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DANIEL TSUI LECTURE BEIJING 2005 SANKAR DAS SARMA UNIVERSITY OF MARYLAND CONDENSED MATTER THEORY CENTER WWW.PHYSICS.UMD.EDU/CMTCPowerPoint Presentation

DANIEL TSUI LECTURE BEIJING 2005 SANKAR DAS SARMA UNIVERSITY OF MARYLAND CONDENSED MATTER THEORY CENTER WWW.PHYSICS.UMD.EDU/CMTC

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### DANIEL TSUI LECTUREBEIJING 2005SANKAR DAS SARMAUNIVERSITY OF MARYLANDCONDENSED MATTER THEORY CENTERWWW.PHYSICS.UMD.EDU/CMTC

### Minimal QC Requirements

“My connection to Dan”

1976

1987-88

1995-2005

2004

Lai WY 1983-85 Beijing

Xie XC 1983-87; 88-91 USTC

Zhang FC 1984-86 Fudan

He S 1988-92 USTC

Li Q 1989-93 USTC

Lai ZW 1990-92 USTC(?);Chicago

Liu DZ 1990-94 USTC

Zheng L (1995-98); Hu J (1997-99) Indiana

Hu XD 1998-2003 Beijing;Michigan

Zhang Y 2002- USTC; Yale

Wang DW Taiwan 1996-2002

Tse GW Hong Kong 2004-

Also B.Y.K Hu; K.E. Khor, …

SC Zhang (Stanford); R. Zia (Virginia Tech.);

DC Tsui (Princeton)….

More than 100 publications with these collaborators!

I was in China (Beijing, Shanghai)

in 1986 as a guest of the Chinese

Academy of Sciences with the Institute of Physics being my host!

TIDBITS ABOUT QUBITSSankar Das Sarma

- QUBITS = TWO-LEVEL QUANTUM SYSTEM
- LINEAR SUPERPOSITION
- QUANTUM ENTANGLEMENT
- QUANTUM PARALLELISM

TOPOLOGICAL QUANTUM COMPUTATION

www.physics.umd.edu/cmtc

A (VERY) BRIEF HISTORY OF COMPUTATION

- UNARY: 10,000 YEARS AGO
- BINARY: 1,000 YEARS AGO; BITS
- ANALOG COMPUTERS: ~ 1000 years
- BOOLEAN ALGEBRA: BITS
- DIGITAL COMPUTERS: ~ 100 years
- QUANTUM MECH.: 100 YEARS AGO
- QUBITS: NOW (PERHAPS)
- QUANTUM COMPUTERS: ??

Spin Quantum Computation in Semiconductor NanostructuresLocalized Spin 1\2 qubits in Semiconductor Nanostructures(Heisenberg Coupling)X. Hu;R. de Sousa;B. Koiller;V. Scarola; W.Witzel ARDA, ARO, UMD, LPS, NSA

SPINTRONICS

- SPIN MATERIALS
Diluted magnetic semiconductors (DMS): ferromagnetic

- SPIN DEVICES
Active control of (nonequilibrium) spin AND charge

- SPIN QUBITS
Scalable solid state spin quantum computation

QUANTUM COMPUTERSHOW TO BUILD A QCPHYSICS OF QC ARCHITECTURE

- SCALABLE and ROBUST
- FAULT TOLERANT
- 100-10,000 COUPLED QUBITS
- Qubit dynamics
- Qubit coupling, entanglement
- Qubit decoherence

Shor algorithm

Exponential speedup

Database search

Grover algorithm

Algebraic speedup

Quantum simulation

Feynman’s dream

Quantum parallelism Entanglement

Universal one and two-qubit gates

Quantum error correction

Boolean vs. Quantum

P/NP some day??

Topological QC

What can a QC do?Why build a QC?Qubits: 2-level quantum systems

Initialization of qubits

Control and manipulation of qubits

Quantum coupling of 2-qubits

1- and 2-qubit gates

Quantum error correction

High fidelity

Qubit specific measurement

Long quantum coherence

Scalability

PROPOSED QC ARCHITECTURES (far too many)

- ION TRAPS
- LIQUID STATE NMR
- NEUTRAL ATOM OPTICAL LATTICE
- CAVITY QED
- SQUIDS, JOSEPHSON JUNCTIONS
- COOPER PAIR BOXES
- ELECTRON SPINS IN SOLIDS (GaAs, Si)
- SOLID STATE NMR
- ELECTRON STATES ON HE-4 SURFACE
- QUANTUM HALL STATES

Quantum computing with spins

Electron/nuclear spin: An ideal qubit?

Quantum algorithms: Factoring, searching...

