Experimental Implementations of Quantum
Information Processing
by
Yaakov Shmuel Weinstein
B.A., Physics and Mathematics
Yeshiva University, (1997)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
OFTECHNOLOGY
at the
SEP 2 0 2000
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LIBRARIES
February 2000
© Massachusetts Institute of Technology 2000. All rights reserved.
A uthor ................
---
.........
Department of Mechanical Engineering
January 14, 2000
/
Certified by........
Seth Lloyd
Associate Professor of Mechanical Engineering
Thesis Supervisor
Accepted by ...............
..........
Ain Sonin
Chairman, Department Committee on Graduate Students
Experimental Implementations of Quantum Information
Processing
by
Yaakov Shmuel Weinstein
Submitted to the Department of Mechanical Engineering
on January 14, 2000, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
Quantum information processing (QIP) is the study of information and information
processing as governed by the laws of quantum mechanics. A theory of quantum
information allows for the development of quantum computers, which could be used
to greatly increase the speed of certain mathematical computations; to construct
quantum simulators, capable of simulating other quantum systems; and to perform
quantum cryptography, whose security is guaranteed by the laws of physics. The
power offered by QIP stems from unique properties of quantum mechanics such as
entanglement and interference. To understand the fundamentals of QIP we will take
a short tour of several aspects of classical information processing highlighting where
QIP differs. Then, nuclear magnetic resonance is introduced as an experimentally
viable way of studying QIP. Finally, a number of experiments of quantum information
processing will be thoroughly explored.
Thesis Supervisor: Seth Lloyd
Title: Associate Professor of Mechanical Engineering
2
Acknowledgments
First and foremost, I would like to thank my advisor Dr. Seth Lloyd. Dr. Lloyd's
brilliance and broad range of knowledge is well known to his colleagues and students.
He is a lecturer par excellance.
As an advisor, Dr. Lloyd challenged me to ask
questions and voice my opinions. Working with Dr. Lloyd has given me a chance to
chart and explore a wide range of fields, from control to chaos, classical to quantum.
Dr. Lloyd is always willing to hear whatever questions may be bothering a young
scientist that particular day with a gentle chiding or a broad smile.
I have had the good fortune that Dr. David Cory has taken me into his group in the
Spatial NMR Lab. Much more than having the opportunity to use his spectrometers,
Dr. Cory has taken the time to allay any fears I faced in implementing experiments.
Dr. Cory constantly keeps his students in mind, and cares for them both in academic
and non-academic areas.
When I first came to MIT, I stood in awe of the knowledge and brilliance of
the two men I have just mentioned. But only now, having had the chance to work
with them and know them, do I really see what brilliant scientists (engineers) and
wonderful people they are. It has been an honor to work with them.
My colleagues in the quantum computer group deserve special mention.
Dr.
Richard J. Nelson took me under his wing when I came to MIT. The postdocs and
researchers have been happy to share their knowledge and experience in NMR and
many areas of study, Drs. Mark Price, Ching-Hua Tseng, Shyamal Somaroo, Yehuda
Sharf, and Tim Havel. The other graduate students in the group, Greg Boutis, Evan
Fortuanto, Marco Pravia, Grum Teklemariam, Hugo Touchette - we have had many
discussions, arguments. I think we have learned a lot from each other, and hope that
we continue to do so.
My family has supported me through thick and thin during my first years at MIT.
My siblings Daniella, Aliza, and Zev Weinstein were always there to listen to my
complaints and just to talk. My parents, Dr. and Mrs. Zelig and Evelyn Weinstein,
have encouraged me and were always there to be proud of my accomplishments even
3
if they may not have known what I was doing. To even begin to list all that they
have done would take more memory than this computer has, so I will just say thank
you.
Finally, I would like to acknowledge my dear wife, Leora Aimee. There are no
words to express the thanks I owe her. To her I would like to dedicate this work.
4
Contents
1 Introduction
1.1
Bits and Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Classical and Quantum Correlations . . . . . . . . . . . . . . . . . . .
10
1.3
Classical and Quantum Information Processing
. . . . . . . . . . . .
12
1.4
N o Cloning
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.5
Power of Quantum Information Processing . . . . . . . . . . . . . . .
15
1.6
Errors and Decoherence
15
1.7
Nuclear Magnetic Resonance as a Paradigm for Quantum Information
1.8
2
9
. . . . . . . . . . . . . . . . . . . . . . . . .
9
Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Experim ents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Quantum Coherent Feedback Control
22
2.1
Feedback Control and Quantum Feedback Control . . .
22
2.2
Implementation . . . . . . . . . . . . . . . . . . . . . .
23
2.3
R esults . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3 The Quantum Fourier Transform
28
3.1
What is the Quantum Fourier Transform and Why is it Important?
28
3.2
Construction of the QFT . . . . . . . . . . . . . . . . .
30
3.3
Implementation of the QFT . . . . . . . . . . . . . . .
30
3.4
R esults . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
4 Quantum Tomography
37
5
5
4.1
Correlation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.2
Tom ography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Quantum Chaos and the Quantum Baker's Map
44
5.1
Classical Chaos and Quantum Chaos . . . . . . . . . . . . . . . . . .
44
5.2
The Quantum Baker's Map
. . . . . . . . . . . . . . . . . . . . . . .
45
5.3
The Baker's Map on a Quantum Computer . . . . . . . . . . . . . . .
45
5.4
Other Quantum Chaotic Maps . . . . . . . . . . . . . . . . . . . . . .
46
6 Conclusions and Future Work
47
6.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
6.2
Future W ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
6
List of Figures
3-1
The NMR spectra of the three carbon-13 atoms of alanine after performance of the QFT. The top spectra are theoretical while the bottom
are experimental. Peaks in NMR spectra are labeled by their resonant frequency. The observables are the phase and amplitudes of the
resonances. Each spin is split into four peaks since its energy level is
dependent on whether the other two spins are up (along the magnet) or
down. The space between the peaks is the strength of the J-coupling.
For example, the two peaks of spin 1 are separated by 54Hz which is
the strength of the J12 coupling. The four peaks show up clearly in
spin 2 which has relatively large J-couples to both the 1 and 3 spins. In
spin 2 the distance between the first and second peaks (and the third
and fourth peaks) is 35Hz, the J 23 coupling strength, and the distance
between the first and third peaks (and second and fourth peaks) is
54Hz, J 12 . The J-coupling between the 1 and 3 spins is very small,
J13
= 1.2Hz. Therefore, the four peaks of the 1 and 3 spins are not
fully resolved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
33
3-2
Theoretical and experimental results of the final deviation density matrix after implementation of the QFT on a thermal state. The left
column shows (from top to bottom) the theoretical, experimental and
difference of the real components of the three spin density matrix. The
right column shows the same for the imaginary terms. The diagonal
of the deviation density matrix can be seen running horizontally from
the left corner to the right corner, the magnitude of all terms on the
diagonal are zero. The states are labeled from 1000) at the left and
count up to 1111) at the back and front corners. . . . . . . . . . . . .
