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Quantum Information Processing and Quantum Error Correction and Quantum Error Correction with Trapped Ca + Ions

Quantum Information Processing and Quantum Information Processing and Quantum Error Correction with Trapped Ca + Ions Quantum Error Correction with Trapped Ca + Ions h h d d Rainer Blatt Institute of Experimental Physics, University of Innsbruck, Institute of Quantum Optics and Quantum Information, p , Austrian Academy of Sciences  Trapped Ca + for quantum information processing  Coherence of multi ‐ particle entangled states  Quantum error correction with trapped ions  Undoing a measurement by quantum error correction  Quantum simulations with trapped ions € IQI FWF MICROTRAP Industrie AQUTE GmbH G b SFB SFB Tirol Tirol $

Level scheme of Ca + qubit on narrow S - D quadrupole transition   1 s P 3/2 854 nm P 1/2 866 nm 866 nm D 5/2 393 nm 397 nm D 3/2 3/2 729 nm 40 C 40 Ca + + S 1/2

Spectroscopy of the S 1/2 – D 5/2 transition Spectroscopy of the S 1/2 – D 5/2 transition D 5/2 D 5/2 P 1/2 P 1/2 2 level system: 2 level system: 2 level system: 2-level-system: 2-level-system: 2-level-system: Fluorescence Fluorescence detection detection S 1/2 S S S 1/2 Zeeman structure in 5/2 non-zero magnetic field: non zero magnetic field: 3/2 1/2 D -1/2 5/2 -3/2 3/2 -5/2  sideband quantum + vibrational cooling state degrees of S S 1/2 1/2 processing 1/2 freedom -1/2

Spectroscopy with quantized fluorescence (quantum jumps) P D absorption and emission cause fluorescence steps cause fluorescence steps (digital quantum jump signal) monitor spectroscopy S detection efficiency: 99.85% 8 8 D-Zustand besetzt S-Zustand besetzt tensity D state occupied S state occupied S 7 ungen ents 6 measureme der Messu scence in 5 4 3 3 Anzahl # of m Fluores 2 1 D D 0 0 20 40 60 80 100 120 Zählrate pro 9 ms time (s) counts per 9 ms

Toolbox: entangling Mølmer ‐ Sørensen gate operation n+1 entangling operation with n bichromatic excitation n-1 + + + n+1 n+1 n n n-1 1 n-1 1 n+1 n n-1 MS ‐ gate generates GHZ states Realizes a two ‐ body Hamiltonian where every ion interacts with every other, e.g., for 3 ions

Toolbox: collective local rotations collective local operations 40 Ca + with resonant excitation D 5/2 + + + resonant manipulation S 1/2 Generate rotations about x/y axis Examples: Examples: 

Toolbox: addressed single ‐ qubit rotations 40 Ca + D 5/2 Single ‐ qubit rotations: Single ‐ qubit rotations: + + + off ‐ resonant manipulation ‐ far detuned laser ‐ AC ‐ stark shift AC stark shift S 1/2 Generate rotations about z axis Examples: p

Quantum gate operations: universal toolbox + + + + + collective local operations, F = 99% single ‐ qubit z ‐ rotations, F = 99% Mølmer ‐ Sørensen entangling operations, F = 98%

Entangling gates with more than two ions Entangling gates with more than two ions Two ‐ body interaction by off ‐ resonant spin ‐ motion coupling Effective spin ‐ spin interaction Many ions: Creation of GHZ states

Mølmer ‐ Sørensen gate: Two ions Mølmer ‐ Sørensen gate: Two ions A Sørensen K Mølmer A. Sørensen, K. Mølmer, PRL 82 , 1971 (1999) … … 1 0 … … 1 1 1 1 0 0 … 1 0 0

Creating GHZ ‐ states with 4 ions Creating GHZ ‐ states with 4 ions DDDD DDDD DDDS DDSD DSDD SDDD DDSS DSDS DSSD SDDS SDSD SSDD n = 1 n = 0 DSSS DSSS SDSS SDSS SSDS SSDS SSSD SSSD |0,SSSS>

Creating GHZ ‐ states with 8 ions Creating GHZ ‐ states with 8 ions DDDDDDDD SSSSSSSS

n ‐ qubit GHZ state generation with global MS gates Fidelity (%) P Parity signal it i l 1 99.5(7) single ion Ramsey fringe 2 98.6(2) 3 3 97.0(3) 4 4 95 7(3) 95.7(3) 6 6 89.2(4) 89 2(4) 81.7(4) 8 Phase  of analysing pulse (  ) T. Monz, P. Schindler, J. Barreiro et al., Phys. Rev. Lett. 106 , 130506 (2011)

n ‐ qubit GHZ state generation with global MS gates genuine N ‐ particle Entanglement P Parity signal it i l by 8 96  40  10 18  18  12 12 17  14 W Dü I Ci W Dü I Ci W. Dür, I. Cirac W. Dür, I. Cirac J. Phys A 34, J. Phys A 34, 6837, (2001) 6837, (2001) T. Monz, P. Schindler, J. Barreiro et al., Phys. Rev. Lett. 106 , 130506 (2011)

