MAGIC IN THE LAB
Trapped Atom-RF Interaction: MAGIC ω L = ω , "Carrier" ω L = ω + ν , "blue sideband" ω L = ω − ν , "red sideband" ω − ν ω ω + ν ω L
MAGIC – Experiment Yb + [3/2] 1/2 2 P 1/2 935nm π - polarized 2 D 3/2 369nm 2 S 1/2 RF-optical double resonance spectroscopy PRL 102 (2009)
MAGIC – Experiment Spin-Motion coupling using RF radiation Zeeman Resonance Yb + D 3/2 state ∂ PRL 102 (2009)
MAGIC – Experiment Spin-Motion coupling using RF radiation Zeeman Resonance Yb + D 3/2 state ∂ ∂ PRL 102 (2009)
MAGIC – Experiment Spin-Motion coupling using RF radiation 171 Yb + Excitation probability 0.8 0.6 0.4 0.2 0 | ↑ › -200 -150 -100 -50 0 50 100 150 Detuning from Carrier (kHz) 12.6 GHz | ↓ ›
Single Qubit Gates Using RF-waves Gate error O(10 -5 ) Gate error O(10 -6 ) K. R. Brown et al. , PRA 84 (2011) T. P. Harty et al. , PRL 113 (2014)
• RF (MW) for all coherent operations • Individual Addressing • Spin-Spin Coupling: • Adjust Magnitude • Simultaneous • On and Off • Change Sign MAGIC QUANTUM TOOLBOX
Addressing a Quantum Byte Magnetic Gradient 19 T/m 1 2 3 4 5 6 7 8 10 µ m
Addressing a Quantum Byte Magnetic Gradient 19 T/m 1 0
Addressing a Quantum Byte 1 2 3 4 5 6 7 8 10 µ m 1 1 excitation probability excitation probability 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 10 20 30 40 0 10 20 30 40 pulse duration ( µ s) pulse duration ( µ s) Nat. Commun. 5 (2014)
Addressing a Quantum Byte Benchmarking 1 2 3 4 5 6 7 8 10 µ m Example: F 5,4 F 5,1 1 1 C 5,4 = 7.6(1.3)·10 -5 state fidelity state fidelity 0.95 0.95 C 5,1 = 1.9(9)·10 -5 0.9 0.9 0.85 0.85 0 500 1000 1500 0 500 1000 1500 sequence length N sequence length N Nat. Commun. 5 (2014)
Addressing a Quantum Byte Measured cross-talk matrix for interacting ions 1 2 3 4 5 6 7 8 C i,j (10 -5 ) addressed qubits (rows); observed qubits (columns) - 3.0(9) 1.9(8) 2.2(9) 2.3(9) 1.0(8) 0.7(6) 0.7(7) 3.8(1.4) - 4.1(1.1) 2.3(9) 2.3(1.1) 1.6(1.1) 0.9(8) 0.9(9) 2.1(1.0) 3.7(1.2) - 4.5(1.2) 1.6(7) 2.1(6) 0.8(7) 1.1(6) 0.9(9) 1.7(6) 2.7(1.1) - 3.1(9) 0.8(7) 0.6(6) 0.6(6) 1.9(9) 1.6(9) 3.1(1.0) 7.6(1.3) - 3.1(1.0) 1.8(9) 0.5(5) 1.5(5) 1.2(8) 1.5(8) 1.0(8) 5.5(1.4) - 3.6(1.3) 0.8(8) 0.8(8) 1.4(8) 1.5(7) 1.2(8) 1.2(8) 2.9(1.1) - 2.6(8) 0.8(6) 1.1(5) 0.6(6) 0.8(8) 2.5(9) 1.1(8) 3.4(1.2) - Nat. Commun. 5 (2014)
• RF (MW) for all coherent operations • Individual Addressing • Spin-Spin Coupling: • Adjust Magnitude • Simultaneous • On and Off • Change Sign MAGIC QUANTUM TOOLBOX
MAGIC: Spin-Spin Interaction B 1. Individual Addressing N − ! ∑ 2. Spin-Spin Coupling σ z,i σ z,j J ij 2 i < j
MAGIC: Spin-Spin Interaction | > ω h | >
MAGIC: Spin-Spin Interaction B | > ω h | >
MAGIC: Spin-Spin Interaction B | > ω h | >