- 1-qubit: Spin rotation
- 2-qubit: Exchange interaction

Quantum gates:

Spin relaxation and manipulation of localized states in semiconductors:Considerations for solid state quantum computer architectures

Si Donor Nuclear Spin

QC Architecture

Quantum Dot

QC Architecture

Semiconductor implementations semiconductors:

GaAs quantum dots

D. Loss and D.P. DiVincenzo, PRA 1998

Silicon donors (P)

B. Kane, Nature 1998

R. Vrijen et al., PRA 2000

Fault tolerant if coherence time

Experiments semiconductors:

GaAs

- Neighboring quantum dots
- Single electron in each dot
- Does a model of this system reproduce the Heisenberg
- model?

Spin Transitions in Few Electron Quantum Dots semiconductors:

Exact Diagonalization Theory

Going beyond perturbative/Heitler-London

exchange gate calculations in coupled dot

QC architectures ATOM to MOLECULE

Vito Scarola

WHEN IS THE 2-ELECTRON QUNTUM DOT A ‘MOLECULE’ WITH TUNABLE EXCHANGE COUPLING?

WHEN IS IT JUST AN ARTIFICIAL 2-ATOM SYSTEM?

Model semiconductors:

Schematic Parameter Space semiconductors:

Dot separation

Modified magnetic length

1

0

Cyclotron energy

Parabolic confinement

Small Exchange

Spin

Hamiltonian

Vortex

Mixing

Level

Crossings

1

(magnetic field)

Conclusion semiconductors:

- Exact diagonalization allows accurate
- treatment of strongly interacting regime

- Exchange splitting (J) oscillates with
- magnetic field

- Trial state analysis implies singlet-triplet
- transitions of Composite Fermions
- Artificial Atom to Artificial Molecule

Two Spins in Two Quantum Dots:Quantum Gates semiconductors:

S1

S2

B

Single spin qubits

Qubit #1

Qubit #2

- Heisenberg
- Hamiltonian:
- Quantum gates:
- Heisenberg interaction + local magnetic field gives
- universal set of quantum gates

Validity of Heisenberg Exchange Hamiltonian semiconductors:

For Spin-Based Quantum Dot Quantum Computers

Our system

Energy spectrum

Exchange splitting

Six electron double dot semiconductors:

Energy spectrum

Exchange splitting

Validity of Heisenberg Exchange Hamiltonian For Six-Electron Double Quantum Dot

Adiabatic Condition semiconductors:

- When the system Hamiltonian is changed adiabatically, the system wavefunction can be expanded on the instantaneous eigenstates:

- System evolution is governed by the
- Schroedinger equation:

- Instantaneous eigenvalues and eigenstates are needed to integrate the Schroedinger equation.

Loss due to non-adiabaticity semiconductors:

In an exchange gate for a double dot

Exchange in silicon-based quantum computer architecture semiconductors:

MOTIVATION

Kane’s proposal for a silicon-based quantum computer

B.E.Kane,

Nature (1998)

P donors in Si

Concern with donor positioning:

Each31P in the array must be

exactly under the A-gate.

From the website of SNF at the

University of New South Walws

Sydney, Australia

1-qubit operations semiconductors:

2-qubit operations

R =

EXCHANGE

- BUILDING BLOCKS OF KANE’S PROPOSAL
- qubits are the 31P nuclear spins (I=½)
- Spin interactions in Si:31P

Hydrogenic model for semiconductors:P donors in Si

~

+

_

P (V) 15 e – 15 p+

Si (IV) 14 e –

14 p+

Asymptotic exchange coupling of two hydrogen atoms

(Herring&Flicker, 1964)

a semiconductors:

Electrons in Si(beyond m* and …)

CONDUCTION

BAND MINIMUM:

Anisotropic and

six-fold degenerate

REAL SPACE:

Diamond

structure

RECIPROCAL SPACE: Brillouin zone

Ground state semiconductors:

Envelope functions:

Bloch wave

functions:

Heitler-London triplet-singlet splitting

Exchange between 31P donors in Si

* semiconductors:

Exchange calculated for two donors along [100]

2 semiconductors:nd neigh.

(12)

*

1stneigh.

PRL 88, 027903 (2002).

(4)

3rd neigh.

(6)

*

Exchange versus donor displacements

within the Si unit cell

The extreme sensitivity of the semiconductors:

exchange coupling to the relative

positioning of the substitutional donor

pair in Si is entirely due to the

six-fold degeneracy of the Si

conduction band minimum.

Dipolar spin coupling ? Dipolar gates?

Qubits are dipolar coupled semiconductors: single electron spins

B

R. de Sousa et al., cond-mat/031140, PRA 70, 052304 (2003)

Si:P SPIN DIPOLAR GATE QC ARCHITECTURE

Gate imperfection in the presence of exchange semiconductors:

- Long-range dipolar ~1/R3 is much stronger than short-range exchange for large inter-donor separation; How large should be the separation so that J can be neglected?

- J0 leads to error of the order of (J/D)2; Hence the criterium for gate error to be within p is:

Gate times and donor separation semiconductors:

- Separations of the order of 300 Å allows easier lithography;
- Gates are 106 times slower than exchange coupling; however there is no need for exchange control and donor positioning with atomic precision.

