Physical implementations of quantum computing

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1 Physical implementations of quantum computing Andrew Daley Department of Physics and Astronomy University of Pittsburgh

2 Overview Introduction DiVincenzo Criteria Characterising coherence times Survey of possible qubits and implementations Neutral atoms Trapped ions Colour centres (e.g., NV-centers in diamond) Quantum dots Superconducting qubits (charge, phase, flux) NMR Optical qubits Topological qubits Summary and comparison

3 DiVincenzo Criteria: Requirements for the implementation of quantum computation 1. A scalable physical system with well characterized qubits 1i 0i 2. The ability to initialize the state of the qubits to a simple fiducial state, such as i 0i 3. Long relevant decoherence times, much longer than the gate operation time (see next section) 4. A universal set of quantum gates (single qubit rotations + C-Not / C-Phase /... ) control target U U 5. A qubit-specific measurement capability D. P. DiVincenzo The Physical Implementation of Quantum Computation, Fortschritte der Physik 48, p. 771 (2000) arxiv:quant-ph/

4 Desiderata for quantum communication 6. The ability to interconvert stationary and flying qubits 7. The ability faithfully to transmit flying qubits between specified locations Quantum Computer A Quantum Computer B Quantum Computer C

5 Characterising coherence times Coherence times for qubits are characterized by the timescales: (1) for a change in the probability of occupation of either qubit state; and (2) for a randomisation of the phase in superposition states 1i T 1 T 2 State Density matrix i = 0i + 1i = ih 2! 2 0i Timescale T 1 characterises changes in 2 and 2 Timescale T 2 characterises decay of, (loss of purity) Typically, T 2 <T 1 For ensemble measurements (e.g., repeated measurements with fluctuating parameters or multiple qubits in inhomogeneous environments), the system may appear to decohere, due to averaging on a timescale T 2 <T 2 This can often be corrected, e.g., by spin-echo experiments that remove the averaging over the fluctuating parameter

6 Neutral atoms

7 Rb:

8 Alkali atoms e.g., Rb, Li, K, Cs,... Qubits encoded on long-lived hyperfine states Alkaline earth (-like) atoms e.g., Sr, Yb,... Metastable electronic states 1 P 1 P 1/2 1i 3 P 2 3 P 1 3 P 0 1 S 0 S 1/2 1i 0i 0i or nuclear spin states 0i 1i

9 Back to the DiVincenzo Criteria: Requirements for the implementation of quantum computation 1. A scalable physical system with well characterized qubits 1i 0i 2. The ability to initialize the state of the qubits to a simple fiducial state, such as i 0i 3. Long relevant decoherence times, much longer than the gate operation time 4. A universal set of quantum gates (single qubit rotations + C-Not / C-Phase /... ) control target U U 5. A qubit-specific measurement capability D. P. DiVincenzo The Physical Implementation of Quantum Computation, Fortschritte der Physik 48, p. 771 (2000) arxiv:quant-ph/

10 Initialisation Optical pumping P 1/2 + S 1/2 1i 0i Decoherence times T 1 is very long (e.g., generated by blackbody radiation, T 1 > 100s) T 2 is limited, e.g., by inhomogeneous magnetic field fluctuations (measured scales ~100ms - 20s)

11 Single-qubit gates Raman transitions or radio frequency / microwave fields P 1/2 S 1/2 1i 0i i = 0i + 1i In appropriate parameter regimes, resonant laser coupling will generate the equations of motion (Schrödinger equation) i d 1 0 = 2 (t) 1 dt 2 (t) 0 where we make a Rotating wave approximation, and write the qubit states in the rotating frame. This treatment is excellent (good to within a factor of 10 8 in typical experiments). We have also chosen the phase of so that it is real at the location of the qubit. R

12 S 1/2 0i 1i i = 0i + 1i i d dt 0 = (t) 2 (t) 0 Introducing = R t (t)dt as a new time variable we can solve this equation exactly: U t = In particular a so-called -pulse R +1 1 (t) (0) = U (t) t (0) cos 1 2 i sin 1 2 i sin 1 2 cos 1 2 (t)dt = inverts the TLS, U t= / = 0 i i 0 apple gi! i ei ei! i gi and a 2 pulse returns the atom to its ground state, U t=2 / = apple gi! gi ei! ei but with a negative sign for the amplitudes (rotation of a spin-1/2 by 2 changes the sign).

13 Experiments:

0 mw Features: Control via magnetic field / laser light

14 Degenerate Bose/Fermi Gases in the laboratory: K/Li/Sr Experiment Innsbruck: 3.45 mw 2.2 mw Cs BEC Innsbruck 1.0 mw Features: Control via magnetic field / laser light Microscopically well understood systems Degenerate Fermi Gas Bose-Einstein Condensate

used to trap individual atoms for quantum computing purposes Optical

15 Dipole traps and optical lattices: laser laser off-resonant laser AC Stark shift V (x) = ( )I = V 0 sin 2 (kx) Dipole traps for single atoms can be used to trap individual atoms for quantum computing purposes Optical lattices allow preparation of a whole register at once, in contrast, e.g., to trapped ions.