Fidelity decay of n ‐ qubit GHZ states (simple model) T. Monz, P. Schindler, J. Barreiro, M. Chwalla, M. Hennrich, Innsbruck (2009) single ‐ ion Ramsey contrast s g e o a sey co as Assumption: Assumption: p p magnetic field fluctuations magnetic field fluctuations cause collective dephasing cause collective dephasing 2 ions o s 3 ions 4 ions o s 8 i 8 ions 6 ions Waiting time (µs)

Dephasing of a single qubit (decay of Ramsey contrast) M. Chwalla, T. Monz, P. Schindler, Innsbruck 2010 5/2 3/2 1/2 ‐ 1/2 ‐ 3/2 3/2 ‐ 5/2 1/2 / ‐ 1/2 

Fidelity decay in randomly fluctuating magnetic field T. Monz et al., Phys. Rev. Lett. 106 , 130506 (2011) Hamiltonian with Fidelity Fidelity with (GHZ state) and denotes an ensemble average over the random fluctions For stationary noise of the magnetic field, characterized by a decay we obtain:

B ‐ field autocorrelation and coherence decay

Coherence of large ‐ scale entanglement T. Monz et al., Phys. Rev. Lett. 106 , 130506 (2011) relative one qubit error probability correlated noise correlated noise ‐ > superdecoherence ‐ > superdecoherence two qubits two qubits six four qubits three qubits qubits

Coherence of large ‐ scale entanglement T. Monz et al., Phys. Rev. Lett. 106 , 130506 (2011) BUT BUT application of a MS gate operation to the initial state yields the symmetric GHZ state the symmetric GHZ state insensitive to global dephasing by magnetic field noise insensitive to global dephasing by magnetic field noise measured coherence time consistent with expectation from lifetime limit:

Coherence of large ‐ scale entanglement T. Monz et al., Phys. Rev. Lett. 106 , 130506 (2011) D 5/2 5/2 S 1/2 consistent with expectation from lifetime (D 5/2 ) limit:

Error Protection: robust entanglement Error Protection: robust entanglement H. Häffner et al., Appl. Phys. B 81 , 151 (2005) prepare qubits i in S states S t t 5/2 5/2 Hiding states in S, S‘ states 3/2 D avoids decoherence from 1/2 -1/2 spontaneous emission spontaneous emission -3/2 -5/2 D 5/2 S‘ S‘ S 1/2 S 1/2 -1/2

Error Protection: Decoherence free subspace Error Protection: Decoherence free subspace Th. Monz et al., Phys. Rev. Lett. 103, 200503 (2009) Idea: use long-lived Bell states as logical qubits Idea: use long-lived Bell states as logical qubits Gate operations by Gate operations by - addressing individual ions - simultaneous addressing of innermost ions Trade off more ions (2x) for much extended coherence times (100-1000x)

Error Protection: Decoherence free subspace Error Protection: Decoherence free subspace Th. Monz et al., Phys. Rev. Lett. 103, 200503 (2009) Idea: use long-lived Bell states as logical qubits Idea: use long-lived Bell states as logical qubits 1 st experiment: Gate operations by Gate operations by - addressing individual ions and MS-gates - simultaneous addressing of innermost ions Analysis by state and process tomography

Process tomography of CNOT gate with logical qubit Process tomography of CNOT gate with logical qubit experiment mean gate fidelity Th. Monz et al., Phys. Rev. Lett. 103, 200503 (2009)

Quantum gate operations: universal toolbox + + + + + collective local operations, F = 99% single ‐ qubit z ‐ rotations, F = 99% Mølmer ‐ Sørensen entangling operations, F = 98%

Collective entangling gates + individual light shifts N ions Basic set of operations: individual light shift gates Mølmer-Sørensen gate + + + + + collective spin flips ● favorable ion addressing by light shifts (~  2 ) ● no interferometric stability between beams required no interferometric stability between beams required  Arbitrary unitary operations can be achieved !  Arbitrary unitary operations can be achieved ! ...but what is an optimal way to do this ?

Optimal control for arbitrary quantum gates V. Nebendahl et al., Phys. Rev. A 79 , Quantum optimal control: 012312 (2009) 012312 (2009) Find Find such that such that Gradient ascent algorithm: g N. Khaneja et al ., J. Magn. Res. 172 , 296 (2005) j , g , ( ) Modification of search algorithm: ● no simultaneous application of several Hamiltonians ● no simultaneous application of several Hamiltonians ● sequence of pulses with variable length Example: quantum Toffoli gate Example: quantum Toffoli gate 1 = = 2 3   

Scalable quantum computation requires error correction

Optimal control : Quantum Error Correction Quantum Error Correction: 3 qubits encode logical qubit (protection against spin flips) spin flip errors spin flip errors spin flip errors spin flip errors encoding reset ancillas error syndrome y correction detection step V. Nebendahl et al., Phys. Rev. A 79 , Implementation : 34 laser pulses (11 entangling pulses) 012312 (2009)

Quantum Error Correction, reduced encoding error decoding corr. reset   requires only three instead of five qubits i l h i d f fi bi  repetition requires re ‐ encoding