Plan

1. Experimental Implementations of Quantum Computing David DiVincenzo, IBMCourse of six lectures, IHP, 1/2006

2. Plan • Criteria for the physical implementation of quantum computing (I,II) • Single-electron quantum dot quantum computing (II, III) • Subtleties of decoherence: a Born approximation analysis (III) • Current experiments on single-electron quantum dots (IV) • Quantum gates implemented with the exchange interaction (IV) • Josephson junction qubits (V, VI) • Adiabatic q.c.; topological q.c. (VI)

3. Liquid-state NMR NMR spin lattices Linear ion-trap spectroscopy Neutral-atom optical lattices Cavity QED + atoms Linear optics with single photons Nitrogen vacancies in diamond Electrons on liquid He Small Josephson junctions “charge” qubits “flux” qubits Spin spectroscopies, impurities in semiconductors & fullerines Coupled quantum dots Qubits: spin,charge,excitons Exchange coupled, cavity coupled (list almost unchanged for some years) Physical systems actively consideredfor quantum computer implementation

4. Liquid-state NMR NMR spin lattices Linear ion-trap spectroscopy Neutral-atom optical lattices Cavity QED + atoms Linear optics with single photons Nitrogen vacancies in diamond Electrons on liquid He Small Josephson junctions “charge” qubits “flux” qubits Spin spectroscopies, impurities in semiconductors & fullerines Coupled quantum dots Qubits: spin,charge,excitons Exchange coupled, cavity coupled (list almost unchanged for some years) Physical systems actively consideredfor quantum computer implementation

5. Well defined extendible qubit array -stable memory Preparable in the “000…” state Long decoherence time (>104 operation time) Universal set of gate operations Single-quantum measurements Five criteria for physical implementation of a quantum computer D. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/9612126; “The Physical Implementation of Quantum Computation,” Fort. der Physik 48, 771 (2000), quant-ph/0002077.

6. Well defined extendible qubit array -stable memory Preparable in the “000…” state Long decoherence time (>104 operation time) Universal set of gate operations Single-quantum measurements Interconvert stationary and flying qubits Transmit flying qubits from place to place Five criteria for physical implementation of a quantum computer& quantum communications

7. Two-level quantum system, state can be 1. Qubit requirement • Examples: superconducting flux state, Cooper-pair charge, electron spin, nuclear spin, exciton • NB: A qubit is not a natural concept in quantum physics. Hilbert space is much, much larger. How to achieve?

8. Possible state of array of qubits (3): 1. Qubit requirement (cont.) • “entangled” state—not a  of single qubits • 23=8 terms total, all states must be accessible (superselection restrictions not desired) • Qubits must have “resting” state in which state is unchanging: Hamiltonian • (effectively).

9. Position of each electron is element of Hilbert space Fock vector basis (second quantization), e.g., |0000101010000….> Looks like large infinity of qubits, but superselection (particle conservation) make this untrue. Additional part of Hilbert space: electron spin --- doubles the number of modes of the Fock space Solid State Hilbert Spaces

10. Strategy to get a qubit: restrict to “low energy sector”. Still exponentially big in number of electrons now Fock vectors are in terms of orbitals, not positions identify states that differ slightly , i.e., electron moved from one orbital to another, or one spin flipped. This pair is a good candidate for a qubit: Fermionic statistics don’t matter (no superselection) decoherence is weak Hamiltonian parameters can (hopefully) be determined very accurately Solid State Hilbert Spaces

11. Initial state of qubits should be 2. Initialization requirement • Achieve by cooling, e.g., spins in large B field • T = D/log (104) = D/4 (D= energy gap) • Error correction: fresh |0 states needed throughout course of computation • Thermodynamic idea: pure initial state is “low temperature” (low entropy) bath to which heat, produced by noise, is expelled

12. T2 lifetime can be observed experimentally Very device and material specific! E.g., T2=0.6 lsec for Saclay Josephson junction qubit (shown) T2 measures time for spin system to evolve from 3. Decoherence times Vion et al, Science, 2002 Kikkawa & Awschalom, PRL 80, 4313 (1998) to a 50/50 mixture of | and |. This happens if the qubit becomes entangled with a spin in the environment, e.g., 1 1 y = ­ñ+ ¯ñ ­ñ Þ ­­ñ+ ¯¯ñ (| | ) | (| | ) 2 2 There is much more to be said about this!!

