Experimental Implementations of Quantum Computing

David DiVincenzo, IBM

Course of six lectures, IHP, 1/2006

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)

Physical systems actively consideredfor 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

(list almost unchanged for some years)

Physical systems actively consideredfor 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

(list almost unchanged for some years)

Five criteria for physical implementation of a quantum computer

1. Well defined extendible qubit array -stable memory

2. Preparable in the “000…” state3. Long decoherence time (>104 operation time)4. Universal set of gate operations5. Single-quantum measurements

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.

Five criteria for physical implementation of a quantum computer

& quantum communications

1. Well defined extendible qubit array -stable memory

2. Preparable in the “000…” state3. Long decoherence time (>104 operation time)4. Universal set of gate operations5. Single-quantum measurements6. Interconvert stationary and flying qubits7. Transmit flying qubits from place to place

1. Qubit requirement• Two-level quantum system, state can be

1|0| ba• 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?

1. Qubit requirement (cont.)• Possible state of array of qubits (3):

...010|001|000| cba

• “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).

0H

Solid State Hilbert Spaces• 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• 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

2. Initialization requirement

• Initial state of qubits should be

...000000| Achieve by cooling, e.g., spins in large B field

• T = /log (104) = /4 (= 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

3. Decoherence times

)|(|2

1

)|(|2

1|)|(|

2

1

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.,

There is much more to be said about this!!

Vion et al, Science, 2002

• T2 lifetime can be observed experimentally

• Very device and material specific!

• E.g., T2=0.6sec for Saclay Josephson junction qubit (shown)

• T2 measures time for spin system to evolve from

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:

)/)(exp( tdtHiTU

• Different Hamiltonians are needed to generate the desired quantum gates:

yixiH ,zjziHcNOT

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

“Ising”

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

6/7. Flying Qubits

Algorithmic significance of criteria 6/7

1-5 are a bare minimum 6,7 involved in distributed quantum processing, cryptography6,7 may be advantageous in an efficient architecture

Quantum-dot array proposal

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

Kane (1998)

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

Quantum dot attached to spin-polarized leads* P. Recher, E.V. Sukhorukov, D. Loss, Phys. Rev. Lett. 85, 1962 (2000)

Spin Read-out via Spin Polarized Leads

* - 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)],

Thus: Is = 0 spin up ( Ic << Is )

Is > 0 spin down

=> single-spin memory device, US patent PCT/GB00/03416

Loss & DiVincenzoquant-ph/9701055

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:

)/)(exp( tdtHiU

• Different Hamiltonians are needed to generate the desired quantum gates:

yixiH ,zjziHcNOT

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))

Quantum-dot array proposal

Gate operations with quantum dots (1):

--two-qubit gate:

Use the side gates to move electron positionshorizontally, changing the wavefunction overlap

Pauli exclusion principle produces spin-spin interaction:

)( 21212121 zzyyxxJSJSH 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.

Making the CNOT from exchange:

a|

a|b|

b|Exchange generates the“SWAP” operation:

More useful is the “square root of swap”, S

S S =

Using SWAP:

S Sz

CNOT

Quantum-dot array proposal

Gate operations with quantum dots (2):

--one-qubit gate:

Desired Hamiltonian is:

)( zzyyxxBB BBBgBSgH

One approach: use back gate to move electronvertically. Wavefunction overlap with magneticor high g-factor layers produces desired Hamiltonian.

If Beff= 1T, t 160 psecIf Beff= 1mT, t 160 nsec

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 arriveat an (Ohmic) spin-boson model…

Spin-Boson Model

BSBSn

nnnzzx HHHbbXH

22

Leggett et al. RMP ’87; Weiss, 2nd ed. ‘99

22 E

constantcoupling:,)(:Coupling 2 n

nnnn cbbcX

nmnmzyx bbi ],[withbosons:Bath,2],[with1/2spin:System

functionspectral,)2/()()J(n

/12 cec sc

snn

s=1 –> Ohmic case

What is the decoherence time?

Master equation for spin boson • Von Neumann eq. for full density matrix ρ(t):

)()()](),([)( ttLittHit

Vλ~

(t)L

Bs LL(t)LL(t)

0

t

t

QtLQdti

BVVBS

t

t

SSSSS

TettU

LQttUQLTrtt

tttdtttLit

0

0

)'('

01

1

),(

)',()',(

)'()',(')()()(

Exact masterequation forsystem state

0)2( 0

BVitL

VBS LeLiTr B

]],[,[ 0)2( 0

AVeViTrA BitL

BSB

AtVee BitHitH BB 0,)(

00

.).()()( ,, ,,

chaegStbStV ijtiij

ji ji

ji

ji

ji

ij

VAtVtVAVtVVAAVtViTrA BBBBBS )()()()( 00)2(

nenegSVtV titiij

ji

jiB

)1()(2

,,

2

11

en

Born approximation:

• small

Evaluate in

Where

Thus:

Markov Approximation (MA) (heuristically)(general remarks, see also Fick & Sauermann)

t

zz dttttt0

')'()'()(

:,"0)(" cczz tfort

ctfordrop

Otttt

'

)('.)()'(

c

t

ttfor

c

c

0 00

• small

Consider pieces like

Environment correlation time

TkE

T

EE

T

B2

20

120

2

2

10

1

22//1

)2/coth(/1

Markov approximation -

Standard route to Bloch equations, exponential decay of coherenceWith transverse and longitudinal relaxation times:

condmat/0304118

Equation of motion: factorized initial conditions, Born approximation

t

t

QtLQdti

BVVBS

t

t

SSSSS

TettU

LQttUQLTrtt

tttdtttLit

0

0

)'('

01

1

),(

)',()',(

)'()',(')()()(

(2)

0)2( 0

BVitL

VBS LeLiTr B

MAKE NO FURTHER APPROXIMATIONS

Solution is algebraic in Laplace space:

…

(For the “prepare – evolve – measure” experiment)C(t) is power lawAt long times!

Structure of solution at low temperature:

1=T1-1 2=T2

-1

- -

alpha=0.01

.

.

“branch cut 2” contribution: prompt loss of coherence

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