4-1
The theoretical density matrix
Ptheory
theoretically perfect pulse sequence,
36
is calculated by applying the
Utheory,
to the initial density ma-
trix, Pinitial. The experimental density matrix, Pexp, is the result of the
experiment, the initial density matrix after application of the actual
pulse sequence, Uexperimental. . . . . . . . . . . . . . . . . . . . . . . .
5-1
38
The action of the baker's map on phase space. The unit square, which
we have divided in two just to show the action of the map, is first
stretched in one direction and squeezed in the other, keeping the total
area constant. Then it is cut vertically down the middle and one side
is placed on top of the other, similar to the way a baker kneads dough.
Two initial conditions arbitrarily close together that are split by the
cut on the map, will end up very far from each other. Continual applications of the map will cause the whole of phase space to act in a
chaotic m anner.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
45
Chapter 1
Introduction
1.1
Bits and Qubits
Information theory forms the mathematical basis for analyzing how physical systems
get, represent, and process information. The mathematics of information theory was
initially worked out by Claude Shannon in the 1940's. One of the fundamental questions addressed by Shannon was how to measure information. Shannon postulated
that information is a function of the number of possible states of a system. He further showed that logarithms define the unique measure of information subject to
reasonable requirements.
The fundamental unit of information, the bit, represents a distinction between two
possibilities: 0
/
1, yes
/
no, true
/
false, heads
/
tails, capacitor charged
/
capacitor
uncharged. Having thus defined a bit, the relation between number of possibilities
and amount of information is I = log2(number of possibilities). When a fair coin is
tossed it generates log2 2 = 1 bit of information. When a balanced six-sided die is
rolled, it generates log 26 bits of information. In general, a system with N equally
likely states can register log 2 N bits of information.
Information can also be defined in contexts where the different possible states are
not equally likely. The information of a random variable X with possible values x
9
each with probability p(x) is defined as
I(X) =
p(x) log 2 p(x)-
-
(1.1)
X
Note that the word 'bit' can be used in two distinct ways, 1) to measure the
amount of information that can be registered by a two state system, and 2) to refer
to a physical system capable of registering one bit of information.
Parallel to classical information theory, a quantum bit, or "qubit" [1], is defined as
the amount of quantum information that can be held by a quantum two state system,
a system with a two dimensional Hilbert space. For example, a quantum mechanical
particle of spin . has two possible states, spin up (m, = +.),
(m =
-j), I 4).
I t),
and spin down
It can therefore register one qubit of information. Unlike a classical
bit, which is in one of two distinct states, a qubit can be in a superposition of states
with varying amplitudes and relative phases. A spin j particle is not limited to the
states I t),
I4),
it may be in a superposition of the two states, al t) +31
and 3 are complex amplitudes such that 1a12 + 11312
4), where a
1.
Like 'bit,' 'qubit' can be used in two distinct ways, 1) to measure the amount of
quantum information that can be registered by a quantum two state system, and 2) to
refer to a physical (quantum) system capable of registering one qubit of information.
1.2
Classical and Quantum Correlations
Conditional information I(XIY) is a measure of how much information remains in a
random variable X given the value of a random variable Y. For example, let X be the
random variable representing the value of a fair die toss, p(1) = p(2) =
The information in X is I(X)
=
1og 2 6
... =
p(6) =
.
bits. Let Y be the random variable connoting
whether the value of that toss was an even or odd number p(even) = p(odd) =
.
The information in Y is 1og2 2 = 1 bit. Learning the value of Y tells us something
about the value of X. If it is known that Y = even, X has only three remaining
possibilities, p(2) = p(4) = p(6) = 1. Hence, the amount of information remaining in
10
the random variable X is log23 = I(XY).
Mutual information, defined as
I(X : Y) = I(X) - I(XIY) = I(Y) - I(YIX) = I(X) + I(Y) - I(X, Y),
(1.2)
is a measure of how much information is shared between X and Y. In the example
above, the mutual information between X, the value of the die toss, and Y whether
the die toss yielded an even or odd number, is I(X : Y) = I(X) - I(XIY) =
10926
-
10923 = 1 bit. All the information in Y is shared with X because learning the
value of X also divulges the value of Y. When there is mutual information between
two systems we say the systems are correlated. The maximal mutual information
between two classical bits is one bit.
When two quantum systems interact quantum mechanically, they become correlated. Quantum systems can exhibit forms of correlation that classical systems do not.
In particular, quantum systems can apparently exhibit greater mutual information
than possible for classical systems. Consider the quantum state IV)AB
)B
4
+
-)A
0
l)B.
It can be shown that I(A) = I(B)
=
=
t)A
AI0
T
1, while, I(A, B) = 0. Ac-
cordingly, the mutual information defined in (1.2) I(A : B) = I(Z)+I(B)-I(A, B) =
2. The quantum mutual information so defined is greater than the maximum possible classical mutual information (even stranger, the quantum conditional information
I(AIB) = I(A, B) - I(B) = -1.
A review of features of quantum information can be
found in Cerf and Adami [2]). The excess of quantum correlation leads to a number
of odd quantum effects. One such effect is entanglement. A quantum state for two
systems A and B is entangled if it cannot be written in the form IW)A 0 IX)B. The
state |4 ')AB described above is entangled, while the state I t)A
t)B
is not.
When systems A and B are in the entangled state 14)AB, measurement of A in
the basis
I t), j 4) will
yield the result I t) with a probability of
probability of i. If A is found to be
I t)A
0
1
I t)
4
)
A
0
1 I)B.
and I4) with a
the two spin system must be in the state
t)B- Likewise, if A is found to be
the state I
}
I 4)
the two spin system must be in
Subsequent measurement of B is now guaranteed with a
11
probability of 1 to be I t) if A was I t), and I 4) if A was 4). That is, the results of
the measurements on A and B are completely correlated.
Entanglement can lead to seemingly paradoxical results. Suppose that spins A
and B are in the entangled state IV)AB described above. B is sent on a spaceship to
Alpha Centauri with Bob, while A stays on Earth with Alice. Alice's measurement
of A in the I t), 4) basis forces the system into the state
I 4-)A
0
4
Bob's measurement of B in the
)B.
I t),I 4)
I t)A
0
1 t)B
or the state
basis is guaranteed to give the
same result as the measurement of A. It seems that information is transferred faster
than the speed of light! This apparent paradox is known as the Einstein-PodolskyRosen (EPR) paradox [3] [4] and two spins in a state such as
J)AB
are known as an
EPR pair.
The paradox can be resolved by carefully studying how to transfer information via
an EPR pair. Suppose Alice and Bob had agreed beforehand to measure their spins
in the I T), I4) basis. When Alice measures A there is a 50% chance she will get I T)
and a 50% chance she will get
IT).
When Bob measures B he learns the state of A.
But Alice could not choose which state A should be in. B (as well as A) is randomly
in the state
I 4) or I 4).
Hence, Alice cannot 'tell' Bob anything; no information is
transferred.
To transfer information Alice and Bob must arrange a different scheme. To send
a 0 Alice will measure A in the I T), 4.) basis, to send a 1 she will measure A in the
I -+), I +-)
basis. When Alice measures A, B does apparently 'jump' into the same
state as A. But Bob does not know in what basis to measure qubit B.Thus he cannot
extract the information from the qubit B.