MAGIC: Spin-Spin Interaction N − ∑ J ij σ z,i σ z,j B 2 i < j | > ω h | > In Laser Physics at the Limit, Springer, 2002, p. 261; also: quant-ph/0111158 Adv. At. Mol. Opt. Phys. 49 , 295 (2003); also: quant-ph/0305129
MAGIC Example: Two Ions B
MAGIC Example: Two Ions B d z J 2 Spin flip ⇒ d z = F z / (m ν 2 ) 2 / (m ν 2 ) ∝ ( ∂ z B / ν ) 2 . J 12 = − F z d z = − F z J. Phys. B 42, 154009 (2009)
MAGIC Example: Two Ions B d z J 2 J 12 ∝ ( ∂ z B / ν ) 2 J. Phys. B 42, 154009 (2009)
MAGIC: Outline of Math (1d) Interaction N ∑ = − F q n σ z (n) … 0 n = 1 N V harm = ∑ A i,j q i q j Potential (external + Coulomb) n = 1 ⇒ N uncoupled normal modes ⇒ Unitary transformation with In Laser Physics at the Limit, Springer, 2002, and p. 261. also: quant-ph/0111158
Magnetic Gradient Induced Coupling: MAGIC B 1. Qubit resonances shifted individually 2. Spin-Spin coupling between individual qubits N − ∑ σ z,i σ z,j J ij 2 i < j
Interlude: Ramsey-type measurements
Ramsey measurement state phase: 0 0 3/2 π π /2 π
Ramsey measurement state phase: π /2 0 3/2 π π /2 π
Ramsey measurement state phase: π 0 3/2 π π /2 π
Ramsey measurement state phase: 3/2 π 0 3/2 π π /2 π
Interlude: Quantum Gates using J-coupling
| ↑ › | ↑ › 12.6 GHz 12.6 GHz | ↓ › | ↓ ›
J- type coupling – CNOT Gate Schematic π ⎛ ⎞ ⎜ ⎟ ⎝ 2 ⎠ y
J- type coupling – CNOT Gate Schematic ⎛ ⎞ i τ = σ σ τ z z U ( ) exp 2 J ⎜ ⎟ gate 12 1 2 gate ⎝ ⎠ π ⋅ τ = J 12 gate 2
J- type coupling – CNOT Gate Schematic π ⎛ ⎞ − ⎜ ⎟ ⎝ 2 ⎠ x ↑↑ ⎯⎯⎯ → ↑↑ CNOT
J- type coupling – CNOT Gate Schematic ⎛ ⎞ i τ = σ σ τ z z U ( ) exp 2 J ⎜ ⎟ gate 12 1 2 gate ⎝ ⎠ π ⋅ τ = J 12 gate 2
J- type coupling – CNOT Gate Schematic π ⎛ ⎞ − ⎜ ⎟ ⎝ 2 ⎠ x ↓↑ ⎯⎯⎯ → ↓↓ CNOT
• RF (MW) for all coherent operations • Individual Addressing • Spin-Spin Coupling: • Adjust Magnitude • Simultaneous • On and Off • Change Sign MAGIC QUANTUM TOOLBOX
Measuring MAGIC Ramsey phase at conditional evolution time 4 ms Ramsey phase at conditional evolution time 4 ms Ramsey phase at conditional evolution time 4 ms 1 1 1 Single ion Control |0> Control |1> 0.75 0.75 0.75 excitation probability excitation probability excitation probability Δ φ = ij J 0.5 0.5 0.5 ij τ 2 0.25 0.25 0.25 0 0 0 0 0 0 1/2 1/2 1/2 1 1 1 3/2 3/2 3/2 2 2 2 PRL 108 , 220502 (2012) p p p Ramsey phase / Ramsey phase / Ramsey phase / (rad) (rad) (rad) Science Advances 2 (2016)
Magnitude of MAGIC Variation of trapping potential 60 2 19 T/m ⎛ ⎞ ∂ z B J ∝ 50 ⎜ ⎟ ν z ⎝ ⎠ 40 2 π (Hz) 30 B J 12 20 10 0 120 130 140 150 160 170 180 ν z (kHz) PRL 108 , 220502 (2012)
MAGIC: Spin-Spin Interaction Q Q Q 3 1 2
MAGIC: Spin-Spin Interaction J 13 σ 1 z σ 3 z Q Q 1 3 J 12 σ 1 z σ 2 z J 23 σ 2 z σ 3 z Q 2
