Quantum Computations By NMR:
Status and Challenges.
Anil Kumar
Indian Institute of Science, Bangalore
Current QC Students
Ms. Jharana Rani Samal*
(* Nov., 12, 2009)
Mr. Ram K. Rao
Mr. V.S. Manu
Collaborators @ IISc:
Prof. K.V. Ramanathan
Prof. N. Suryaprakash
Prof. Apoorva Patel
And
Prof. Malcolm H. Levitt - UK
Former QC Students
Dr. Arvind
Dr. Kavita Dorai
Dr. T.S. Mahesh
Dr. Neeraj Sinha
Dr. K.V.R.M.Murali
Dr. Ranabir Das
Dr. Rangeet Bhattacharyya
- IISER Mohali
- IISER Mohali
- IISER Pune
- CBMR Lucknow
- IBM, Bangalore
- NCIF/NIH USA
- SUNY Stony Brook
Dr.Arindam Ghosh
Dr. Avik Mitra
Dr. T. Gopinath
- SUNY Buffalo
- Varian Pune
- Univ. Minnesota
DSX
300
AV
700
NMR Research Centre, IISc
DRX
500
AV
500
AMX
400
Introduction to QC
All present day computers (classical computers) use binary (0,1)
logic, and all computations follow this Yes/No answer.
Feynman (1982) suggested that it might be possible to simulate the
evolution of quantum systems efficiently, using a quantum simulator
What is Special about Quantum Systems?
Coherent Superposition
Classical
bit
Qubit
|0 
0
1
|1 
c1|0  + c2|1 
New Possibilities
Entanglement
EPR States:2(-1/2)(00+ 11) NOT
resolvable into tensor product of
individual particles (01+ 11)
(02+ 12)
Quantum
Computation
Teleportation
Quantum Algorithms
1. PRIME FACTORIZATION
Classically :
exp [2(ln c)1/3(ln ln c)2/3]
400 digit
1010years (Age of the Universe)
Shor’s algorithm : (1994)
(ln c)3
3 years
2. SEARCHING ‘UNSORTED’ DATA-BASE
Classically
:
Grover’s Search Algorithm : (1997)
N/2 operations
N operations
3. DISTINGUISH CONSTANT AND BALANCED FUNCTIONS:
Classically
:
( 2N-1 + 1) steps
Deutsch-Jozsa(DJ) Algorithm : (1992) . 1 step
4. Quantum Algorithm for Linear System of Equation:
A.W. Harrow, A. Hassidim and Seth Lloyd; PRL, 103, 9 Oct . (2009).
Exponential speed-up
Experimental Techniques for Quantum Computation:
1. Trapped Ions
Ion Trap:
2. Cavity Quantum
Electrodynamics (QED)
1kV
Linear Paul-Trap
Laser
Cooled
Ions
T ~ mK
~
16 MHz
1kV
http://heart-c704.uibk.ac.at/linear_paul_trap.html
3. Quantum Dots
4. Nuclear Magnetic Resonance
5. Josephson junction qubits
6. Fullerence based ESR quantum computer
NUCLEAR SPINS
1. Nuclear spins have small magnetic
moments (I) and behave as tiny
magnets.
2. When placed in a large magnetic
field B0 , spins are oriented either
along the field (|0 state) or opposite
to the field (|1 state) .
B1
0
3. A transverse radiofrequency field (B1) tuned at the Larmor frequency of
spins can cause transition from |0 to |1 (NOT operation).
4. Spins are coupled to other spins by indirect spin-spin (J) coupling, and
controlled (C-NOT) operations can be performed using J-coupling.
SPIN IS QUBIT
ANZMAG-2008
Why NMR?
> A major requirement of a quantum computer is that the
coherence should last long.
> Nuclear spins in liquids retain coherence ~ 100’s millisec
and their longitudinal state for several seconds.
> A system of N coupled spins (each spin 1/2) form an N
qubit Quantum Computer.
> Unitary Transform can be applied using R.F. Pulses and
J-evolution and various logical operations and quantum
algorithms can be implemented.