1.3
Classical and Quantum Information Processing
Classical information processing and computing can be expressed as a sequence of
one and two-bit gates. These include the NOT gate, which flips 0 to 1 and 1 to
12
0, and AND gate, which gives the output 1 if and only if its two inputs are both
1. For universal computation all that is necessary is the ability to perform NOT,
OR, AND, and COPY operations. Similarly, quantum computing and processing can
be described by a sequence of one and two-qubit gates [5]. Unlike classical gates,
quantum gates must be reversible since, as will be explained later on, only unitary
operators are allowed in quantum mechanics. Reversibility means that the input state
can be uniquely determined from the output state. This condition does not effect a
NOT gate which is reversible, but a classical AND is not reversible. If the output of
the AND gate is 0 the input may have been 00, 01, or 10. To construct and AND
that is reversible a three bit gate, such as a controlled-controlled NOT or Toffoli gate,
is required.
The Toffoli gate is a three input, three output gate that flips the third bit if and
only if the first two bits are 1. The truth table for the Toffoli gate is as follows
input
output
0
0
0
0
0
0
0
1
0
01
0
1
0
0
1
0
0
1
1
0
1
1
1
0
0
1
0
0
1
0
1
1
0
1
1
1
0
1
1
1
1
1
0
11
0
Note that the input state of the three bits can be determined from the output
state; the Toffoli gate is reversible. It is easily seen that the Toffoli performs a NOT
operation on the third bit if the first two bits are both one. It also performs and
AND gate from the first two bits onto the third bit if the third bit is a zero. An
OR gate can be constructed from two Toffoli gates and the Toffoli gate performs a
COPY of the first bit onto the third bit if the second bit is initially one and the third
bit initially zero. Accordingly, a Toffoli gate is a universal logic gate in the sense
13
that circuits to compute any Boolean functions can be constructed from Toffoli gates
alone.
1.4
No Cloning
In contrast to classical systems, in which an unknown bit can be copied at will, an
unknown state of a quantum system cannot be copied onto another quantum system
[6]. This is known as the 'no cloning' rule of quantum mechanics. 'No cloning' states
that an unknown quantum state cannot be copied or cloned onto another system. If
cloning were possible many copies could be made of one of the spins of an EPR pair.
All of them could be sent with Bob to Alpha Centauri. After Alice's measurement
of A, Bob could measure some of these copies in the I t), I 4) basis and some in the
I -+), I +-) basis. Bob could now determine in which basis Alice measured her spin
by noting the basis that the measurement always give the same result. This would
allow for superluminal communication.
To prove that 'no cloning' is true, let us imagine that we know of such an operator,
U, that would copy any state. Then U(Ia)l0)) for any state, U([6)10)) -
1,3)10).
Ia)la)
and, since U must work
Consider the state 1y) = (1a) + |/3))/V'.
the linearity of quantum mechanics, U(j-y)j0)) -+ (Ia)Ia) +
l(a/3)I
#
1.
IA#)|#))/Vf
By
0 17)1Y) if
Thus, cloning an arbitrary quantum state is not possible. We will
see the effects of 'no cloning' on quantum information processing when we discuss
quantum feedback control.
At first 'no cloning' seems counterintuitive. The Toffoli gate shown above performs
the classical COPY operation even though 'no cloning' should exclude this ability.
To resolve this apparent contradiction, apply the Toffoli gate on an input where the
first input bit is a|O) + ,31) (the second bit 11) and the third bit
10)).
The Toffoli
gate will leave the third bit zero if the first bit is zero and flip the third bit if the first
bit is a one. The output of these two bits is a 00) 1 ,3 +,|111)1,3.
(|010)3
+
011)3)
This is not equal to
0 (1a10)1 + fl11)1). Though the Toffoli gate can copy states 0 and 1,
it cannot copy a superposition of states. This is in accordance with the 'no cloning'
14
theorem.
1.5
Power of Quantum Information Processing
The advantages of QIP over classical information processing stem from a quantum
systems ability to be in superpositions of states and to entangle with other quantum
systems. The power inherent in superpositions is demonstrated as follows. While a
classical bit can process information on one state only, a qubit can process information
on both states of the superposition. For example, if Ix) 0 |0) goes to |x) ® If(x)),
than, using as input a qubit in superposition -(10)
If(0)) +11)
+ 11)) 0 10), yields 1(10) 9
0 If(1))). When the initial (or program) bit is in a superposition, the
computer performs a superposition of computations. Thus, information is processed
simultaneously on both states of the superposition. Note, that such superpositions
typically give rise to entanglement. Suppose f(x) in the above example is NOTX.
Then, f(0) = 1 and f(1) = 0. The output state is then
1(10)11) + 11)10)), an
entangled state. If two input qubits are in the state ao 00) +
#101)
+ 7110) + 6111)
the computer performs a superposition of four computations. With three qubits, the
computer performs a superposition of eight computations. 100 qubits would allow
the quantum computer to perform a superposition of 2100 computations giving more
processing power than if all the particles in the universe were classical bits. This
combination of superposition and entanglement is what allows quantum computers
to factor [27], search [7], and simulate [8][9], faster than their classical counterparts.
1.6
Errors and Decoherence
While these phenomenon may give QIP an advantage over classical information processing, they also introduce new challenges. Quantum information can go wrong in
ways not possible classically. Such peculiarly quantum errors go under the name of
decoherence. To understand decoherence we must first review quantum measurement.
When a measurement is made on a quantum system in a superposition of states,
15
the system "collapses" into one state of the superposition. The probability of collapsing into a particular state is the value of the complex amplitude squared. For
example, given a quantum spin in the state al
t)
+ 3| 4) the probability of the state
being measured as I t) is Jal 2 and the probability of being measured as
Since the probabilities must add to 1,
Ja12
+ 1)312
=
I 4)
is 1/312.
1
Measurement of this kind is irreversible and nonunitary. Once the measurement
has taken place there is no way to return the system to its original state. The
information inherent in the relative phase between a and , is lost. Successful QIP can
transfer information without measurement by making two quantum systems interact
in a coherent, non-destructive manner. QIP can also instruct how to make a proper
measurement, a measurement that will reveal the information that is sought.
Evolution of an isolated quantum system is described by the Schr6dinger equation
in
= H10)
dt
(1.3)
where H is a Hermitian operator, H = Ht, called the Hamiltonian. The time evolution induced by the Schrddinger equation is 'unitary' 10(x)) = U(t) [)) where
U(t)I'1P) -+ e -iHt/
and UtU = UUt = 1 since H = Ht. However, no quantum
system can be said to be truly isolated (except perhaps the universe as a whole).
Let us single out the quantum system we are interested in
else the environment E. The interaction between
Q and
two systems, almost as if E is performing measurements on
attention to
Q alone
Q,
and call everything
E typically entangles the
Q. If we now
restrict our
its evolution looks non-unitary due to its interactions with E.
This non-unitary evolution includes both conventional errors (bit-flipping) and purely
quantum errors (dephasing and decoherence) and tends to destroy the superpositions
within
Q. A
major barrier to QIP is overcoming noise, decoherence, and to effectively
isolate quantum system of interest.