13. Quantum algorithms are specified as sequences of unitary transformations U1,U2, U3, each acting on a small number of qubits Each U is generated by a time-dependent Hamiltonian: 4. Universal Set of Quantum Gates • Different Hamiltonians are needed to generate the desired quantum gates: “Ising” 1-bit gate • many different “repertoires” possible • integrated strength of H should be very precise, part in 10-4, • from current understanding of error correction

14. Ideal quantum measurement for quantum computing: For the selected qubit: if its state is |0, the classical outcome is always “0” if its state is |1, the classical outcome is always “1” (100% quantum efficiency) If quantum efficiency is not perfect but still large (50%), desired measurement is achieved by “copying” (using cNOT gates) qubit into several others and measuring all. If q.e. is very low, quantum computing can still be accomplished using ensemble technique (cf. bulk NMR) Fast measurements (10-4 of decoherence time) permit easier error correction, but are not absolutely necessary 5. Measurement requirement

15. 6/7. Flying QubitsAlgorithmic significance of criteria 6/7 1-5 are a bare minimum 6,7 involved in distributed quantum processing, cryptography 6,7 may be advantageous in an efficient architecture

16. Quantum-dot array proposal

17. Kane (1998)  Concept device: spin-resonance transistorR. Vrijen et al, Phys. Rev. A 62, 012306 (2000)

18. Ideal quantum measurement for quantum computing: For the selected qubit: if its state is |0, the classical outcome is always “0” if its state is |1, the classical outcome is always “1” (100% quantum efficiency) If quantum efficiency is not perfect but still large (50%), desired measurement is achieved by “copying” (using cNOT gates) qubit into several others and measuring all. If q.e. is very low, quantum computing can still be accomplished using ensemble technique (cf. bulk NMR) Fast measurements (10-4 of decoherence time) permit easier error correction, but are not necessary 5. Measurement requirement

19. Spin Read-out via Spin Polarized Leads Quantum dot attached to spin-polarized leads* P. Recher, E.V. Sukhorukov, D. Loss, Phys. Rev. Lett. 85, 1962 (2000) Thus: Is = 0 spin up ( Ic << Is ) Is > 0 spin down => single-spin memory device, US patent PCT/GB00/03416 * - magnetic semiconductors [R. Fiederling et al., Nature 402, 787 (1999); Y. Ohno et al., Nature 402, 790 (1999)] - Quantum Hall Edge states [M. Ciorga et al., PRB 61, R16315 (2000)],

20. Loss & DiVincenzo quant-ph/9701055

21. Quantum algorithms are specified as sequences of unitary transformations U1,U2, U3, each acting on a small number of qubits Each U is generated by a time-dependent Hamiltonian: 4. Universal Set of Quantum Gates • Different Hamiltonians are needed to generate the desired quantum gates: 1-bit gate • many different “repertoires” possible • integrated strength of H should be very precise, 1 part in 10-4, • from current understanding of error correction • (but, see topological quantum computing (Kitaev, 1997))

22. Quantum-dot array proposal Gate operations with quantum dots (1): --two-qubit gate: Use the side gates to move electron positions horizontally, changing the wavefunction overlap Pauli exclusion principle produces spin-spin interaction: Model calculations (Burkard, Loss, DiVincenzo, PRB, 1999) For small dots (40nm) give J 0.1meV, giving a time for the “square root of swap” of t 40 psec NB: interaction is very short ranged, off state is accurately H=0.

23. Making the CNOT from exchange: Exchange generates the “SWAP” operation: More useful is the “square root of swap”, = Using SWAP:  CNOT

24. Quantum-dot array proposal Gate operations with quantum dots (2): --one-qubit gate: Desired Hamiltonian is: One approach: use back gate to move electron vertically. Wavefunction overlap with magnetic or high g-factor layers produces desired Hamiltonian. If Beff= 1T, t 160 psec If Beff= 1mT, t 160 nsec

25. small Decoherence analysis, Loss/DiVincenzo: the spin-boson model: General system-bath Hamiltonian: If we single out the lowest two eigenstates of H_S, then we arrive at an (Ohmic) spin-boson model…

26. Spin-Boson Model Leggett et al. RMP ’87; Weiss, 2nd ed. ‘99 s=1 –> Ohmic case What is the decoherence time?

27. Master equation for spin boson • Von Neumann eq. for full density matrix ρ(t): Exact master equation for system state

28. Evaluate in Born approximation: • small Where Thus:

29. Markov Approximation(MA)(heuristically)(general remarks, see also Fick & Sauermann) Consider pieces like • small Environment correlation time

30. Markov approximation - Standard route to Bloch equations, exponential decay of coherence With transverse and longitudinal relaxation times:

31. condmat/0304118

32. Equation of motion: factorized initial conditions, Born approximation (2) MAKE NO FURTHER APPROXIMATIONS

33. Solution is algebraic in Laplace space: … C(t) is power law At long times! (For the “prepare – evolve – measure” experiment)

34. Structure of solution at low temperature: . . - - alpha=0.01 G1=T1-1 G2=T2-1

35. “branch cut 2” contribution: prompt loss of coherence

36. Comments: --System-environment Hamiltonian can be deduced for proposed solid state qubits --Reduced dynamics of system can be derived (master equation) --”Standard Approach”: Born Markov theory, gives simple predictions, (exponential decay of coherence, relaxation times) --The Standard Approach is not the full story – non-exponential components of decay of coherence are expected. --Big gaps between theory and experiment remain