Achievements of NMR - QIP
 2. Quantum Logic Gates
 10. Quantum State Tomography
 11. Geometric QC
 12. Adiabatic QC
 3. Deutsch-Jozsa Algorithm
 13. Quantum State discriminator
 1. Preparation of
Pseudo-Pure States
 4. Grover’s Algorithm
 5. Hogg’s algorithm
14. Error correction
15. Teleportation
 6. Berstein-Vazirani parity algorithm
16. Quantum Simulation
 7. Quantum Games
17. Quantum Cloning
 8. Creation of EPR and GHZ states
18. Shor’s Algorithm
 9. Entanglement transfer
Maximum number of qubits achieved in our lab: 8
NMR sample has ~ 1018 spins.
Do we have 1018 qubits?
No - because, all the spins can’t be
individually addressed.
Progress so far
Spins having different Larmor frequencies can be
individually addressed
as many “qubits”
One needs resolved couplings between the spins
in order to encode information as qubits.
NMR Hamiltonian
H = HZeeman + HJ-coupling
Two Spin System (AM)
=  wi Izi +  Jij Ii  Ij
i
|bb = |11
i<j
Weak coupling Approximation
wi - wj>> Jij
H =  wi Izi +  Jij Izi Izj
i
i<j
A2= |1M
M2= |1A
|ab = |01
|ba = |10
M1= |0A
A1= |0M
|aa = |00
Spin States are eigenstates
A2 A1
Under this approximation all spins
having same Larmor Frequency
can be treated as one Qubit
wA
M2 M1
wM
An example of a three qubit system.
A molecule having three different nuclear spins having
different Larmor frequencies all coupled to each other
forming a 3-qubit system
13CHFBr2
Homo-nuclear spins having different Chemical shifts
(Larmor frequencies) also form multi-qubit systems
3 Qubits
2 Qubits
1 Qubit
CHCl3
111
11
1
0
10
01
00
011 110
010
000
001 100
101
The two methods
Coupling (J) Evolution
Examples
XOR
I1
Transition-selective
Pulses
I1z+I2z
p
y
x
y
11
I1z+I2x
(1/2J)
I1z+2I1zI2y
I2
00
x
1/4J
1/4J
01
10
I1z+2I1zI2z
11
I1
I2
NOT1
10
p
00
p
01
11
y
x
-y
I1
SWAP
y
x
-y
10
p2
I2
p1
01
p3
00
y
y
-x
-x
111
I1
010
I3
I3
001
100
000
y
I2
101
Toffoli
I2
I1
110 p
011
y
-x
x
non-selective p pulse
+ a p on 000  001
011
OR/NOR
010
111
110
001
000 p
101
100
Pure States:
Tr(ρ ) = Tr ( ρ2 ) = 1
For a diagonal density matrix, this condition requires
that all energy levels except one have zero populations.
Such a state is difficult to create in NMR
Pseudo-Pure States
Under High Temperature Approximation
ρ = 1/N ( α1 + Δρ )
Here α = 106 and U 1 U-1 = 1
We create a state in which all levels except one have
EQUAL populations. Such a state mimics a pure state.
Pseudo-Pure State
In a two-qubit system:
(i) Equilibrium:
ρ = 106 + Δρ = {2, 1, 1, 0}
(ii) Pseudo-Pure
Δρ = {4, 0, 0, 0}
How to Create ?
i) Spatial averaging
ii) Temporal averaging
iii) Logical labeling
iv) Spatially averaged- Logical labeling
0
| 11
1
| 01
1
| 10
2
| 00
0
| 11
0
| 01
0
| 10
4
| 00
Preparation of Pseudo-pure states
• Spatial Averaging
• Logical Labeling
• Temporal Averaging
Cory, Price, Havel, PNAS, 94, 1634 (1997)
N. Gershenfeld et al, Science, 275, 350 (1997)
Kavita, Arvind, Anil Kumar, PRA 61, 042306 (2000)
E. Knill et al., PRA, 57, 3348 (1998)
• Pairs of Pure States (POPS)
B.M. Fung, Phys. Rev. A 63, 022304 (2001)
• Spatially Averaged Logical Labeling Technique (SALLT)
T. S. Mahesh and Anil Kumar, PRA 64, 012307 (2001)
1 Spatial Averaging:
