introduction to quantum computing lecture 3: qubit technologies rod van meter...

Introduction to Quantum Computing

Lecture 3: Qubit Technologies

Rod Van [email protected]

June 27-29, 2005WIDE University School of Internet

with help from K. Itoh, E. Abe,and slides from T. Fujisawa (NTT)

Course Outline

● Lecture 1: Introduction● Lecture 2: Quantum Algorithms● Lecture 3: Devices and Technologies● Lecture 4: Quantum Computer

Architecture● Lecture 5: Quantum Networking &

Wrapup

Lecture Outline

● Brief review● DiVincenzo's criteria & tech overview● Technologies:

All-Si NMR, quantum dots, superconducting, ion trap, etc.

● Comparison

Superposition and ket Notation

● Qubit state is a vector– two complex numbers

● |0> means the vector for 0;|1> means the vector for 1;|00> means two bits, both 0;|010> is three bits, middle one is 1;etc.

● A qubit may be partially both!(just like the cat, but stay tuned for measurement...)complex numbers are wave fn amplitude; square is probability of 0 or 1

1-qubit state and Bloch sphere(Phase)

relativephase

ei cos2

0 e i sin2

1

cos2

0 e i sin2

1

global phase has no effect

1-qubit basis states

01

01

0

1

0 1

2 2=1 , C

superposition

x

y

z

1

0 1

2

0

0 i 1

2

Bloch sphere

Reversible Gates

Xor

x x

NOTControl-NOT

Control-Control-NOT(Toffoli gate)

Control-SWAP(Fredkin gate)

This set can be used for classical reversible computing, as well;brings thermodynamic benefits.Bennett, IBM JR&D

Nov. 1973

Quantum Algorithms● Deutsch-Jozsa(D-J)

– Proc. R. Soc. London A, 439, 553 (1992)

● Grover's search algorithm– Phys. Rev. Lett., 79, 325 (1997)

● Shor's algorithm for factoring large numbers– SIAM J. Comp., 26, 1484 (1997)

D. Deutsch R. Jozsa L. K. Grover P. W. Shor

Grover's Search

nN 2 items

N 2n unordered items, we're looking for item “β”

Classically, would have to check about N/2 items

To use Grover's algorithm, create superposition of all N inputs

Nrepetitions of the Grover iterator G will produce “β”

1

N x 0

N 1

x

Hard

task!!

β

Shor's Factoring Algorithm

66554087 ? 6703 9929An “efficient” classical algorithm for this

problem is not known.

Using classical number field sieve, factoring anL-bit number is

O ek3

L log2 L O L3

Using quantum Fouriertransform (QFT), factoring an L-bit number is

Superpolynomial speedup! (Careful, technicallynot exponential speedup, and not yet proven thatno better classical algorithm exists.)

Summary: Characteristics of QC

● Superposition brings massive parallelism● Phase and amplitude of wave function

used● Entanglement● Unitary transforms are gates● Measurement both necessary and

problematic when unwanted

So, can we surpass a classical computer with a quantum one?

Depends on discovery of quantum algorithms

and development of technologies

Lecture Outline



● Comparison

DiVincenzo’s Criteria1. Well defined extensible qubit

array2. Preparable in the “000…” state3. Long decoherence time4. Universal set of gate operations5. Single quantum measurements

Qubit initialization

Read the result

Must be done within decoherence time!

Execution of an algorithm

Qubit Representations

● Electron: number, spin, energy level● Nucleus: spin● Photon: number, polarization, time,

angular momentum, momentum (energy)

● Flux (current)● Anything that can be quantized and

follows Schrodinger's equation

Problems● Coherence time

– nanoseconds for quantum dot, superconducting systems

● Gate time– NMR-based systems slow

(100s of Hz to low kHz)● Gate quality

– generally, 60-70% accurate● Interconnecting qubits● Scaling number of qubits

– largest to date 7 qubits, most 1 or 2

A Few Physical Experiments

● IBM, Stanford, Berkeley, MIT(solution NMR)

● NEC (Josephson junction charge)● Delft (JJ flux)● NTT (JJ, quantum dot)● Tokyo U. (quantum dot, optical lattice, ...)● Keio U. (silicon NMR, quantum dot)● Caltech, Berkeley, Stanford (quantum dot)● Australia (ion trap, linear optics)● Many others (cavity QED, Kane NMR, ...)