16 Quantum Register 0 1 Requirements: Long lived storage of qubits Addressing of individual qubits Single and two-qubit gate operations Array of singly occupied sites Qubits encoded in long-lived internal states (alkali atoms - electronic states, e.g., hyperfine) Entanglement via Rydberg gates or via controlled collisions in a spin-dependent lattice

17 Rydberg Blockade Gates: FIG. 1. (a) Setup: A constant electric field along the z direction is applied to alkali atoms trapped in microtraps. (b) Level scheme: Two ground states jg and je (qubits), and laser excitation to the Rydberg state jg!jr. Excitation of atoms to Rydberg states (highly excited electronic states), with long-range interactions Jaksch, Cirac, Zoller, Rolston, Coté, Lukin, PRL 2000

18 Rydberg Blockade Gates: Excitation of atom 1 ( pulse g $ r) Excitation and deexcitation of atom 2 (2 pulse g $ r) Deexcitation of atom 1 ( pulse g $ r) $ Initial Step 1 Step 2 Step 3 eei eei eei eei gei i rei i rei gei egi egi egi egi ggi i rgi i rgi (not rri) ggi FIG. 2. Schematics of the ideal scheme. The internal state Jaksch, Cirac, Zoller, Rolston, Coté, Lukin, PRL 85, 2208 (2000) E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, Nature Phys. 5, 110 (2009). A. Gäetan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, Nature Phys. 5, 115 (2009).

19 Gates by controlled collisions 0 1 State - dependent lattice: particles move only if they are in state 0 If two particles are present on the same site, their energy shifts by U due to collisions. This leads to a phase accumulation Controlled collisions Initial Final 00i 00i 01i e iut/~ 01i 10i 10i 11i 11i e iut/~ D. Jaksch, H.-J. Briegel,J. I. Cirac, C. W. Gardiner, and P. Zoller, PRL 82, 1975 ( 99)

20 Collisional Gates (simple example): Gate: controlled collisions D. Jaksch et al., PRL 82, 1975 ( 99) Operation performed in parallel for whole system Simple preparation of a cluster state Ideal setup for measurement-based quantum computing. Atom 1 Atom 2 1 S 0 3 P 0 internal states move collision by hand

21 Back to the DiVincenzo Criteria: Requirements for the implementation of quantum computation 1. A scalable physical system with well characterized qubits 1i 0i 2. The ability to initialize the state of the qubits to a simple fiducial state, such as i 0i 3. Long relevant decoherence times, much longer than the gate operation time 4. A universal set of quantum gates (single qubit rotations + C-Not / C-Phase /... ) control target U U 5. A qubit-specific measurement capability D. P. DiVincenzo The Physical Implementation of Quantum Computation, Fortschritte der Physik 48, p. 771 (2000) arxiv:quant-ph/

22 New Possibility: In-situ measurement of atoms, correlation functions: Experiments: Harvard, Chicago, Munich, Oxford, Toronto,... e.g., Markus Greiner s Microscope at Harvard:

23 HARVARD UNIVERSITY n MIT CENTER FOR ULTRACOLD ATOMS

Varian Experiments: Harvard, Chicago, Munich, Oxford, Toronto,.

org on April 4, 201 podd 0.6 esults ithm 0.25 B T/U = 0.05 row. 0.4 T/U = 0.

0 s ex0 20 40 e 2D U/J abatw. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I.

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15 (solid black line), and 0.05 471, 319-324 (2011).

New Possibility: In-situ measurement of atoms, correlation functions: 1.

24 Varian Experiments: Harvard, Chicago, Munich, Oxford, Toronto,... Quantum Gas Microscopes at Harvard / Garching: Downloaded from on April 4, 201 podd 0.6 esults ithm 0.25 B T/U = 0.05 row. 0.4 T/U = 0.15 A icate T/U = , re0.2 site. n=2 shell 0.10 fluo s ex e 2D U/J abatw. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Foelling, L. Pollet, and M. Greiner, and the transverse confinement (averaged over five shots and binned over , (2010). value of podd versus the interaction-to-tunneling ratio U/J. Data sets, withscience 1s error s that formc. partweitenberg, of the n = 1 (squares) n = 2 (circles) shells in a deep M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, M.andEndres, J.Mott F. Sherson, d on finite-temperature Monte Carlo simulations in a homogeneous system at Nature eraction ratio (T/U) of 0.20 (dotted red line), 0.15 (solid black line), and , (2011). s on the right is the corresponding odd-even variance given by podd(1 podd). New Possibility: In-situ measurement of atoms, correlation functions: 1.0 B odd A p ng of the insulator. w podd on averaging e images. and the 45 Hz. As ased, the insulator. The valred shells numbered tom numu imaging plane, are B), 870 T (E and F) an be corppropriate onto the y circular f 20 ana-shot raw 0.0 C D a c b y x 2 μm e E he occupied sites were as a result of atoms lost F x1000 counts of background gas collisions. The average occupation numbers and error bars shown in Figure Fig. 2 Single-site addressing. a, Top, experimentally obtained and

Quantum information processing with individual neutral atoms in optical tweezers. Philippe Grangier. Institut d Optique, Palaiseau, France

Quantum information processing with individual neutral atoms in optical tweezers Philippe Grangier Institut d Optique, Palaiseau, France Outline Yesterday s lectures : 1. Trapping and exciting single atoms