I1z = 1/2
10 0 0
0 1 0 0
0 0 -1 0
0 0 0 -1
Cory, Price, Havel, PNAS, 94, 1634 (1997)
I2z = 1/2
I1z + I2z + 2I1zI2z = 1/2
(2)
(p/3)X
Eq.= I1z+I2z
1
1 0
0 -1
0 0
0 0
0 0
0 0
1 0
0 -1
2I1z I2z = 1/2
3 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1
2
(1)
p
3
(p/4)Y
4
1/2J
Gx
0
0
0
1
Pseudo-pure
state
(1)
(p/4)X
10 0
0 -1 0
0 0 -1
0 0 0
5
I1z + I2z + 2I1zI2z
6
2. Logical Labeling
•N. Gershenfeld et al,
Science,
1997, 275, 350
Kavita, Arvind,
and Anil Kumar
Phys. Rev. A,
2000, 61, 042306
DRX-500
SIF
3. Temporal Averaging
0
| 11
1
| 01
1
p | 11
1
| 10
+
0
| 01
2
| 00
1
| 10
+
1
| 01
2
| 11
=
0
| 10
2
| 00
2
| 00
2
| 01
1
| 11 p
2
| 10
Pseudo-pure
state
6
| 00
E. Knill et al., PRA, 57, 3348 (1998)
4. Pseudo Pure State by SALLT:
(Spatially Averaged Logical Labeling Technique)
This method does not scale with number of qubits
Subsystem
Pseudo-pure
states of
2 qubits
T. S. Mahesh
and Anil Kumar,
PRA 64, 012307 (2001)
Relaxation of Pseudopure states
1
-1
-1
-1
-3
1
3
1
1
-3
1
-3
1
1
1
1
Cross Correlations
retard the relaxation
Open circles 00; Filled circles 11 PPS
of some PPS
Arindam Ghosh and Anil Kumar, J. Magn. Reson., 173, 125 (2005).
Logic Gates
By 1D NMR
Logic Gates
XOR2
(Exclusive OR or C-NOT)
e1 , e2 
UXOR2
NOT1
e1 , e1  e2 
e1 , e2 
e2
e1
e2
e1
e2
e1
e2
e1
INPUT OUTPUT
0 0
0 0
0 1
0 1
1 0
1 1
1 1
p
1 0
1
01
0
11
2
00
1
10
0 0
0 1
e1 , e2 
N
INPUT
p
UNOT1
OUTPUT
p
1 0
1 1
1 0
p 0 0
1 1
0 1
0
p 11
1
01
2 p
00
1
10
Logic Gates
Using 1D NMR
NOT(I1) 
XOR2 
XOR1 
Kavita Dorai, PhD Thesis, IISc, 2000.
Logical SWAP
INPUT
OUTPUT
0 0
0 0
e1
0 1
1 0
e2
1 0
e1 , e2 
e2 , e1
1 1
p
0 1
1 1
0
| 11
p
1
| 01
p
2
XOR+SWAP
1
p1
2
| 00
| 10
p3
Kavita, Arvind, and Anil Kumar
Phys. Rev. A, 2000, 61, 042306
Toffoli Gate = C2-NOT
e1 , e2 , e3 
e1 , e2 , e3  (e1^e2)
e1
e2
e3
Input Output
AND
NAND
000
000
001
001
010 010
011
111
100
100
101 p 101
110
110
111
011
Eqlbm.
Toffoli
Kavita Dorai, PhD Thesis, IISc, 2000.
e1
e2
e3
Logic Gates
By 2D NMR
2D NMR Quantum Computing Scheme
Preparation Evolution
y
Detection
Mixing
y
-y
I0
t1
I1, I2 etc.
t2
Computation
Gz
Creation of
Initial States
Labeling
of the
initial
states
Computation
Ernst & co-workers, J. Chem. Phys., 109, 10603 (1998).
Reading
output
states
Two-Dimensional Gates
A complete
set of 24
Reversible,
One-to-one,
2-qubit
Gates
T. S. Mahesh,
Kavita Dorai,
Arvind and
Anil Kumar,
JMR,
148, 95, (2001)
NOP
Three-qubit 2D-Gates:
NOT1
111
110
101
100
I1
I3
I2
INPUT
I0
w1
011
010
001
000
TOFFOLI
OR/NOR
111
110
101
100
011
010
001
000
OUTPUT
111
110
101
100
w2
011
010
001
000
111
110
101
100
011
010
001
000
T. S. Mahesh,et al,
JMR,148, 95, 2001
2D Gates by Hadamard Spectroscopy
I
F (Io)
C=C
F (I1)
F (I2)
Total time: Less than 2 min.
2D Method takes 2 Hours
T. Gopinath and Anil Kumar, J. Magn. Reson., 183, 259 (2006).
Quantum Algorithms
(a) DJ
(b) Grover’s Search
DJ algorithm on ONE qubit with one work bit:
Constant
f1 (x)
f2 (x)
0
1
0
1
x
0
1
Balanced
f3 (x)
f4 (x)
0
1
1
0
Uf
|x  |y  |x |y  f(x)
Cleve Version
|x is input qubit and |y is work qubit
I/P
O/P
Pulses
Operator
x y f1 (x)
0 0
0
0 1
0
1 0
0
1 1
0
1
0
Uf
0
0
Constant
y f1 (x) f2 (x)
0
1
