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Trapped-ion quantum computing:
A Coherent Control Problem
x = 1000x2 + 3x = 5R(Β΅)R(Β΅)R(theta)
Roee Ozeri
Weizmann Institute of Science
Quantum Science Seminar. April 2020
Laser-cooled Crystals of trapped atomic ions
Separation of few mm
Temperature few mK
0
1
Qubit encoded in internal levels of ion
Interface with qubit using lasers or microwaves
Trap by NIST Trap made by Sandia
Trapped ion interaction with Radiation
Coupling between the two qubit levels using EM radiation.
For a single ion:
Where:;
and
Trapped ion interaction with Radiation
When d = swm, only , n and , n + s will be resonantly coupled (another RWA).
In the interaction representation and within the Rotating Wave Appr. (RWA)
Carrier: s = 0
Blue sideband (BSB): s = +1
Red sideband (RSB): s = -1
Sideband Spectroscopy
S1/2
D5/2
β’ Scan the laser frequency
across the S βD transition
axialradial
carrier
Blue sidebandsRed sidebands
T β 2 mK
Sideband cooling to the ground state
Pulse on red sideband together with optical-pumping
Tom Manovitz; MSc thesis (2016)
Single qubit gates β Carrier rotations
Randomized benchmarking
microwaveLaser (Raman)
e 10-4e 10-6
Lucas, Oxford
Use a bi-chromatic drive:
πΒ± = π0 Β± π + π0
MΓΈlmer-SΓΈrensen gates
9
π0
πβπ+
π
| ββ, π
| ββ, π
| ββ, π β 1
| ββ, π + 1| ββ, π
| ββ, π β 1
| ββ, π + 1| ββ, π
A. SΓΈrensen & K. MΓΈlmer, PRL (1999) & PRA (2000)
Spectrally:
Hamiltonian (appx.) β spin dependent force:
π» = βΞ© π½π¦ π π‘ ΰ·π₯ + π π‘ ΰ·π
Global spin operator:
απ½π¦ =ΰ·π1π¦+ ΰ·π2
π¦
2
Harmonic oscillator
operators
ππ0 π1 π2βπ1βπ2
π+πβ
Entanglement in trapped ion systems
π π‘; 0 = πβππ΄ π‘ΰ·ππ¦β¨ΰ·ππ¦
2 πβππΉ π‘ απ½π¦ ΰ·π₯πβππΊ π‘ απ½π¦ ΰ·π
Bi-chromatic drive:
A. SΓΈrensen & K. MΓΈlmer, PRL (1999) & PRA (2000)
πΒ± = πππ· Β± π + ππ
πΊ π‘ , πΉ π‘
απ½π¦ ΰ·π₯
απ½π¦ ΖΈπ
π΄ π‘
MS gate: π΄ π = Ξ€π 2 , πΉ π = πΊ π = 0
πππ(π; 0)ββββ β π ββ
2
MΓΈlmer β SΓΈrensen gate
π0
πβπ+
π
| ββ, π
| ββ, π
| ββ, π β 1
| ββ, π + 1| ββ, π
| ββ, π β 1
| ββ, π + 1| ββ, π
Laser-driven entanglement
Benhelm et. al. Nat. Phys. 4 463 (2008 )
Akerman, Navon, Kotler, Glickman, RO, NJP 17, 113060 (2015)
SS; DD; SD + DS
Great, but sensitive to errors in gate parameters:
Time, trap frequency, laser detuning, laser intensity
πππ(π; 0) ππ =ππ β π π·π·
2
Quantum Engineering of Robust Gates
ri - amplitudes
Valid gate if:
Multi-tone MΓΈlmer β SΓΈrensen gates
π΄ π = Ξ€π 2 πΉ π = πΊ π = 0
Multi-tone MΓΈlmer β SΓΈrensen gates
Continuous family of gates:
β’ Choose your favorite shape through phase-space
β’ Extra degrees of freedom:
Overdetermined system; add constraints
Optimization of Quantum Gates
πΉπ = 1 +1
2
π2πΉπ ππ , ππ , ππ π=1π
ππΌ2απΌπππππ
πΏπΌ2 +β―
Null errors order-by-order:
Analytic expression for gate fidelity:
Gate timing errors β Cardioid gates
Solution is a Cardioid
MΓΈlmer-SΓΈrensen gate Cardioid gate
Gate timing errors β Cardioid gates
Gate fidelity:
Shapira, Shaniv, Manovitz, Akerman, RO, Phys. Rev. Lett. 121, 180502 (2018)
Trap frequency errors β CarNu gates
Shapira, Shaniv, Manovitz, Akerman, RO, Phys. Rev. Lett. 121, 180502 (2018)
Trap frequency and timing errors:
MS CarNu(2,3,7)
Two-qubit gate fidelity
NIST Oxford
Entanglement gates in multi-ion crystals
β’ Multiple modes of motion: choose one
β’ Slower because of heavier effective weight
β’ Increasing laser-power: off-resonance coupling to other modes or carrier transition
β’ Still works for a small number of ions
Small Scale Quantum computers
Quantum Algorithms demonstrated:
β’ Quantum Chemistry Calculations Phys. Rev. X (2018)
β’ Grover Search algorithm Nature Comm. (2017)
β’ Topological Quantum error-correction Science (2014)
β’ Shor Factoring Algorithm Science (2016)
β’ Many moreβ¦
Architectures for scaling up: # 1
Shuttling ions between different trapping regions
Kielpinski, Monroe and Wineland Nature 417, 709 (2002)
Architectures for scaling up: # 2
Photon interconnects between elementary Logic units (ELUβs)
Hong-Ou-Mandel interference
Monroe et. al. Phys. Rev. A 89, 022317 (2014)
NISQ approach: Large Crystals
10 βs -100βs ions
β’ Exponential speed-up for quantum computing ~ 60 ions sufficient, provided high fidelity
β’ Quantum simulators
β’ Can we stretch previous ideas to larger ion-numbers?