Physical Realization

Cavity QED

Magnetic

resonance

Ion trap

Superconductor

Examples of Qubits

Trapped ions

Neutral atoms

in optical lattices

Solution NMR

Crystal lattice

Flux states in

superconductors

Charge states in

superconductors

Impurity spins in

semiconductors

Single-spin

MRFM

Atomic

cavity QEDOptically driven

electronic states

in quantum dots

Electronically

driven

electronic states

in quantum dots

Optically driven

spin states in

quantum dots

Electronically driven

spin states in quantum dots

Electrons

floating

on liquid

helium

Solid-state

systems

Others: Nonlinear

optics, STM etc

All-silicon quantum computer

Shor (7)

Entanglement (4) Rabi (1)

Lecture Outline



● Comparison

Technologies Reviewed

● Liquid NMR● Solid-state NMR● Quantum dots● Superconducting Josephson junctions● Ion trap● Optical lattice● All-optical

Liquid Solution NMR

Billions of molecules are used, each one a separate quantum computer. Most advanced experimental demonstrations to date, but poor scalability as molecule design gets difficult and SNR falls. Qubits are stored in nuclear spin of flourine atoms and controlled by different frequencies of magnetic pulses.Used to factor 15 experimentally.

Vandersypen, 2000

All-Silicon Quantum Computer

105 29Si atomic chains in 28Si matrix work

like molecules in solution NMR QC.

A large field gradient

separates Larmor frequencies

of the nuclei within each

chain.

Many techniques

used for solution

NMR QC are

available.

No impurity

dopants or

electrical contacts

are needed.

Copie

s

12

3

4

5

6

8

7

9

T.D.Ladd et al., Phys. Rev. Lett. 89, 017901 (2002)

Overview

dBz/dz = 1.4 T/mB0 = 7 T Active region

100m x 0.2m

L 400 Î ¼m

W 4 Î ¼m

H 10 Î ¼m s 2 .1 Î ¼m

l 300 Î ¼m

t 0 .25 Î ¼m

w 4 Î ¼m

Keio Choice of System and Material

Two incompatible conditions to realize quantum computers:

1. Isolation of qubits from the environment

2. Control of qubits through interactions with the

environmentSystem: NMR1. Weak ensemble measurement2. Established rf pulse techniques for

manipulationMaterial: silicon

1. Longest possible decoherence time2. Established crystal growth, processing and

isotope engineering technologies

Fabrication: 29Si Atomic Chain

28Si29Si

J.-L.Lin et al., J. Appl. Phys 84, 255 (1998)

Regular step arrays on

slightly miscut 28Si(111)7x7

surface (~1º from normal)

Steps are straight, with

about 1 kink in 2000 sites.

STM image

Kane Solid-State NMR

Qubits are stored in the spin of the nucleus of phosphorus atoms embedded in a zero-spin silicon substrate. Standard VLSI gates on top control electric field, allowing electrons to read nuclear state and transfer that state to another P atom.

Kane, Nature, 393(133), 1998

Semiconductor nanostructure

High electron mobility transistor (HEMT)

http://www.fqd.fujitsu.com/

Quantum dot structure

N < 30

# of electrons

Quantum dot in the Coulomb blockade regime

Coulomb blockade region, (the number of electrons is an integer, N)

orbital degree of freedom&

spin degree of freedom

double quantum dot

bonding & anti-bonding orbitals(charge qubit)

(spin qubit)

Single-electron transistor (SET)

Quantum dot artificial atom

S. Tarucha et al., Phys. Rev. Lett. 77, 3613 (1996).

1s 2p 3d

|(r)|2

500 nm

Environment surrounding a QD

coherency of the system

dissipation (T1), decoherence (T2) & manipulation

Double quantum dot device

A double quantum dotfabricated in AlGaAs/GaAs 2DEG

(schematically shown by circles)