1
1
0
1
1
1
0 0 0
0
1 0 0
1
0 1 0
0
0
0 0 1
No px pulse
y f2 (x) f3 (x)
1
0
0
0
1
1
0
1
1 0 0
1
0 0 0
0
0 0 1
0
0 1 0
0
px pulse on
work qubit
Balanced
y f3 (x) f4 (x)
0
1
1
1
1
0
0
0
0 0 0
0
1 0 0
1
0 0 1
0
0 1 0
0
y f4 (x)
1
0
0
1
1 0 0
0 0 0
0 1 0
0 0 1
px pulse on one of the
transitions of the work qubit
Deutsch-Jozsa Algorithm
Experiment
One qubit DJ
Eq
Kavita, Arvind, Anil Kumar,
Phys. Rev. A 61, 042306 (2000)
Two qubit DJ
Steps:
Grover’s Algorithm
|00..0
Pure State
Superposition
Selective
Phase Inversion
Avg
Grover
iteration
Inversion about
Average
N times
Measure
Grover search algorithm using 2D NMR
Ranabir Das and Anil Kumar, J. Chem. Phys. 121, 7603(2004)
Typical systems used for NMR-QIP using J-Couplings
Chuang et al,
quant-ph/0007017
2 qubits
7 qubits
6
3 qubits
7
5 qubits
Marx et al,
Phys. Rev. A
62, 012310 (2000)
4 qubits
Knill et al, Nature,
404, 368 (2000)
7 qubits
How to increase the
number of qubits?
Use Molecules Partially Oriented (~ 10-3)
in Liquid Crystal Matrices
(i) Quadrupolar Nuclei (spin >1/2). Reduced
Quadrupolar Couplings
(ii) Spin =1/2 Nuclei. Reduced Intramolecular
Dipolar Couplings
Quadrupolar
Systems
Using spin-3/2 (7Li) oriented
system as 2-qubit system
Neeraj Sinha,T. S. Mahesh, K. V. Ramanathan,
and Anil Kumar, JCP, 114, 4415 (2001).
Pseudo-pure states
2-qubit Gates
using 7Li
oriented system
Neeraj Sinha,
T. S. Mahesh,
K. V. Ramanathan,
and Anil Kumar,
JCP, 114, 4415 (2001).
133Cs
system – spin 7/2 system
Equilibrium
-7/2
-5/2
-3/2
-1/2
1/2
3/2
5/2
7/2
[Cs pentadeca-fluorooctonate + D2O]
Half-Adder
Conventional Labeling
111
7
Half-Adder
110
6
p(5,6,5,7,6)
101
5-pulses
5
100
4
011
Subtractor
3
010 p(3,2,3,5,4,3,2,5,7)
2
9-pulses
001
1
000
Subtractor
Optimal Labeling
-7/2
-5/2
-3/2
-1/2
1/2
3/2
5/2
7/2
7
6
5
4
3
2
1
000
010
011
001
101
110
111
100
Half-Adder
p(1,3,2)
3-pulses
Subtractor
p(6,4,2)
3-pulses
Murali et.al.
Phys. Rev. A.
022313 (2003)
R. Das et al, PRA
012314 (2004)
Collins version of 3-qubit DJ implemented on the 7/2 spin of Cs-133.
C
B
B
B
B
B
B
B
B
B
B
There are 2 constant
and 70 Balanced
functions. Half differ
in phase of the
Unitary transform.
12 are shown here.
1-Constant and 11Balanced
B
Gopinath and Anil Kumar,
JMR, 193, 163 (2008)
Dipolar
Systems
Advantages of Oriented Molecules
• Large Dipolar coupling - ease of selectivity - smaller Gate time
• Long-range coupling - more qubits
Disadvantage
For Homo-nuclear spin system
Weak coupling Approximation
wi - wj>> Dij
Spins become Strongly coupled
A spin can not be identified as a qubit
Solution
2N energy levels are collectively treated as an N-qubit system
3-Qubit Strongly Dipolar Coupled Spin System
Bromo-di-chloro-benzene
C2-NOT gate
(|110  |111)
POPS
(|000 000| |001 001| )
GHZ state (|000+ |111)
populations
Z-COSY (90-t1-10-τm-10-t2) was
used to label the various transitions
coherences
T.S. Mahesh et.al., Current Science 85, 932 (2003).
5-qubit system
HET-Z-COSY spectrum for labeling
(in liquid crystal)
Eqlbm
E-level diagram
Proton transitions
Fluorine transitions
|0
|0
|0
|0
Starting from 4-qubit PPS
prepared by SAALT:
Entanglement between 2nd and 3rd qubit
|0000+ |0110
H