16
1.7
Nuclear Magnetic Resonance as a Paradigm
for Quantum Information Processing
A prototypical system on which to test many aspects of QIP is liquid-state nuclear
magnetic resonance (NMR). The idea of using nuclear spins for the realization of a
quantum computer was first suggested by Lloyd [10] in 1983. Later, Cory et al [11]
and Gershenfeld and Chuang [12] realized that liquid state NMR is an ideal way to
explore many aspects of quantum computing and QIP. Instead of operating on just one
quantum system, liquid state NMR operates on an ensemble of Avogadro's number
of such systems. This allows for direct observation of expectation values without an
actual measurement, and subsequent loss of information. In addition, NMR has a
decoherence times on the order of seconds, allowing for the implementation of many
operations.
In liquid-state NMR, a liquid sample is placed in a large magnetic field which, by
convention, points in the z direction. The atoms in the sample that have non-zero
spins (for example hydrogen or carbon-13 both of which have a spin of 1/2) tend
to line up with the magnetic field. However, because liquid state NMR cannot be
done at absolute zero temperature not every spin lines up with the magnetic field. A
fraction 1+E align with the field while j--E align against the field; here E =
LB::
10-6
for nuclear spins at room temperature in a 10 Tesla magnet. Because of the large
number of spins, the excess magnetization aligned with the magnet can be detected,
and it is this signal that is observed by NMR.
In general, ensembles require dealing with interactions between the individual
systems. But in liquid-state NMR tumbling of the molecules averages out interactions
between the systems leaving each individual quantum system effectively isolated from
its surroundings. Each molecule can be treated as an individual quantum system. The
signal from each of these systems makes up the NMR signal.
A system that is either in the state
i0
1)
with pi, or in the state
|02)
with probabil-
ity P2 is described by a density matrix o = pI|ib1)(0k1|+p 210 2 )(0 2 1. The density matrix
produces the proper statistics for such a system as follows. The expectation value of
17
an operator, A, on such a system is (A) = tr A = pi(01 1A 102 ) +p 2 (4( 2 1A 102 ). In gen-
eral, the density matrix is a non-negative Hermitian operator with trace 1. Since the
wavefunction, IV)), for a closed system obeys the Schr6dinger equation ihdI)dt = H|b),
the density matrix obeys the equation
dL
= i[e, H]
dt
(1.4)
the quantum version of the Louiville equation for classical distributions on phase
space.
A single spin which has a probability of
matrix L =
1' (1
+ }| 4)(4
.
j
spin up and j spin down has a density
A two spin system with equal probability of being in
any of the four possible states can be represented by a density matrix L =
|+
4T) (14
4)(14I+ 4 4.)(41I+ 4 44)(44. Similarly, N spins having equal probability
for any of the possible states can be described by a density matrix Q =
)(TiT2
---
TNI +
1TI2 ---
N) (tIT2
---
I
+
---
+ -L
1
2 ---
N) (
1
2 --
1 TT2
-.. TN
N-
An NMR system is in a mixed state, an ensemble in which the systems may be
in any of several orthonormal 'pure' states
I1),
IV)
2 ), ... ION).
describing such a state is L = p,1041)(011+ p 2 IV2 )(0 21 +
...
The density matrix
+ PNIN) ()N I-
Because intermolecular interactions are averaged out by the tumbling of the
molecules, the individual quantum systems are indistinguishable. Therefore, the state
of the NMR ensemble can be described by a reduced density matrix of size 2', where
n is the number of spin 1/2 nuclei in the molecule, instead of a full density matrix of
size
2 N,
where N is the number of spins in the entire ensemble.
In NMR it is customary to subtract from the reduced density matrix the trace due
to the large identity term. This term comes from the molecules of the ensemble that
cancel out because they are in thermal equilibrium. Only the traceless part of the
reduced density matrix undergoes unitary evolution and, therefore, subtracting the
trace will not effect our continued description of the state as it evolves. The remainder
is then scaled to have integral elements. From this point on, 'density matrix,' p, will
refer to the reduced, shifted, scaled density matrix.
18
The general Hamiltonian governing a three spin liquid NMR system is
H = w1 I, + w 2 4, + w3 I + 27rJ
12 11
- 2 + 27rJ
13
1 -3
+ 27r J 2 31 2 .13
(1.5)
where w is the Larmor frequency, the frequency of the spins precession about the
z-axis. w = 7B where - is the gyromagnetic ration and B is the strength of the
magnetic field. I refers to 2! where - are the Pauli spin matrices. J is the coupling
constant between two of the spins.
In the weak coupling regime,
differentiated,
WA -
WB
WA
~
WB
>>
# 0, the terms I,^f and
JAB.
I
When the two spins can be
do not significantly effect the
energy levels, Iz (because they do not commute with I7 - It), and can be ignored.
We are left with the Hamiltonian
H = wl
+ w 2 1+2
3 Izf
+ 27rJ 12 IzI + 27rJ 1 3 IzIz + 2 J 23 JI.
(1.6)
The first three terms in the Hamiltonian correspond to three qubits. The last three
terms are the interaction terms. They allow for conditional rotations, such as those
necessary for controlled-NOT and Toffoli gates, between the qubits. The particular
interaction described by the liquid state NMR Hamiltonian is known as J-coupling.
Radio frequency (RF) pulses applied at a spin's Larmor frequency will rotate the
spin's magnetization. For example, a 90-degree RF pulse along the x-axis will rotate
a spin originally along the z-axis onto the y-axis (following the left-hand rule). In an
unfortunate twist of language, quantum observables (not necessarily terms that can be
seen on an NMR spectra) such as I, are used to denote both operators and expectation
values. When describing an expectation value (the system is in the 'state' I,) what
is in fact meant is Tr(Ixp) which is the expectation value of the magnetization along
the x axis.
Rotations, and therefore RF pulses, may be symbolized by unitary operators, U.
An RF pulse along x for example is described as (Slichter)
U=e
19
.
(1.7)
The effect of a rotation on a density matrix is described by
UpU-
1
(1.8)
For example an RF pulse along x acting on 1, is described as
eiIxOIze-iIzo = Izcos(O) + Iysin(9).
(1.9)
This method of describing NMR experiments is known as the product operator formalism.
Allowing the system to evolve under the Hamiltonian will couple the spins. This
can be described in a way similar to rotations. Recalling that the Pauli spin matrices
square to 1,
e ia-
I1cos
( ±)
+ I I22sin (a)(1.10)
where t, is the amount of time under Hamiltonian evolution and a = 27r J.
Not all of the terms of the density matrix are observable in an NMR spectra.
Detection coils in the spectrometer wrap around the sides of the sample, perpendicular
to the direction of the magnetic field. When an RF pulse rotates the spins into the
transverse plane (the x-y plane), the spins' rotation induces an electric current in the
wire which is the signal that makes up the NMR spectra. Therefore, the observable
terms are I, Iy which are known as 'single quantum' terms. 'Double quantum' terms
such as I',,
and other higher order quantum terms are not observable since they
are quadrapolar and higher. We will discuss how to rotate all terms into observables
when we discuss quantum tomography.
As explained, an NMR qubit is just a spin - atom. The qubit can be placed in any
superposition of
I )
and I 4) by an RF pulse of the proper phase and degree. In ad-
dition, qubits can be entangled by allowing evolution of the Hamiltonian, specifically
the II,
terms, to act upon the system.
