Bulk Spin Resonance Quantum Information Processing

transcript

Yael Maguire

Physics and Media Group (Prof. Neil Gershenfeld)

MIT Media Lab

ACAT 2000

Fermi National Accelerator Laboratory, IL

17-Oct-2000

Why should we care?• By ~ 2030: transistor = 1 atom, 1 bit = 1 electron,

Fab cost = GNP of the planet• Scaling: time (1 ns/ft), space (DNA computers

mass of the planet).• Remaining resource: Hilbert Space.

• Classical bit

• Analog “bit”

• Quantum qubit

b { , }0 1 0 1

q q 11

a [ , ]0 1 0 1

• 2 Classical Bits

• 2 Quantum Bits

b b1 0 00 01 10 11{ , , , }

q 00 01 10 11

• N Classical Bits–N binary values

• N Quantum Bits–2N complex numbers

–superposition of states

–Hilbert space

More Bits

• correlated decay

• project A

• hidden variables?

• action at a distance?

• information travelling back in time?

• alternate universes (many worlds)?

• interconnect in Hilbert space – O(2-N) to O(1)

12 01 10 01 10

A A or

Entanglement

• Examples:

– Shor’s algorithm (1000 bit number):• O((logN)2+) vs. O(exp(1.923+

(logN)1/3(loglogN)2/3)• O(1 yr) @ 1Hz vs. O(107 yrs) @ 1

– Grover’s algorithm (8 TB):• O( ) vs. O(N)• 27 min. vs. 1 month @ same clock

speed.

The Promise

What do you need to build a quantum computer?• Pure States

• Coherence

• Universal Family

• Readout

• Projection Operators

• Circuits

Previous/Current Attempts

•spin chains • quantum dots

•isolated magnetic spins • trapped ions

•Optical photons • cavity QED

•Coherence!

Breakthroughs:•Bulk thermal NMR quantum computers

–quantum coherent information bulk thermal ensembles

•Quantum Error Correction–Correct for errors without observing. –Add extra qubits syndrome

What do you need to build a quantum computer using NMR?

• Pure States– effective pure states in deviation density matrix

• Coherence– nuclear spin isolation, 1-10s

• Universal Family– arbitrary rotations (RF pulses) and C-NOT (spin-spin interactions)

• Readout– Observable magnetization

• Projection Operators– Change algorithms

• Circuits– Multiple pulses are gates

Gershenfeld, Chuang, Science (1997)Cory, Havel, Fahmy, PNAS (1997)

• wave function

• observables

• pure state

• mixed state

• Hamiltonian (energy)

• evolution

• equilibrium

*A A c c Amn n m

( ) ( ) ( ) / / t U t U e t eiHt iHt

Quantum Mechanics

• ~1023 spin degrees of freedom– rapid tumbling averages inter-molecular interactions

• ~N effective degrees of freedom– decoherence averages off-diagonal coherences

p k kk

1 2 10

N spins I (1/2)

B0 B1Bulk Density Matrix

• high temperature approximation

• identity can be ignored

• ensemble molecule deviation

NMR: “reduced” density matrix

e H kT

U U U U U UN N

Deviation Density Matrix in NMR

• magnetic moment

• angular momentum

• spin precession

• Zeeman splitting

• 2 spin interaction Hamiltonian

I E B 1

H I I I IA zA B zB AB zA zB chemical shifts~ 100 MHz

scalar coupling~ 100 Hz

Spin Hamiltonian

• apply a z field:

• evolve in field:

• two spins, scalar coupling:

• evolution = 3 commuting operators

H B B Iz z z

iHt i B tI i I

z z z /

cos sin

2 21 2

H I I I IA zA B zB AB zA zB

e R t R t R tiHtzA A zB B zAB AB

/ ( ) ( ) ( )

R tzAB ABAB( ) cos sin AB

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1Arbitrary single qubit operations

Magnetic Field and Rotation Operators

CAB RyA RzB RzA RzAB RyAi

( ) ( ) ( ) ( ) ( )

90 90 270 90 90 90

1 1 0 0

0 0 1 1

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 1 0 0

0 0 1 1

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

Ry-1PRy

ARyA(-90)

RyA(90)

/ 2 Bt=

/ 2 At=3

RxA(180)

RxB(180)

RxA(180)

RxB(180)

• ENDOR (1957)– electron-nuclear

double resonance

• INEPT (1979)– insensitive nuclei

enhanced by polarization transfer

The Controlled-NOT Gate

The Controlled-NOT Gate Input thermaldensity matrix

CNOT output

Ground State Preparation• We want:

where• How? Use degrees of freedom to create an

environment for computational spins. – 1. Logical Labeling (Gershenfeld, Chuang)