Transfer
Entanglement transfer
Entanglement between 1st and 4th qubit
|0000+ |1001
Ranabir Das , Rangeet Bhattacharya and Anil Kumar, JMR. 170, 310-321 (2004).
w2+ w40
w2+ w27
(p)27(p)40
(p)27
8-qubit system
H
Molecule: 1-floro naphthalene
(in liquid crystal ZLI1132)
Equilibrium spectrum
1H
spectrum
(1-512)
19F
spectrum
(513-630)
F
H
H
H
H
H
H
HET-Z-COSY spectrum
R. Das, R
Bhattacharyya
and Anil
Kumar,
Quantum
Computing –
Back Action,
AIP Proc.,
864, 313 (2006)
Energy-level diagram
(256 levels)
19F
transitions
Proton transitions
in A domain
Proton transitions
in B domain
C7-NOT [p1]
POPS(1) [Eq- p1]
POPS(40) [Eq- p40]
POPS(1)+C6SWAPPOPS(40)
R. Das, R Bhattacharyya and Anil Kumar, Quantum Computing – Back
Action, AIP Proceedings 864, 313 (2006)
Geometric Quantum Computing
• Geometrical phase is robust, since it depends only on the
solid-angle enclosed by the path and is independent of
the details.
• This fact can be used to perform fault-tolerant
Quantum Information Processing
What is Geometric Phase ?
• When a vector is parallel transported on a curved surface, it acquires a
phase. A part of the acquired phase depends on the geometry of the path.
• A state in a two-level quantum system is a vector which can be
transported on a Bloch sphere. The geometrical part of the acquired
phase depends on the solid angle subtended by the path at the centre of
the Bloch sphere and not on the details of the path.
Adiabatic Geometric Phase
M. V. Berry, Quantum phase factors accompanying adiabatic changes,
Proc. R. Soc. Lond. A, 392, 45 (1984)
When a quantum system, prepared in one of the eigenstates of
the Hamiltonian, is subjected to a change adiabatically, the
system remains in the eigenstate of the instantaneous
Hamiltonian and acquires a phase at the end of the cycle.
The phase has two parts. The dynamical and the geometric part.
D = exp( -iEt / )
g = exp( i )
 = D +  g
(Dynamical Phase)
(Geometric Phase)
(Total Phase)
Pancharatnam, S.
Proc. Ind. Acad. Sci. A 44 , pp. 247-262 (1956).
Non-Adiabatic Geometric Phase
Aharovov and Anandan later showed that adiabaticity is not a
necessary condition. When the density matrix corresponding to
a quantum system completes a cyclic path, in the density
operator space through different intermediate stages, the system
acquires the same geometric phase depending upon the
geometry of the path.
    /  /   //  //  ....   
  ei 
Y. Aharonov and J. Anandan, Phase change during Cyclic Quantum Evolution,
Phys. Rev. Lett. 58(16), 1593 (1987)
Non-adiabatic Geometric phases in
NMR QIP using transition selective pulses
1. Geometric phases using slice & triangular circuits.
- Use of above in DJ & Grover algorithms.
Ranabir Das et al, J. Magn. Reson., 177, 318 (2005).
2. Experimental measurement of mixed state geometric phase by
quantum interferometry using NMR.
Ghosh et al, Physics Letters A 349, 27 (2005).
3. Geometric phases in strongly dipolar coupled spins.
- Fictitious spin- ½ subspaces.
- Geometric phase gates in dipolar coupled 13CH3CN.
- Collins version of 2-qubit DJ algorithm.
- Qubit-Qutrit parity algorithm.
Gopinath et al, Phys. Rev A 73, 022326 (2006).
Geometric phase acquired by a slice circuit
The state vector of the two level sub space cuts a slice on the Bloch
sphere.
z
The slice circuit can be achieved
by two transition selective pulses
10
A.B = (p)q
11
.
10
(p)
11
q+p+
The resulting path encloses a solid
angle W = 2 .
(p)x
(p)-x
A spin echo sequence t-p-t is applied
to refocus the evolution under internal
Hamiltonian (the dynamic phase).
Second (p) pulse is applied to restore the
state of the first qubit altered by the (p) pulse.
Unitary operator associated with the slice circuit:
A.B =
(0
(
)
A. B
+ 1 ) 0 
0 + ei 1 0
Solid angle W = 2.
13C
Spectra of 13CHCl3
Ranabir Das et al, J. Magn. Reson., 177, 318 (2005).
Geometric phase acquired by a triangular circuit
The state vector of the two level subspace traverses a triangular path
on the Bloch sphere.
The solid angle subtended by the
Triangular circuit is, W = .
Pulse sequence:
(p)x
(p)-x