An NMR system is also subject to natural decoherence. This takes on many forms
two of which are, T1 , know as the spin-lattice relaxation, and T2 , known as the spin20
spin relaxation. T is a time constant for the decay of the magnetization along I,. T 2
measures decay of the transverse magnetization.
1.8
Experiments
In the following chapters, a number of important QIP experiments will be presented
and their implementations described. The first is the quantum coherent feedback
loop (QCFL). The QCFL differs from 'classical' quantum feedback control in that the
system to be controlled and the controller are both quantum systems. This allows
the controller to interact coherently with the system, avoiding the loss of information
that accompanies the measurement of a classical controller on a quantum system.
In addition, it allows for the transfer of states that cannot be transferred through a
classical feedback loop.
The second chapter describes the quantum Fourier transform (QFT). The QFT
acts like a classical Fourier transform in that it takes a 'position' state to the corresponding 'momentum' state. The QFT also picks out the periodicity of the input
functions which, in the quantum realm, are wavefunctions. The QFT plays an integral role in many quantum algorithms, such as Shor's algorithm for factoring large
numbers, and quantum simulations. The implementation of the QFT is a first step
towards realizing these algorithms and simulations.
The third chapter presents ways of measuring the accuracy with which an operator
is performed on an NMR system, and describes quantum tomography, the process by
which all terms of the density matrix in an NMR system can be read out. A method
of performing complete tomography of three bits is offered.
The fourth chapter introduces quantum chaos. A quantum computer gives us
the ability to study quantum chaos on an experimental level. This can be done via
the quantum baker's map, for which the complete pulse program is given, and other
quantum chaotic maps, that will be introduced.
21
Chapter 2
Quantum Coherent Feedback
Control
2.1
Feedback Control and Quantum Feedback Control
A feedback controller is a device used to stabilize and control a system. In classical
feedback control, a sensor makes a measurement on the system and feeds the result
back to a controller. The controller processes the results of the measurement and
compares it to the input signal. Should there be a difference between the responses,
an actuator corrects the error by guiding the system into the desired state.
Quantum feedback control works in a similar fashion. However, quantum systems, unlike classical ones, may be in a superposition of states. When a feedback
loop is implemented on a quantum system, the measurement decoheres the system into one state of the system's superposition. This makes the first part of the
feedback loop probabilistic and destructive.
It also causes a loss of information.
[13][14][15][16][17][18][19][20][21] For example, a single spin may be in a superposition
of spin up and spin down, al t) +,31
4). A sensor's measurement would probabilis-
tically force the quantum system into either the state | t) or the state
losing the information of the relative phase between the states.
22
J4),
thereby
For a quantum feedback loop to function without decohering the system it must
receive information from the system without making a measurement. At first this
sounds implausible, how is it possible to learn the state of the system without a
measurement? This can be done if the sensor itself is a quantum system. In such a
case, the sensor can interact coherently with the quantum system to be controlled,
avoiding decoherence, and allowing for transfer of quantum information without a
measurement. If the controller and actuator are also quantum systems, the complete
feedback loop may be implemented without any destructive measurements and no
loss of information [22] [23].
No cloning prohibits the copying of the controller state onto the system. For the
system to evolve into the state of the controller, the controller must evolve into some
other state. Thus, the controller can only control the system once. After controlling
the system once the controller is no longer in the desired state. No cloning also
prohibits making copies of the controller state before it enacts control of the system.
If the controller was in a known state no cloning would not be in effect and it would
be possible to make multiple copies of the controller state. The copies could than be
used for continuous control of the the system.
2.2
Implementation
A complete quantum feedback loop has been implemented on an NMR system. In this
implementation the quantum system to be controlled is a nuclear spin (spin 1) and
the controller is a second nuclear spin (spin 2). In addition, the controller spin was
correlated with a third spin (spin 3) such that it is in a mixture of Einstein-PodolskyRosen (EPR) states, a state that cannot be transferred through a classical feedback
loop. The goal of the quantum feedback loop is to transfer this non-classical correlation from the controller spin to the system spin. This was successfully implemented
using NMR techniques.
Being able to control quantum systems coherently is especially important in the
area of quantum information processing, since non-coherent control would cause a
23
loss of information. In quantum computing, coherent control is necessary for effective
quantum error correction.
The Hamiltonian of a three spin (qubit) NMR sample in the weak coupling regime
is
H
=
1If +W 212+W 3
2 -r(Ji,2I I2 + J1,3 1JIU + J 2 ,3 IIZ)
(2.1)
where Ii = o/2.
The experiment was performed on three carbon-13 spins of enriched alanine,
initially in a thermal state.
The resonant frequency of carbon-13 at 9.4 Tesla is
approximately 100.617MHz.
Spin 1 is the system to be controlled, spin 2 is the
controller, and spin 3 is the ancilla. Frequency differences between the spins are
1w 1 -
w2 1 = 12, 587Hz between spins 1 and 2, 1w 2
and 3, and JWi
-
-
w3 1 = 3, 435Hz between spins 2
w3 1 = 16,022Hz between spins 1 and 3. Coupling constants between
the three spins are J2= 54Hz, J23
=
35Hz, and J 13 = 1.2Hz. Relaxation time T for
alanine is approximately 1.56s and T2 is about 420ms.
The first step is to create a quantum correlation between the controller and ancilla
spins. This is done with the following pulse sequence,
1(w
23 1
2.2
(2.2)
)(7r)
This pulse sequence reads: apply a pulse along the y-axis that rotates spins 2 and
3 by 90 degrees, allow a period of spin evolution under the natural Hamiltonian for
a time 1/4J 23 , apply a pulse along the x-axis that rotates spin 1 180 degrees, allow
another period of spin evolution for a time 1/4J 23 , and apply a final pulse along the
y-axis that rotates spin 2 by 90 degrees. Spins 2 and 3 are now in a mixture of EPR
states,
-I
+ 21213 - 223
(2.3)
where, as explained above, I = o-/2. Had the experiment been performed at absolute
zero, spins B and C would be in a pure, entangled EPR state. Due to the large
24
identity term in the complete NMR density matrix, o, an NMR version of EPR does
not produce an entangled state [24]. However, the deviation density matrix, p, does
transform as if it were an entangled EPR state. Therefore, it cannot be transferred
via a classical feedback loop without decoherence of the state. Using the quantum
coherent feedback loop, this state can be transferred. The quantum coherent feedback
loop is not limited to only unentangled states. Even an entangled EPR state can be
transferred via the quantum coherent feedback loop.
The first step in a feedback loop has the controller (spin 2) acquire information
from the system (spin 1). This is done using a logic gate known as a controlled-NOT
gate. The controlled-NOT flips a bit if and only if another bit is 1 (or up). The
following pulse sequence
-
4J2
-Fr) X
.
(2.4)
implements a controlled-NOT (modulo extra phase) between spins 1 and 2. It says,
flip spin 2 if and only if spin 1 is in the state
I t)1.
