Bulk Spin Resonance Quantum Information Processing
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 0 1
q q 11
0
a [ , ]0 1 0 1
Bits
• 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
01
10
A A or
o BA
AB
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
GFLOP
– Grover’s algorithm (8 TB):• O( ) vs. O(N)• 27 min. vs. 1 month @ same clock
speed.
The Promise
N
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
c nnn
*A A c c Amn n m
nmnm
Tr
pkk
k k
H
( ) ( ) ( ) / / t U t U e t eiHt iHt
/
e
Z
H kT
Quantum Mechanics
HA
HB Br
Br
S
• ~1023 spin degrees of freedom– rapid tumbling averages inter-molecular interactions
• ~N effective degrees of freedom– decoherence averages off-diagonal coherences
p k kk
k
210
1 2 10
23
23
( )
/
/
/
/
e
Z I
e
e
e
H kT
N
E kT
E kT
E kT
N
I N
1
2 1
0
0
1
2
2 1
1 2
N spins I (1/2)
B0 B1Bulk Density Matrix
• high temperature approximation
• identity can be ignored
• ensemble molecule deviation
NMR: “reduced” density matrix
E
kT
e H kT
N N
102
1
26
/
U U U U U UN N
1
2
1
2
Deviation Density Matrix in NMR
• magnetic moment
• angular momentum
• spin precession
• Zeeman splitting
• 2 spin interaction Hamiltonian
H B
J J I
I E B 1
H I I I IA zA B zB AB zA zB chemical shifts~ 100 MHz
scalar coupling~ 100 Hz
d
dtB
A-B
Spin Hamiltonian
• apply a z field:
• evolve in field:
• two spins, scalar coupling:
• evolution = 3 commuting operators
H B B Iz z z
e e e
i I
R
iHt i B tI i I
z
z
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
2 21
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
i
i
i
i
i
i
i
i
i
( ) ( ) ( ) ( ) ( )
/
90 90 270 90 90 90
1
25 2
1 1 0 0
1 1 0 0
0 0 1 1
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
1 1 0 0
0 0 1 1
0 0 1 1
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
i
i
i
Ry-1PRy
CAB
ARyA(-90)
B
RyA(90)
ABt=
/ 2 Bt=
/ 2 At=3
/ 2
RxA(180)
RxB(180)
RxA(180)
RxB(180)
B
A
B
A
• 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
xx0
A
x
A
x
AM M M HPHij iiN N
2 21,
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
NN BN
BNN
M
2/cosh
2/sinh
2
ˆˆTr
0
0
max
222
4
sx
NM
Scaling
• Is it quantum? Schack, Caves, Braunstein, Linden, Popescu, …
• Initial conditions vs quantumevolution
• But, Boltzmann limit is not scalable
catcatN 2
1̂2
1ˆ
22221
1
NN
is separable if
3.8x10-610
1.5x10-59
6.0x10-58
2.4x10-47
9.1x10-46
3.4x10-35
1.2x10-24
0.043
0.112
0.251
N
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
zBzA
zBBzAA
IIJ
IIH
ˆˆ
ˆˆˆ
zBzA
zBBzAA
IIDJ
IIH
ˆˆ2
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
-d6
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
n
Tk
BOm
B
o2
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, [email protected]• % 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
