Overview of Quantum

Gates for Ion Trapped

Quantum ComputersAndrew Schaffer

OPTI 646

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Most successful approaches so far

The Molmer-Sorensen (MS) gate

Proposed in 1999 by Anders Sorensen and Klaus Molmer

Physically realized for clock-state qubits with GHz laser frequency modulation

Can achieve high fidelities with extremely well-controlled hardware and error

correction

The Light-Shift (LS) gate

Not yet developed using clock-state qubit (not yet a universal gate)

Entanglement gate demonstrated this year on clock-state qubits by Honeywell

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Ion Traps

20 ions cooled and trapped in a

linear ion trap

Traps are nearly harmonic

Use quantum harmonic oscillator

to approximate ground state

motional energy and subsequent

energy levels

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MS gate theory: Raman coupling of

two-qubit states in harmonic trap

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc2.html

Motional States Electronic States

ν

ν

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Molmer-Sorensen Hamiltonian

Sørensen, Anders, and Klaus Mølmer. "Quantum computation with ions in thermal motion." Physical review

letters 82.9 (1999): 1971.

ν is the trapping frequency

ωeg is the difference in frequencies of the

internal up and down states

Ωi and ωi are the Rabi oscillation and

frequency of the laser on the ith ion

σi are the internal degrees of freedom of

the ith ion

ηi is the Lamb-Dicke parameter which is

appropriate for light-atom interaction at

very cold temperatures (proportional to

the wavevector of the light)

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Molmer-Sorensen gate energy levels

Choose detuning δ = ε + ν = ωeg - ωr = ωb -

ωeg

Allow δ to be large enough that

intermediate states are not populated

In the diagram to the left, motional energy is

conserved between the down, down and

up, up states

can be defined with no dependence on

motional state, nRoos, Christian F. "Ion trap quantum gates with amplitude-modulated laser beams." New

Journal of Physics 10.1 (2008): 013002.

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Molmer-Sorensen gate energy levels

Choose detuning δ = ε + ν = ωeg - ωr = ωb -

ωeg

Energy is conserved for , but

not

Account for energy shift by using both ± δ

light, ωr and ωb

Roos, Christian F. "Ion trap quantum gates with amplitude-modulated laser beams." New

Journal of Physics 10.1 (2008): 013002.

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Molmer-Sorensen gate on two ions

Can be altered to similarly

couple the down, up and up,

down states

Physically realizable way to

create the Bell states in clock-

state, trapped ions

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Recent implementation of MS gate on

trapped Ions

Use of 9Be+ ions encoded on

*Gaebler, John P., et al. "High-fidelity universal gate set for Be 9+ ion

qubits." Physical review letters 117.6 (2016): 060505.

Coupling down and up states via the 2P1/2 transition

Δ = 2𝜋 ∗ 900 𝐺𝐻𝑧

𝜂 = Δ𝒌 ∗ z0 ≈ .25

𝐺𝑎𝑡𝑒 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 30 𝜇𝑠 (applies 𝜋

2pulse)

𝐹 = .9992 for creation of Bell-state from initial

ground state:

𝑙𝑎𝑠𝑒𝑟 𝜆 ≈ 313 nm

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Drawbacks of the MS-gate

Large detunings (~GHz) can be difficult to access and control reliably

Gates very sensitive to optical field phase, easily leads to decoherence if

not controlled precisely as shown by Lee et al.

Optical transitions in the UV (313nm) can be difficult and/or expensive to

reliably create and control over a variety of platforms (ex, atom chips and

optical fibers)

Light-Shift gates can potentially avoid these issues if implementation on

clock-state qubits is possible!

Lee, Patricia J., et al. "Phase control of trapped ion quantum gates." Journal of Optics B: Quantum and Semiclassical Optics 7.10

(2005): S371.

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Honeywell Light-Shift (LS) Gate

To realize a LS-gate for clock-state

qubits, use of a long-lived excited

state is necessary (use of couple

through D state rather than P state)

Honeywell accomplishes this using 171Yb+ ion entangling qubits through

use of the D3/2 manifold encoding the

spin up and spin down states in the

S1/2 manifold:The linear H0/H1 ion trap; https://www.honeywell.com/us/en/news/2020/10/get-to-

know-honeywell-s-latest-quantum-computer-system-model-h1

[1] Baldwin, C. H., et al. "A high fidelity light-shift gate for clock-state qubits." arXiv preprint arXiv:2003.01102 (2020).