Spin 2 gains information about
the state of spin 1. Spins 1 and 2, the system and the controller, are now coherently
correlated and are in the state
2
-I z - - 4'1I
I
41z
-
2121J3
xY
(2.5)
The controller than processes the information it received and feeds it back into
the system via a controlled-NOT gate between controller, spin 2, and the system,
spin 1
-
4J12
3
412
\(y.
This sequence flips spin 1 if and only if spin 2 is in the state
(2.6)
4)1. Spin 2 processes
the information acquired in the previous step and feeds it back onto spin 1. However,
spins 1 and 2 are still correlated.
25
The system and controller are decorrelated with a final controlled-NOT gate
2
3
1
S2
1
(7).(7r)
4J12
4J12
r)2
2
.
(2.7)
(
putting the three spin in the state
-Iz
+ 2I I
-
2I IY3.
(2.8)
The system spin, spin 1, is now correlated with the ancilla spin, spin 3. The system is
in the state that the controller was originally in. The controller, meanwhile, is in the
state that the system was originally. Since the controller is no longer in its original
desired state it can no longer control the system. This is what was expected due to
the no cloning theorem. Had we been able to clone the controller before the implementation of the quantum coherent feedback loop, we could have kept controlling the
system via the clones. Because cloning of an unknown quantum state is impossible,
it is no longer possible to control the state of the system.
2.3
Results
It is useful to measure how accurately this quantum dynamical process was performed.
An appropriate measure of how close the actual final state is to the desired final state
is
1 - [Tr(pf
-
p,) 2 /Trp?]1 / 2
(2.9)
where pi is the initial density matrix of spins 2 and 3 after they have been correlated.
This is the correlation that was transferred. pf is the density matrix of spins 1 and 3
after the correlation from spin 3 has been transferred from spin 2 (the controller) to
spin 1 (the system). The fidelity of transfer of the quantum correlation was 91.5%,
where 100% shows complete correlation and 0% shows know correlation whatsoever.
Errors arose mainly because of imperfect flip angles, miscalculation of refocusing
pulses and noise. Relaxation contributed little to the error since the entire program
26
required 42.3ms compared to the T2 time of 420ms.
The demonstrated quantum coherent feedback loop is non-destructive and deterministic because no destructive measurement takes place. In addition, it can transfer
quantum correlation not possible for a classical controller. The transfer could have
only taken place due to quantum interactions between spins 1 and 2. Additionally,
the weak interaction between spins 1 and 3 preclude the possibility that a direct
correlation from 1 to 3 took place.
The quantum feedback loop promises to have many applications in stabilizing
quantum dynamics [25], quantum trajectory tracking, and quantum error correction
[26}.
27
Chapter 3
The Quantum Fourier Transform
3.1
What is the Quantum Fourier Transform and
Why is it Important?
A key subroutine of the quantum algorithms for factoring [27] and simulations [28]
[29] [30] is the quantum Fourier transform (QFT) [31] [32] [33]. In essence, the QFT
takes a 'position' state jx) to the corresponding 'momentum' state and is defined as
follows:
QFTqIx)
1
-- i1
e 2 x7xY/qly).
-
(3.1)
Sy=o
In general the QFT transforms the input amplitudes as,
QFTq E f(x) Ix) -+ E f(y)y).
x
y
(3.2)
Where the coefficients f(y) are
f(y) =
E e2 rixy/qf (x).
28
(3.3)
For example, the two qubit QFT corresponds to the unitary operator,
QFT4 =
1 1
1
1
1
1
i
-1
-i
2
1
-1
1
-1
1
-i
-1
i
-
(3.4)
The operator separates the input states by 0 degrees in the first row and column, and
then by 90 degrees, 180 degrees and 270 degrees, multiples of 1.
Equation (3.4) shows that the QFT, which is discrete, has effects similar to that of
the classical discrete Fourier transform. In particular, if f(x) is periodic with period
r, then f(y) will exhibit a spike at y = r. The classical Fourier transform reveals
the periodicity in functions, the QFT reveals periodicity of wavefunctions. This is
the key to Shor's algorithm which allows a quantum computer to factor very large
numbers in polynomial time. The factoring problem can be reduced to finding the
period of the function
f (x) = axmodN
(3.5)
where N is the number to be factored and a is a number coprime with N.
The QFT is also an integral component of quantum simulations [28]. This is because the QFT can be used to transform the wavefunction back and forth between the
position and momentum basis. For example, when simulating the time evolution of
the Schroedinger equation [29] [30] one can input the wavefunction into the quantum
computer in the position basis. Pictured from phase space a QFT will transform the
wavefunction into the momentum basis allowing for evolution along the momentum
trajectory. A second QFT transforms the wavefunction back into the position basis
and evolution may proceed along the position trajectory.
29
3.2
Construction of the QFT
As formulated by Coppersmith, the QFT can be constructed from two basic unitary
operations, the Aj or Hadamard gate, operating on the jth qubit
Aj
=
--
\/-
1
(3.6)
-1)
and Bik, a conditional phase shift, operating on the jth and kth qubits
1
=
Bik
0 0
0 1
0
0
0
0 0 1
0
(3.7)
0 0 0 e2 k ;
where
0
jk
=
r/2-
To implement the QFT, the gate sequence,
Bj,+ 1 Bj,+2.. .Bj,L-1A
is implemented on the lead bit,
j
(3.8)
= L - 1. The sequence is repeated on all L bits
as j is indexed from L - 1 to 0. A bit reversal will than complete the QFT. The bit
reversal can be achieved by relabeling the bits.
3.3
Implementation of the QFT
The above sequence of gates can be realized via NMR. As described previously, the
Hamiltonian of a NMR three qubit system with weak coupling is
H =
+W21 +w 3 I3+
27r(J, 2 1f I + Ji,3 II3 + J 2 ,3 I2I3)
where I? = -i/2.
30
(3.9)
The three bit QFT was implemented on an NMR quantum computer. The three
qubits were three carbon-13 spins of an alanine sample whose characteristics have
been described above.
The pulse program is conveniently derived from an idempotent or projection operator description of the propagators via geometric algebra [34]. The operators E± are
defined as (1±o-)/2. The A, matrix can be broken down into E+-E_ +U'(E++E_).
The pulse sequence of the A, gate is,
Aj= (
-(r)-.
(3.10)
This pulse program reads: apply a pulse along the y-axis that rotates spin j 90 degrees,
followed by a pulse along the x-axis that rotates j 180 degrees. Magnetization along
the z-axis would be rotated to the positive x-axis. Since this experiment starts with
the spins at thermal equilibrium (pointing along the z-axis), the first instance of the
A, gate can be replaced by the more easily implemented (1)
pulse along the positive
y-axis.
The Byk gate, can be implemented using the coupling between qubits. In terms of
projection operators the Bjk gate is 1- E1E
2 +eOEE 2,
and can be implemented
by the following pulse sequence:
(r)
Bjk
- (2 .
(3.11)
=
(Iy?~k
-
7
j~
-7~~
The complete pulse program is the combination of A, and Bjk gates described
above plus a bit reversal. The bit reversal on three bits would effectively swap spins
1 and 3. The necessity to perform this swap has been removed by reordering the bits.