-i 2
( 0 + 1 ) 0  0 + e 1  0


A.C . B
Ranabir Das et al, J. Magn. Reson., 177, 318 (2005).
Deutsch-Jozsa algorithm using geometric phases (by slice circuit):
1H
13C
of
13CHCl
3
Constant
Uf(00) is identity matrix
Constant
Uf(11) is achieved by applying
p pulse on the second qubit
Balanced
Balanced
=
Ranabir Das et al, J. Magn. Reson.,
177, 318 (2005).
Grover search algorithm using geometric phases (by slice circuit):
Pseudo |0
pure
state |0
H
H
Superpo
sition
H
Cij
H
Selective
Inversion
Selective Inversion:
C00
H
| i
H
| j
Inversion about
Average
Using Geometric phase gates
x = 00
x = 01
=p
x = 10
Inversion about Average C00 = H U00 H
U00 = C00(p)
x = 11
Ranabir Das et al, J. Magn. Reson., 177, 318 (2005).
Measure
Experimental Measurement of Mixed State Geometric Phase by NMR
Creation of |00
PPS
Preparation of Mixed state by the a
degree pulse and a gradient.
Implementation of Slice circuit to
introduce Geometric Phase.
Measurement
Results:

W 
Geometric Phase (Shift) = - tan -1  r tan   

 2 
The shift of the interference pattern as a function of W and r.
The shift directly gives the geometric phase acquired by the spin qubit.
Ghosh et al, Physics Letters A 349, 27 (2005).
Quantum Games
Game Theory