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Light-Shift energy levels and coupling

Use of quadrupole transitions rather than dipole

transitions

Have a polarization, propagation direction and a

frequency dependence to be allowed

Two frequencies, red and blue-detuned, also used

for LS-gate

Equal and opposite detunings cancel out average

AC-Stark shifts of the qubit frequency which would

decohere the system

[1]

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System interaction diagram

Use of quadrupole transitions rather than dipole

transitions

Oppositely and linearly polarized beams form 90°

angle at ions creating polarization gradient along

linear trapping direction

Only 𝑚𝑓 = ±1 quadrupole allowed in this

configuration

Ions separated by integer multiples of 𝜋/|Δ𝒌|experience an in-phase spin dependent force

[1]

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Hamiltonian of the system

Lamb-Dicke

parameter (wave-

vector and

trapping strength)

and Rabi

Oscillation (laser

intensity and

detuning)

Anti-symmetry

of gate mode

Optical (Raman)

phase of jth ion

Motional

dependent

phase of jth ion

[1]

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Results of Hamiltonian of the system

Only and driven, not driven by anti-symmetry, not driven

because far from resonance

Motional phase and Optical phase decouple at 𝑡 = 2𝜋/𝛿

Results in unitary:

is a small unwanted spin rotation

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Laser pulses to implement LS gate

Remove unwanted spin rotation through spin-echo

technique

Use two pulses to “undo” the unwanted spin rotation

Choose Φ = 2𝜋Ω𝜂

𝛿, result is a unitary which acts as

a gate for us to maximally entangle the ions

[1]

[1]

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Measuring Fidelity

The LS gate together with arbitrary, global rotations give access to a two-

qubit subspace spanned by { , ( + )/ 2), }

Such rotations can be implemented by microwaves

Using this method Honeywell estimated fidelity by using their subspace

randomized benchmarking protocol with a 𝐹 = .9974 [1]

Note: this is not yet a universal gate set

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Quantifying sources of error

Technical sources of error include

Polarization imperfections – residual population in D states at end of gate due

to small coupling of up-state to 𝑚𝑓 = 0 excited state

Laser intensity variability – fluctuations in laser power can lead to AC-Stark shifts

that are not perfectly canceled by the spin-echo

Non-technical sources of error

Deviations from Lamb-Dicke regime

Spontaneous emission

Failures of the rotating wave approximation

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Maximum theoretical fidelities and

needed improvements

Using ideal calculated pulse shapes, detunings and powers, non-

technical sources of error reveal an attainable fidelity of greater than

.9999

With 𝐹 = .9974, it is assumed technical errors dominate the error in the LS

gate and can thus be improved with modest improvements to

parameters such as trap noise, laser line width, laser power stability,

polarization purity, ect.

The LS gate is not yet universal which will be needed for its use in future

quantum computing designs

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Questions

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Implementation of CNOT with single

qubit operations and MS gate

𝑃1 → 𝑃2−1 → 𝐻2 → 𝑅 → 𝑃1 → 𝐻1 → 𝑃1 → 𝑅 → 𝑃2

𝑅 → 𝑀𝑆 𝑔𝑎𝑡𝑒 𝑑𝑒𝑠𝑐𝑟𝑖𝑏𝑖𝑛𝑔 𝑅𝑎𝑏𝑖 𝑓𝑙𝑖𝑝𝑝𝑖𝑛𝑔 𝑤𝑖𝑡ℎ 𝑡 = 𝜋/(2 |Ω~|)

𝑃𝑖 →𝜋

2𝑝ℎ𝑎𝑠𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 |𝑒 > 𝑖𝑛 𝑖𝑜𝑛 𝑖

𝐻𝑖 → 𝐻𝑎𝑑𝑎𝑚𝑎𝑟𝑑 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑛 𝑖𝑜𝑛 𝑖

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Sources

Baldwin, C. H., et al. "A high fidelity light-shift gate for clock-state qubits." arXiv

preprint arXiv:2003.01102 (2020).

Gaebler, John P., et al. "High-fidelity universal gate set for Be 9+ ion

qubits." Physical review letters 117.6 (2016): 060505.

Sørensen, Anders, and Klaus Mølmer. "Quantum computation with ions in

thermal motion." Physical review letters 82.9 (1999): 1971.

Roos, Christian F. "Ion trap quantum gates with amplitude-modulated laser

beams." New Journal of Physics 10.1 (2008): 013002.

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