The complete pulse program is,
31
2
-Sin(!')X+COS( !'-)
QFT3 =
(7r)2X
-
-(61)-()
(1~3)-(ir)l
-(3.12)
-
-
-
(7r)
-
(
-
-
(r)~
-
(7r)2
-
(7r)22(
-
This sequence includes a number of (7) pulses to refocus couplings.
3.4
Results
The purpose of the experiment reported here was to demonstrate an accurate performance of the QFT. Accordingly, a thermal state was chosen as the input state. This
state has the advantage that it consists of a mixture of all possible states. Therefore,
if any particular state gives highly anomalous or wrong results, it will be reflected in
our result. In order to verify the operation of the QFT on arbitrary inputs (assuming
the linearity of quantum mechanics) would require 64 distinct experiments, each with
64 distinct outputs, one for each independent entry in the density matrix which would
be experimentally impractical. However, the accuracy which the present experiment
attains strongly suggests that these experiments, too, would give the results predicted
by theory.
Taking advantage of the knowledge of our starting state allows us to not phase
correct the Hadamard gates, replacing the two pulse gate with a E pulse. This was
32
OFT theoretical
QFT experimental
12650
12600
12550
50
0
-50
-3400
-3450
-3500
Figure 3-1: The NMR spectra of the three carbon-13 atoms of alanine after performance of the QFT. The top spectra are theoretical while the bottom are experimental.
Peaks in NMR spectra are labeled by their resonant frequency. The observables are
the phase and amplitudes of the resonances. Each spin is split into four peaks since
its energy level is dependent on whether the other two spins are up (along the magnet) or down. The space between the peaks is the strength of the J-coupling. For
example, the two peaks of spin 1 are separated by 54Hz which is the strength of the
J1 2 coupling. The four peaks show up clearly in spin 2 which has relatively large
J-couples to both the 1 and 3 spins. In spin 2 the distance between the first and
second peaks (and the third and fourth peaks) is 35Hz, the J 23 coupling strength,
and the distance between the first and third peaks (and second and fourth peaks) is
54Hz, J12 . The J-coupling between the 1 and 3 spins is very small, J13 = 1.2Hz.
Therefore, the four peaks of the 1 and 3 spins are not fully resolved.
done for the first and third Hadamards.
For a general QFT the full Hadamards
should be performed which is possible in the way described above, and was done for
the second Hadamard.
Figures 1 and 2 show theoretical and experimental spectra and density matrices
following the quantum Fourier transform of the state I, + I + I (a thermal state)
on the three qubit NMR quantum computer.
A measure of accuracy appropriate for almost fully mixed density matrices such
as those found in NMR is [35]
33
C=
Tr(ptheoypexp)
Tr(p .,)
r ( pzial)
T
r(pexp)\
VT
| thr eory)
Here, p is the deviation density matrix, the density matrix, o, of the system, minus the
large identity term. A measure of 1 shows that the theoretical and experimental deviation density matrices are totally correlated. A measure of -1 means anti-correlated
and 0 means totally uncorrelated. A more detailed explanation of this measure is
found in the next chapter.
In NMR, only dipolar transverse magnetization can be detected by the coil of the
NMR spectrometer. This means that only single spin single quantum terms, those
terms involving a single spin flip, are observed in the spectra. To see the other terms
of the deviation density matrix, it is necessary to perform readout pulses after the
experiment whose objective is to rotate the unobservable terms into observable ones.
In this implementation of the QFT the experiment was performed a number of times
using various readout pulses at the completion of the experiment. This is done in
order to readout the entire deviation density matrix and obtain an accurate value of
C.
The accuracy of the implementation of the QFT is 74%. This measure reflects both
imperfections in the applied pulses and delays, as well as decoherence. Spin lattice
relaxation (T1 ) is not an important factor over the time scale of the experiment.
To a first approximation, decoherence during the course of the QFT attenuates the
entire deviation density matrix as seen in figure 2. Even though off diagonal terms
do not necessarily relax at the same rate, during the course of implementation of the
QFT all of these terms are mixed so that to a first approximation they attenuate at
the same rate. In fact, as the QFT is performed on more spins, this averaging will
become more exact. Therefore, we can approximately separate the errors caused by
experimental imperfections by renormalizing pep to its attenuated average. Using
this the correlation of the operations themselves is above 97% over the 6 gates in
(11).
The correlation of 74% corresponds to an error rate of .036 per operation over
34
the six gates which, while low, does not attain the error rate of 104 required for
robust quantum computation [36]. We believe these errors arise primarily from spatial
inhomogeneities in the radio frequency fields which can be improved.
Using NMR, the QFT has been implemented on a three bit quantum system and
the fidelity with which we can transform an initially diagonal state has been measured.
Although the correlation does not reach that required for fault tolerant computing, it
is easily high enough to permit studies on small quantum systems including quantum
simulations.
35
Figure 3-2: Theoretical and experimental results of the final deviation density matrix
after implementation of the QFT on a thermal state. The left column shows (from
top to bottom) the theoretical, experimental and difference of the real components of
the three spin density matrix. The right column shows the same for the imaginary
terms. The diagonal of the deviation density matrix can be seen running horizontally
from the left corner to the right corner, the magnitude of all terms on the diagonal
are zero. The states are labeled from I000) at the left and count up to 1111) at the
back and front corners.
36
Chapter 4
Quantum Tomography
4.1
Correlation
As explained above, room temperature NMR is always very close to the fully mixed
state. All the information describing the state is contained in the deviation from
the fully mixed state which has a relative weight one millionth the weight of the
fully mixed state. Therefore, when trying to find an appropriate measure with which
to describe the accuracy of implementing a unitary operator, the conventional measure of fidelity Tr
1ptheorypep
Pheo,,y [37] [38] gives an anomalously high value. A
more appropriate measure of accuracy for almost fully mixed density matrices is the
correlation [35]
C=
Tr(p2,)
Tr(ptheorypexp)
Tr(p4heory) Tr(pe,) \ Tr(pinii)(.
As defined above, p is the deviation density matrix.
The first term in C measures the correlation between the density matrices Ptheoy
and Pexp. This is modified by the second term, the reduction in signal over the course
of the experiment.
C = 1 shows that the theoretical and experimental deviation
density matrices are totally correlated. C = -1
means the matrices are fully anti-
correlated, and C = 0 means the matrices are totally uncorrelated.
37
Pinitial
eory
Uexperimental
Figure 4-1: The theoretical density matrix Ptheory is calculated by applying the theoretically perfect pulse sequence, Utheory, to the initial density matrix, Pinitial. The
experimental density matrix, Pexp, is the result of the experiment, the initial density
matrix after application of the actual pulse sequence, Uexperimental.
4.2
Tomography
In NMR, one cannot calculate every term of the density matrix from one spectrum.
Only transverse dipolar magnetization can be detected by the detection coil of the
spectrometer. This allows only 2n*(binomial expansion to n terms) out of the
2 2n
(where n is the number of spins) terms to be observed in any given spectra. To see
every term it is necessary to repeat the experiment followed by a number of different
readout pulses. The procedure of reading out all the terms of a density matrix is
known as quantum tomography.