Von Neumann and Morgenstern
“Theory of games and Economic behaviour”
- J. von Neumann and O. Morgenstern.
(Princeton University Press, 1947 2ed)
 John Forbes Nash (Princeton University) Robert Aumann (Hebrew University)
 Nobel Prize in Economics ‘1994’
 Nobel Prize in Economics ‘2005’
Evolutionary Games
Quantum Games by NMR
(i) Ulam’s Quantum Game – Guessing a number
(i) Three player Dilamma Game- Optimum strategy for
going to a Bar with two stools
(i) Battle of Sexes Game – Should Bob and Alice together
watch TV or go to a football game
Avik Mitra
Ulam’sGame
 Two player game
 Bob thinks of a number between 1 and 10n
Alice tries to find out the number in minimum number of queries.
Classical: n queries
Quantum: 1 Query
Ulam’sGame
 Classical Algorithm
0011010
Alice
Total number of queries = log2
2n
=n
Bob
Quantum Version of Ulam’sGame
The quantum system consists of two registers.
Register A
• consists of n qubits
Register B
• consists of 1 qubit
Alice makes a superposition of all the qubits.
Bob performs a unitary transform and stores the result in register B.
Alice again interferes the qubits and then performs a measurement.
Total number of queries = 1
--S. Mancini and L. Maccone, quant-ph/0508156
Protocol
Creation
of
PPS
Application
of
Hadamard
gate
Unitary
Transform
of
BOB
Application
of
Hadamard
gate
M
E
A
S
U
R
E
Experimental implementation
 Preparation of Pseudo-pure state(PPS).
111
011
001
000
110
101
010
100
Results
111
011
001
110
101
010
population
100
000
coherence
Avik – Thesis –I.I.Sc. 2007
Adiabatic Quantum
Algorithms
Adiabatic Quantum Computing
 Based on “Adiabatic Theorem” of Quantum Mechanics: A quantum system in
its ground state will remain in its ground state provided that the Hamiltonian
H under which it is evolved is varied slowly enough.
U
The system evolves
from Hi to Hf with a
probability (1-e2)
provided the evolution
rate satisfies the
condition
dH (s)
max 1; s
0; s
0 s 1
dt
e
2
g min
E
(s)
i
f
Where e << 1
s
Farhi et al, PRA, 65,012322 (2002), Science, 292,
472 (2001), quant-ph/0001106; 0007071;
0208135
Adiabatic Quantum Computing
 Evolve the initial state under a slowly varying Hamiltonian so that it acts as
though a unitary transformation occurred on the initial state, bringing it to a
final state during some time T.
H(s) = (1-s)HB + sHF
HB  beginning Hamiltonian
HF  Final Hamiltonian
 Initialize register to desired input qubits.
 Vary the Hamiltonian towards the final Hamiltonian whose eigenstates
encodes the desired final states.
N
 f = U n  i , where U n = exp( -iH nt )
n =0
Adiabatic Algorithms
by NMR
(i) NMR Implementation of Locally Adiabatic
Algorithms
(a) Grover’s search Algorithm
(b) D-J Algorithm
(ii) Adiabatic Satisfibility problem using Strongly
Modulated Pulses
Avik Mitra
Adiabatic SAT
Algorithm by Strongly
Modulated Pulses
In a Homonuclear spin systems
spins are close
(~ kHz) in frequency
space
Pulses are of
longer duration
Decoherence
effects cannot
be ignored
Strongly Modulated Pulses circumvents the above problems
Strongly Modulated Pulses.
• Numerically optimized pulses.
• system Hamiltonian is taken into consideration while designing the
pulses.
• This leads to precise unitary transformation.
n
System Hamiltonian of weakly
coupled spin system
n
H int =  w I + 2p  J jk I zj I kz
j= 1
j
j z
n
H ext (wrf , ,w , t ) =  e
k =1
j k ,
k=2
- i (w rf t + )
 (- wI )  e
k
x
i (w rf t + )
Hamiltonian representing
the radio frequency pulse
H tot = Hint + Hext
In the rotating frame: H eff = Hint +  - w I kx cos( ) + I ky sin( )
n
k =1
 (t ) = e - iH t   (0)  eiH
eff
eff t
Fortunato et al, JCP 116 (2002) 7599, Mahesh et al, PRA 74 (2006) 062312
Strongly Modulated Pulses.
• Back transformation to the original frame is done.
 (t ) = U -z 1  e - iH t   (0)  eiH t  U z
eff
eff
Unitary operator for rotating
frame transformation
U z (t ) = e
USMP =   l ( l )  U (t l )e
-1
z
(
n