Approximate number of electrons: 10 ~ 30Charging energy of each dot: ~ 1 meVTypical energy spacing in each dot: ~100 ueVElectrostatic coupling energy: ~200 ueV

Lattice temp. ~ 20 mK (2 ueV)Electron temp. ~100 mK (9 ueV)

cf. CPB charge qubit (Y. Nakamura ’99)

Measurement system

1 mm

dilution refrigerator

Tlat ~ 20 mK

Telec ~ 100 mK

B = 0.5 T

Double quantum dot

AlGaAs/GaAs 2DEG

EB litho., fine gates,

ECR etching

“thin coax”filtering

Josephson Junction Charge (NEC)

Two-qubit device

Pashkin et al., Nature, 421(823), 2003

One-qubit device can control the number of Cooper pairs of electrons in the box, create superposition of states.Superconducting device, operates at low temperatures (30 mK).

Nakamura et al., Nature, 398(786), 1999

JJ Phase (NIST, USA)

J. Martinis, NIST

Qubit representation is phase of current oscillation.Device is physically large enough to see!

JJ Flux (Delft)

The qubit representation is a quantum of current (flux) moving either clockwise or counter-clockwise around the loop.

Ion Trap

NIST, Oxford, Australia, MIT, others

Ions (charged atoms) are suspended in space in an oscillating electric field. Each atom is controlled by a laser.

Ion Trap

NIST, Oxford, Australia, MIT, others

Number of atoms that can be controlled is limited, and each requires its own laser. Probably <100 ions max.

Optical Lattice (Atoms)

Deutsch, UNM

Neutral atoms are held in place by standing waves from several lasers.Atoms can be brought together to execute gates by changing the waves slightly.Also used to make high-precision atomic clocks.

All-Optical (Photons)

All-optical CNOT gate composed from beam splitters and wave plates. O'Brien et al., Nature 426(264), 2003

Lecture Outline



● Comparison

Comparison

● NMR (Keio, Kane): excellent coherence times, slow gates– nuclear spin well isolated from environment– Kane complicated by matching VLSI pitch to

necessary P atom spacing, and alignment● Superconducting: fast gates, but fast

decoherence● Quantum dots: ditto

– electrons in solid state easily influenced by environment

Comparison (2)

● Ion trap: medium-fast gates, good coherence time (one of the best candidates if scalability can be addressed)

● Optical lattice (atoms): medium-fast gates, good coherence time; gates and addressability of individual atoms need work

● All-Optical (photons): well-understood technology for individual photons, but hard to get photons to interact, hard to store

By the Numbers

Technology Decoherence Time Gate Time GatesAll-Si NMR 25 seconds high millisecs 10000?Kane NMR seconds high millisecs 1000?Ion Trap seconds low millisecs 1000?Optical Lattice seconds low millisecs ?Quantum Dot low microsecs nanosecs 100?J osephson J unction microseconds nanosecs 10-10000?All-Optical N/A N/A N/A

Apples-to-apples comparison is difficult, and coherence times are rising experimentally.

Quantum Error Correction and the Threshold Theorem

Our entire discussion so far has been on “perfect” quantum gates, but of course they are not perfect.

Various “threshold theorems” have suggested that we need 10^4 to 10^6 gates in less than the decoherence time in order to apply quantum error correction (QEC). QEC is a big enough topic to warrant several lectures on its own.

Wrap-Up

● Qubits can be physically stored on electrons (spin, count), nuclear spin, photons (polarization, position, time), or phenomena such as current (flux); anything that is quantized and subject to Schrodinger's wave equation.

● More than fifty technologies have been proposed; all of these described are relatively advanced experimentally.

Wrap-Up (2)

● Many technologies depend on VLSI● Most are one or two qubits● Have not yet started our own Moore's

Law doubling schedule● Several years yet to true, controllable,

multi-qubit demonstrations● 10-20 years to a useful system?

Upcoming Lectures

● Quantum computer architecture– how do you scale up? how do you build a

computer out of this? what matters?● Quantum networking

– Quantum key distribution– Teleportation

● Wrap-Up and Review

introduction to quantum computing lecture 3: qubit technologies rod van meter...

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