Monroe, JQI
Challenges:
β’ 10βs β 100βs of motional modes; spectral crowding
β’ With 2-qubit gates many concatenated operations are required (N3 in Shorβs)
β’ In the MS gate and its variants the gate time T β 1/ π
β’ Infidelity, πΉπ~1βπ
π2π2
β’ Increasing laser-power: off-resonance coupling
β’ Possible remedy: parallel gates
Use Multiple modes of motion
24J. I. Cirac & P. Zoller, PRL 74, 4091 (1995). [3] T. Monz et al. PRL 106, 130506 (2011).
NISQ approach: Large Crystals
β’ We generalize, πΒ± β πΒ±,π = π0 Β±ππ at amplitude ππ and phase ππ.
β’ The Hamiltonian generalizes:
π» = β
π=1
π
π½π¦ ,π ππ π‘ ΰ·π₯π + ππ π‘ ΰ·ππ , π½π¦ ,π β‘π2ππ β ππ¦
β’ With: ππ π‘ + πππ π‘ =2 2
πππΞ©Οπ=1
π ππ cos πππ‘ + ππ πππππ‘
Multi-tone MS
25
25
πππ0 π1 π2βπ1βπ2 π3 π5π4 π6 πβ¦βπ3βπ5 βπ4βπ6βπβ¦
Normal-mode
vector
Normal-mode
frequency
Quantum Engineering of multi-mode Gates
β’ Similar unitary:
π π‘; 0 =ΰ·
π=1
π
πβππ΄π π‘ απ½π¦,π2
πβππΉπ π‘ απ½π¦,π ΰ·π₯ππβππΊπ π‘ απ½π¦,π ΰ·ππ
β’ Same phase-space intuition
26
26
ππ0 π1 π2βπ1βπ2 π3 π5π4 π6 πβ¦βπ3βπ5 βπ4βπ6βπβ¦
Multi-tone MS
Quantum Engineering of multi-mode Gates
πΊπ π‘ , πΉπ π‘
απ½π¦,π ΰ·π₯
απ½π¦,π ΖΈπ
π΄π
Constraints β phase space closure
β’Motion closes in all phaseβspaces at the gate time.
πΉπ π = πΊπ π = 0 for all π = 1,β¦ ,π.
β’Geometric phases for all-to-all coupling:
A1 T β π΄πβ₯2 π =π
2
27
πΊ1 π‘ , πΉ1 π‘
απ½π¦,1 ΰ·π₯
απ½π¦,1 ΖΈπ
π΄1 = π/2
πΊπβ₯2 π‘ , πΉπβ₯2 π‘
απ½π¦,πβ₯2 ΰ·π₯
απ½π¦,πβ₯2 ΖΈπ
π΄πβ₯2
Linear constraint
Quadratic constraint
The quadratic constraint optimization problem is NP-Hard
Blue β Multi-mode gate, Yellow β regular MS, Red β multi-tone adiabatic
28
Gate optimization results β 6 ions, axial modes
Quantum Engineering of multi-mode Gates
Gate simulation results β 6 ions, axial modes
22 tone pairs, robust to: timing errors, normal-mode frequency drifts, heating
29
Normal-mode
frequenciesAll motion closes
at gate time
π΄1 π
π΄πβ₯2 π
π
2
π β 5.82π
π1,
Quantum Engineering of multi-mode Gates
β’ π β 62π
π1, 41 tone pairs, robust to: timing errors, normal-mode frequency drifts,
heating. Resulting fidelity is 0.9987.
30
Gate simulation results β 12 ions, axial modes
Quantum Engineering of multi-mode Gates
π
2
31C. Figgatt et al. Nature 572, 368 (2019). Y. Lu et al. Nature 572, 363 (2019)
MonroeJQI and IonQ
Experimental implementation of similar ideas
In Summary
β’ Trapped-ion quantum computers:
High-fidelity quantum gates
Small universal quantum machines
β’ Increasing to larger ion-qubit numbers:
Scale-up architectures: interconnected small modules
Large-ion crystals: Coherent control optimization problem - hard, yet possible
β’ Work towards ~100 qubit trapped-ion universal quantum computers
32
The Weizmann trapped ion team
33
Roee Ozeri Ravid Shaniv Lior Gazit Lee Peleg Tom Manovitz Yotam Shapira Nitzan Akerman
Ady Stern
David Schwerdt
Sr+ ions
Or KatzVidyut KaushalMeirav PinkasHaim NakavRuti Ben-ShlomiMeir AlonAvi Gross