate	—
Scalability	Multi-ion entangling	Via pulse shaping; demonstrated for 4+ ions	Sackett et al. 2000

References

Original proposals

• K. Mølmer and A. Sørensen, “Multiparticle Entanglement of Hot Trapped Ions,” Phys. Rev. Lett. 82, 1835 (1999)
• A. Sørensen and K. Mølmer, “Quantum Computation with Ions in Thermal Motion,” Phys. Rev. Lett. 82, 1971 (1999)

Experimental demonstrations

• C. A. Sackett et al., “Experimental entanglement of four particles,” Nature 404, 256 (2000)
• J. Benhelm et al., “Towards fault-tolerant quantum computing with trapped ions,” Nature Phys. 4, 463 (2008)
• C. J. Ballance et al., “High-Fidelity Quantum Logic Gates Using Trapped-Ion Hyperfine Qubits,” Phys. Rev. Lett. 117, 060504 (2016)

Linked Papers

• sorensen-1999-ms-gate

Related Entries

• trapped-ion-qubit — qubit platform where the MS gate is standard
• cirac-zoller-gate — earlier trapped-ion entangling gate requiring ground-state cooling
• barium-137-ion-qubit — Ba-137 hyperfine qubit using MS gates in Helios
• beryllium-9-ion-qubit — high-fidelity MS gate demonstrated on Be-9 (Gaebler et al. 2016)