 - iw rf
I kzt 


k =1



)
- iHeff w lwrfl  l t l
l
wrf
F=
2


Tr U T  U SMP
N
Nedler-Mead
Simplex Algorithm

t

(fminsearch)
Avik Mitra et al, JCP, 128, 124110 (2008
k-SAT Problem
1. Let B={x1,x2,….,xn} be a set of ‘n’ Boolean variables.
2. Let Ci be a disjunction of ‘k’ elements of B
Ci = x1  x2  .......  xk
1. F is the Boolean function that is the conjunction of m such
clauses.
F = C1  C2  ......  Cm
Find out all the assignments of Boolean variable in F that simultaneously
satisfies all the clauses i.e. F=1.
For k.≥ 3 The problem is hard to solve.
 Three variable 1-SAT problem
• B = {x1,x2,x3} , set of three variable. Classically – polynomial time
Hogg’s Quantum Algorithm
• Each clause (Ci) has one variable. – Single step. ( NMR
implementation by Peng et
• e.g. F1= x1x2x3. True for 111,
al., PRA, 65, 042373 (2002)
•
Zero otherwise
Adiabatic Version suggested by Farhi et al, quant-ph/0001106
NMR Implementation, using a 3-qubit system.
 The Sample.
Iodotrifluoroethylene(C2F3I)
Fb
Fc
C
C
Fa
 Equilibrium Specrum.
I
J ab = 68.1 Hz
J ac = 48.9 Hz
J bc = -128.8 Hz
Step 1. Preparation of PPS
1 p 
p 
 4  - 2 J -  4  -G z
 x
ik   y
k
k
1 k
I + I      I + (I z + 2I iz I kz )
2
i
z
k
z
i
z
1
2
3
Equilibrium:I z + I z + I z
1 1
2
3
1 2
1 3
2 3
1 2 3
PPS: (I z + I z + I z + 2I z I z + 2I z I z + 2I z I z + 4I z I z I z )
4
Step 2. Creation of Equal Superposition.
p 
• can be achieved by pseudo-Hadamard on all three spins
 
 2 y
• single high power p/2 pulse has offset effect.
1, 2 , 3
• The high power pulse is replaced by an SMP.
soft pulses on
~400 s
all three spins
SMP = 100.5s
Avik Mitra et al,
JCP, 128, 124110
Step 3. Implementation of Adiabatic Evolution
H B = I1x + I 2x + I 3x = I x
H F = I1z + I 2z + I 3z = I z
m
m

H(m ) =  1 -  H B +
HF
M
M

Um  e
 m  180
- iI x  1 - 
 M  2p
a m 
1, 2 , 3
 2 
 x
pulse
e
 m  180
- iI z  
M p
b m 1z,2,3 pulse
mth step of the interpolating
Hamiltonian .
e
 m  180
- iI x  1 - 
 M  2p
a m 
 2 
 x
mth step of
evolution operator
1, 2 , 3
pulse
•Pulse sequence for adiabatic evolution
• Total number of iteration is 31
• time needed = 62 ms
(400s x 5 pulses x 31 repetitions)
Using Concatenated SMPs
Duration: Max 5.8 ms, Min. 4.7 ms
Avik Mitra et al, JCP, 128, 124110 (2008)
Results for all Boolean Formulae
Avik Mitra et al,
JCP, 128, 124110
Conclusions:
NMR QIP is at a Cross -Road
Many Operations
and Algorithms
can be
Implemented
A Very Good
Tool for
Learning and
Explorations
Scaling is Difficult
If
All Spins are to be coupled to
each other with nondegenerate Transitions
Possible to obtain high
number of qubits
If
Only nearest neighbor
couplings are sufficient
Backbone of a C-13, N-15 labeled protein
with side chain protons deuterated
How to do QIP?
Future Directions
1. To increase the number of Qubits in NMR.
Synthesis molecules with several hetero-nuclei.
2. To use strongly modulated pulses (SMPs) and Control Theory
for performing
Unitary operations in short times and accurately.
3. To search for systems with large coherence times.
For example singlet states in equivalent spins or spins
systems with symmetry with symmetry preserving relaxation.
4. Develop protocols which can be carried out using spins
with nearest neighbor couplings.
Thank You