Using 28Si we expect T2~T1~ seconds for B~1T

Si Dipolar QC semiconductors:

- Long range couplings are corrected with no overhead in gate time (ability to -pulse within 5s is required).
- Dipolar implementation is reliable, its advantages/disadvantages should be compared with other proposals without exchange (for example, Skinner, Davenport, Kane, PRL 2003, which requires electron shuttling between donors);
- Dipolar coupling insensitive to electronic structure: No inter-valley interference, interstitial defects are also good qubits;
- “Top-down” construction schemes based on ion implantation can be used even though they lack atomic precision in donor positioning.
- Can be scaled up

Electron spin coherence in semiconductors:

semiconductor QC’s

Bound orbital states T1 ~ 1ms (GaAs Quantum dot)

(B=1T, T<<1K) 10 s (Si:P)

Decoherence is dominated by spin-spin interactions:

SPECTRAL DIFFUSION

Electron’s Zeeman frequency

fluctuates due to

nuclear dipolar flip-flops

B

RESULTS:

T2 ~ 50 ms GaAs-QD

>1000 ms Si:P

- Spin-orbit + phonons semiconductors:
- Hyperfine + phonons
- Spin-orbit + photons

- Spectral diffusion(nuclear spins, time dependent magnetic fields)
- Dipolar / exchange coupling between “like” spins

- Unresolved hyperfine structure
- Different g-factors
- Inhomogeneous fields
- Dipolar / exchange between “unlike” spins.

Spectral diffusion of a Si:P spin semiconductors:

B

Nuclear induced spectral diffusion semiconductors:

- Nuclear spins flip-flop due to their dipolar interaction;
- Electron’s Zeeman frequency fluctuates in time due to nuclear hyperfine field.

Theory

- Nuclear pairs are described by Poisson random variables;
- Flip-flop rates are calculated using the method of moments, a high temperature expansion.

The Hamiltonian semiconductors:

Dependency with semiconductors:29Si density, sample orientation

TM increases very fast when we remove 29Si !

Spin-1/2 theory of nuclear spectral diffusion: Comparison with experiment

GaAs quantum dots with experiment

Spectral diffusion is very important: Ga and As do not have I=0 isotopes !

DYNAMIC NUCLEAR POLARIZATION ? with experiment

Quantum theory of spectral diffusion: Cluster expansion results

[W.M. Witzel, R. de Sousa, S. Das Sarma, cond-mat/0501503]

Conclusions results

- Electrical control of single spin dynamics is promising for III-V quantum dots because of spin-orbit coupling;
- The spin of localized states interact weakly with the phonons at low T: Nuclear induced spectral diffusion if the dominant decoherence mechanism;
- Isotopically purified Si:P donor spins can be coherent for ~1000 s (B = 0.3 Tesla); 60 ms already measured ! (S.A. Lyon, 2003)
- GaAs quantum dots (or donors) coherent for only 1 – 100 s, but TM /J > 106 !

Making a quantum computer resultsWhat is the right analogy?

Aviation?

Manhattan project?

Controlled fusion?

Integrated circuits (“chips”)?

Toy paper airplane to 747 jumbo aircrafts

10-15 YEARS FOR <100 QUBITS RESEARCH QC

PERHAPS 50 YEARS FOR A ‘COMMERCIAL’ QC

BASED ON LINEAR EXTRAPOLATION

Solid state spin quantum computation in semiconductors results

CMTC/UMD Spin Quantum Computation Group

- Sankar Das Sarma
- Vito Scarola
- Kwon Park
- Belita Koiller
- Xuedong Hu
- Rogerio De Sousa, Wayne Witzel
- Juan Delgado, Magdalena Constantin
Supported by NSA, LPS, ARO, ARDA, UMD

SDS, Michael Freedman, Chetan Nayak

cond-mat/0412343 (PRL 2005)

with results

Expansion in t:

Expansion in l:

Quantum theory of spectral diffusion: Two possible series expansions[W.M. Witzel, R. de Sousa, S. Das Sarma, cond-mat/0501503]

Exact expression:

Large sets D are mainly composed of disconnected clusters S resultsi; If clusters are far enough, neglect inter-cluster coupling to get

Quantum theory of spectral diffusion: non-perturbative cluster expansionDefine “set D” contribution recursively:

Additive version of cluster exp.:

Examples of |D|=10:

Need 10-site exact solution

Need 2-site solution only!

resultsconfigurations

Lowest order cluster expansion: Product of pairsExact solution for pair nm

Cluster expansion interpolates between t and l expansions at the lowest order!

Spin-orbit coupling in semiconductor heterojunctions results

Spin-orbit coupling

Dresselhaus:

Rashba:

Electrical control of g-factor results

D.D. Awschalom, Nature 2001

E. Yablonovitch, PRB 2001

Electron penetration into AlGaAs barrier

But even without barrier penetration:

Dresselhaus!

Rashba!

for E~105 Volt/cm !

g-factor control results T1 control !!

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