• ancilla spins - submanifolds act as pure states - exponential signal

– 2. Spatial Labeling (Cory, Havel, Fahmy)• field gradients dephase density matrix terms -

exponential space

– 3. Temporal Labeling (Knill, Chuang, Laflamme)• use randomization and averaging over set of

experiments - exponential time

),...,,,(ˆ

)12/( N

Algorithms - Grover’s Algorithm

• find xn | f(xn) = 1, f(xm)=0

• Initialize L bit registers• Prepare superposition of states• Apply operator that rotates

phase by if f(x) = 1 • Invert about average

• Repeat O(N1/2) times• Measure state

AM M M HPHij iiN N

H P P Pijn i j

ij ii 2 1 0 1 1200

/ ( ) , ,

NMR Implementation

• Pure state preparation

• Superposition of all states

H = RyA(90) RyB(90) - RxA(180) RxB(180)

• Conditional sign flip (test for both bits up)

C = RzAB(270) - RzA(90) - RzB(90)

• Invert-about-mean

M = H - RzAB(90) - RzA(90) - RzB(90) - H

Experimental Implementation ofFast Quantum Searching,

I.L. Chuang, N. Gershenfeld, M. Kubinec,Physical Review Letters (80), 3408 (1998).

Quantum Error Correction

• 3-bit phase error correcting code - Cory et al, PRL, 81, 2152 (1998) - alanine

Quantum Simulation• Feynman/Lloyd - quantum simulations more

efficient on a quantum computer• Waugh - average Hamiltonian theory• Dynamics of truncated quantum harmonic

oscillator with NMR- Samaroo et al. PRL, 82, 5381.

Scaling Issues

• Sensitivity vs. System resources

• Decoherence per gate

• Number of qubits

Scaling

2/cosh

2/sinh

ˆˆTr

Scaling

• Is it quantum? Schack, Caves, Braunstein, Linden, Popescu, …

• Initial conditions vs quantumevolution

• But, Boltzmann limit is not scalable

catcatN 2

is separable if

3.8x10-610

1.5x10-59

6.0x10-58

2.4x10-47

9.1x10-46

3.4x10-35

1.2x10-24

Polarization Enhancement - Optical Pumping

• Error correction as well (or phonon)

Decoherence per gate• Steady state error correction - 10-4 - 10-6

C. Yannoni, M. Sherwood, L. Vandersypen, D. Miller, M. Kubinec, I. Chuang,Nuclear Magnetic Resonance Quantum Computing Using Liquid Crystal Solvents

quant-ph/9907063, July 1999

zBBzAA

ˆˆˆ

zBBzAA

ˆˆˆ ''

0.7 sT2 (1H)7 s

0.2 sT2 (13C)0.3 s

1.4 sT1 (1H)19 s

2 sT1 (13C)25 s

1706 HzJ+2D

J215 Hz

ZLI-116713C1HCl3solvent

acetone

Number of Qubits

• Seth Lloyd, Science, 261, 1569 (1993) - SIMD CA– D-A-B-C-A-B-C-A-B-C....– at worst linear, but may be polylogarithmic

• Shulman, Vazirani (quant-ph/980460) - using SIMD CA– can distill qubits where SNR independent of

system size

Our goals

• Develop the instrumentation and algorithms needed to manipulate information in natural systems

• Table-Top (size & cost)• investigate scaling issues

$50,000

$500,000

$5,000

Magnet Design

• Halbach arrays using Nd2Fe14B: 1.2T 2.0T

• Fermi Lab - iron is a good spatial filter

Compilation• Multiplexed Add:• function program = madd(cnumif0, cnumif1, enabindex, selindex, inputbits, outputbits,• BOOLlowisleft) % outputbits MUST be zeros• %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%• % madd.m• % Implements adding a classical number to a quantum number, mod 2^L.• % If N is the thing we want to factor, then selindex says whether N-cnum is less than or• % greater than B: N-cnum>b --> add cnum, else N-cnum<b --> add cnum - N + 2^L• % Enabindex must all be 1, else choose the classical addend to be zero.• % Edward Boyden, e@media.mit.edu• % INPUT• % cnum classical number to be added• % indices column vector of indices on which to operate• % carryindex carry qubit that you're using• %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%• L = length(outputbits); %It's an L-bit adder: contains L-1 MUXFAs and 1 MUXHA• if (L!=length(inputbits)) %MAKE SURE OF THIS!• program = 'Something''s wrong.';• return;• end;• cbitsif0 = binarize(cnumif0); % BINARIZE!• cbitsif1 = binarize(cnumif1);• cL = length(cbitsif0);• if (cL>L)

Can you implement?

gcc grover.c -o chloroform

Nature is a Computer

IBM Dr. Isaac Chuang Dr. Nabil AmerMIT Prof. Neil Gershenfeld Prof. Seth LloydU.C. Berkeley Prof. Alex Pines Dr. Mark KubinecStanford Prof. James Harris Prof. Yoshi Yamamoto

Bulk Spin Resonance Quantum Information Processing

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