Simultaneous Coupling Q 1 , Q 3 |00> Q Q |01> 1 3 |10> |11> Phase shift of qubit 2 due to qubit 1 and qubit 3 Phase shift of qubit 2 due to qubit 1 and qubit 3 Phase shift of qubit 2 due to qubit 1 and qubit 3 Phase shift of qubit 2 due to qubit 1 and qubit 3 180 180 180 180 Q 2 120 120 120 120 phase angle (deg) phase angle (deg) phase angle (deg) phase angle (deg) 60 60 60 60 0 0 0 0 -60 -60 -60 -60 -120 -120 -120 -120 -180 -180 -180 -180 0 0 1 1 2 2 3 3 4 4 5 5 6 6 0 0 1 1 2 2 3 3 4 4 5 5 6 6 conditional evolution time (ms) conditional evolution time (ms) conditional evolution time (ms) conditional evolution time (ms) Science Advances 2 (2016)
Turn coupling off Recoding 1 → → Q 0 0 and 1 0' 0' 1 1 1 1 1 F=1 Q + mQ pQ 2 2 S ψ → ψ 1/2 2,3 2,3 Q 3 F=0 0 t
Qubit 1: Isolation by Recoding Q Q 1 3 Q 2 120 Q 3 : phase angle (deg) 60 0 0 1 -60 -120 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 conditional evolution time (ms) Science Advances 2 (2016)
+ 1 = F 1 Sign of Coupling − 1 = 0 F 0 Q Q 1 3 Q 2 ⎡ ⎤ 34 ( 7 ) 27 ( 7 ) J ⎢ ⎥ ij = 34 ( 7 ) 39 ( 7 ) Hz ⎢ ⎥ π 2 ⎢ ⎥ 27 ( 7 ) 39 ( 7 ) ⎣ ⎦ Science Advances 2 (2016)
+ 1 = F 1 Sign of Coupling − 1 = 0 F 0 Q Q 1 3 Q 2 − − ⎡ ⎤ 39 ( 5 ) 27 ( 5 ) J ⎢ ⎥ ij = − 39 ( 5 ) 34 ( 7 ) Hz ⎢ ⎥ π 2 ⎢ ⎥ − 27 ( 5 ) 34 ( 7 ) ⎣ ⎦ Science Advances 2 (2016)
• RF (MW) for all coherent operations • Individual Addressing • Spin-Spin Coupling: • Adjust Magnitude • Simultaneous • On and Off • Change Sign MAGIC QUANTUM TOOLBOX
Coherent QFT Using Multiple Coupling • single-qubit gates: rotations and Hadamard gate • conditional dynamics: all mutual couplings • conditional dynamics: selected coupling Science Advances 2 (2016)
Coherent QFT Using Multiple Coupling § Total time 8.6 ms ≈ one CNOT gate Science Advances 2 (2016)
Coherent QFT: Experiment qubit 1 ___ Fit qubit 2 - - - Theory qubit 3 |000> |001> |010> |011> 1 1 1 1 0.75 0.75 0.75 0.75 excitation probability 0.5 0.5 0.5 0.5 0.25 0.25 0.25 0.25 0 0 0 0 0 1/2 1 3/2 2 0 1/2 1 3/2 2 0 1/2 1 3/2 2 0 1/2 1 3/2 2 |100> |101> |110> |111> 1 1 1 1 0.75 0.75 0.75 0.75 0.5 0.5 0.5 0.5 0.25 0.25 0.25 0.25 0 0 0 0 0 1/2 1 3/2 2 0 1/2 1 3/2 2 0 1/2 1 3/2 2 0 1/2 1 3/2 2 ( π rad) Ramsey phase Science Advances 2 (2016)
Coherent QFT: Period Finding SSO = 0.84(3) probability SSO = 0.99(3) SSO = 0.64(2) SSO = 0.54(2) probability ( ) 2 ∑ Science Advances 2 (2016) S ( p , q ) = p i q i See also: i J. Chiaverini et al., Science 308 (2005) P. Schindler et al. , New. J. Phys . 15 (2013)
MAGIC using Dressed State Gates Nature 476 , 185 (2011) University of Sussex (W. Hensinger): S. C. Webster et al. PRL 111 (2013) I. Cohen et al., NJP 17 (2015) J. Randall et al., PRA 91 (2015) High-Fidelity 2-Qubit Gate: S. Weidt et al., PRL 117 (2016)
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