1. Experimental Implementations of Quantum Computing David DiVincenzo, IBM Course 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)
Some very hard things have to happen in the laboratory to make even rudimentary quantum information processing a reality.  I will give a report "from the trenches" to give some idea of how you start from scratch -- in a state of the art solid state physics lab -- and try to make a working qubit.  I will also give a point of view on progress on other fronts where things seem to be going better, in particular in the atomic physics lab.

3. (list almost unchanged for some years)
Physical systems actively considered for quantum computer implementation
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

4. (list almost unchanged for some years)
Physical systems actively considered for quantum computer implementation
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

5. Five criteria for physical implementation of a quantum computer
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
D. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/ ; “The Physical Implementation of Quantum Computation,” Fort. der Physik 48, 771 (2000), quant-ph/

6. Five criteria for physical implementation of a quantum computer & quantum communications
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

7. 1. Qubit requirement Two-level quantum system, state can be
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. 1. Qubit requirement (cont.)
Possible state of array of qubits (3):
“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. Solid State Hilbert Spaces
Position of each electron is element of Hilbert space
Fock vector basis (second quantization), e.g., | ….>
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

10. Solid State Hilbert Spaces
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

11. 2. Initialization requirement
Initial state of qubits should be
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. 3. Decoherence times 1 1 y = (| ­ñ+ | ¯ñ ) | ­ñ Þ (| ­­ñ+ | ¯¯ñ ) 2 2
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
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. 4. Universal Set of Quantum Gates
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:
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. 5. Measurement requirement
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

15. 6/7. Flying Qubits Algorithmic 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 transistor R. Vrijen et al, Phys. Rev. A 62, (2000)

18. 5. Measurement requirement
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

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. Quantum Dot as Spin Filter and Spin Read-out
Quantum Dot weakly coupled to leads
in the Coulomb blockade regime,
in the presence of a magnetic field B
Spin-polarized current Is
vs. leakage current Ic:
>>1 !

21. Loss & DiVincenzo
quant-ph/

22. 4. Universal Set of Quantum Gates
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:
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))

23. 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.
Quantum-dot array proposal

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

25. 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
Quantum-dot array proposal

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

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

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

29. Evaluate in
Born approximation:
Where
Thus:
small

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

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

32. condmat/

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

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

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

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

37. 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