There are an infinite number of readout pulse sequences that can be used to
discover the exact quantum state of the system. On one extreme, tomography may
be done by repeating the experiment over and over again with the proper readout
pulse or other operations at the end of the pulse sequence. The experiment must be
repeated a sufficient amount of times for all the terms of the density matrix to be
rotated into observables. Using only E pulses as readouts, a three qubit system would
require 11 experiments, one without a readout pulse, and another 10 each followed
by one readout E pulse. The readout pulses are listed in the tables below. The tables
show what terms are rotated into observables, in other words, what terms make up
each spectrum in each of the 11 experiments.
38
product operator
none
2
2x
2 y
ix,
ix,
I.V
-ii
--
IV,
I2
J3
2y
ix,
ix,
ix,
IZ,
_[2
JV2
x
ix,
2
I2
_[2
J2
J2
J,2
JV2
J2
J3
J3
I3
J3
3
JV
JV3
J3
IXI J2
IXI J2
IXI J2
J2
J71
.j
IV J2
ly J2
IZI J2
'y z
/ z 1.711 2
IZI J2
IZI IV2
i, jr 2
I z
11 I3
x
IX1 173
izl i.v22
Jxl J3
Jyl J3
Jxl J3
IZI J3
IZI J_3
Jyl
J2
Jzl J2
IZI I2
IYI z
IZI JV2
Jyl J2
1 J2
_[Z
IXI I2
1 J3
_ Ix
Iz Ix
- IZ
IXI J3
Jxl J3
x
IV JV3
171 Iz3
1XI J2
IXI J2 -f Ix J2
IZI J2
IXI I2
_ ly I3
IZ1 T3Jy I3
111
IZI J3
IZI J3
IZI 3
171
1z
z 11
Jzl 1, 3
IZI J3
z Ix
Jxl J3
l z 1J3
I2 J3
J2 J3
J2 J3
J2 J3
J2 J3
3
J2 J3
JV2 JV3
J2 J3
J2 JV3
z
z3
IT'
z J3
x
172 J3
J2 J3
J2 173
I2 J3
J2 J3
J2 J3
_I
J2 J3
IM2
39
- --
f -Ti- -T
'
z
none
product operator
IXI I2 J3
Jxl J2 173
.1
IlL T2 T3
X.1 x Az
ixl V 13
I/ x
IX1 J12.1 _[V3
.
IXI 172 3
/ Iz
IX1 J2 I3
m
2)x
(0
020
2
z 12J3 x
IXI J2 I3
- IZI 2 J3
V
_ IZI I2 J3
-Tl--z ---
IX1 J2 I3
J2 J3
Jxl
x 1713
z
IXI J2 J3
-
9
I' I J2 J3
IV1 J2 3
Jzl J2 J3
JU J2 I3
J2 I3
JV IV2 IV3
Jy I2 13
111.1 Iz 2 J3
IZI I2 J3
IZ1 I2 J3
Jz J2 I3
Jyl J2 3
J2 13
-"IZIz
izi J2 J3
I yl J2 I3
J2 J3
z z
Jyl J2 J3
iz,
171
Iz
2 I3
Jzl I2 J3
Jzl J2 Jz3
Jzl J2 Jz3
IZI J2 J3
Jzl 172 3
IZI J2 JV3
IZI J2 JV3
Iz. 17121z 3
T3
Il.LT2
z ZAX
IZI J2 J3
Jzl J2 J3
2 13
iz,
Jul J2 J3
40
1XI J2 J3
Iz 1 J2 J3
izi
IZI J2 J3
Jzl J2 J3
produ
ct operator
J2
IV2
I2
J2
J3
IU2
I3
IX
7
I2
IX
J2
IM2
IM2
JV2
J3
J73
_I
1XI J2
1XI J2
Jz J2
J73
P
y
- IZI IY2
1XI J2
1 J2
-T
IXI I2
- IZI J2
_ Ju Jz2
1 J2
Jz J2
-1711
IV1 IV2
Jyl J2
-1 11,.V2
z
IV J2
Jzl J2
- IZI J3
IX1 JV3
- IZI JV3
IZI J2
1XI J3
1XI Jz3
IXI I3
_ Jzl
_ IZ1 I3
Jxl I3
-Il J3
Jz J3
z y
Jzl J2
Iz
1XI lz2
Jxl J3
l
iv z
TI T2
x -L z
Jzl 2
V
IllZI F2
z
IXI J3
171 J3
J3
JV3
J3
_ IZI J2
- IZ1
Jxl I2
III J2
2
V
J2
J3
J3
I3
1XI 2
I2
J2
3
173
.1
I3
ix,
ii
y
ll
I/
11
y
izi
(E)
1xi
ix,
ixi
ix,
2,3
2,3
2
(2
x
2
y
2
2
Ill 3
Iz
_ ly J3
z
3
z
iv z
Jzl J3
Jzl J3
Jxl I3
izi 3
IZ1 IV3
- J2 13
J2 J3
J2 J3
_ I2 I3
I2 J3
172 3
I Iz
J2 J3
172 3
.1 Ix
172 IV3
- I2 J3
J2 J3
JV2 I-3
I2 JV3
J2 173
112 3
,j Iz
I2 J3
J2 13
V
J2 j3
X-.
J2 lz3
_ J2 IV3
J2 I3
Jzl
Jzl J 3V
Jxl J3
_ J2 I3
z lz3
J2 I3
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J2 I3
-- jTI3
x
J2
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J3
z
J2 I3
J2 JV3
J2 J3
I2 J3
I
41
2
product operator
1112Z3
1
r1T
2 33
2__(2
J1J2J
zyz
JJ2 J3
3
1
11__2_3
_______
J1J2J
111213
st
11m2e3
xtT12 dy
T3z
3
__
1121J3
______
no
ae
T2_P___
J1J3
______
iz
____
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___
___
___
1I
_
_ _ _ _ _
IZI172
bevb1 3
_ _
yay
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3
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_
_[___3
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zx
z
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111J21I3
y
13_______Z 1
1-21
_ _ _
1
111213
fis2er.Thsvrfista
11/a lJ
_____
______Z
112 13
I3lz
_ _
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__
11_________
als
esetateeyon
aboveit taerms woed ee
1 21 3
are__
___________3
_
_
__
_X
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IV2JZ
fte63trs(
o thosvblyat every
1
X3
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___
Zyx
Fromg the
_
3__
J1 12 J3
Frmteabv
__
11121
11121
JZ 13
_
-x- x
I
!
-I
2 IzgI
3
A zx
A___
_
_
J1J2J
__
J f JI
_
adzt
_re
IZIJ2 I3_________
IV1JI2JI3_______________
I I2
1
Iten
r e a s t_(23*_23
o n e _f t- h e
a tzzl
1
__ _ __ _ __ _
II Ix
__ __
6
1
______________
___
ft
_
_
_
_
AA11121
3
IX I2 IZ
1'2'J
er
______[_____111213
x1z z
z
xz
x y t J~~~
Jul
_
jjjji3
1 11213
I
(E2)1J3
1121
*___3_ 1j
one of the 63trs(r
thon
,tedone
pulses. In addition, every pulse rotates at least one term into an observable that is
not made observable by any of its peers. This verifies that all 11 experiments are
required for complete tomography.
At the other extreme, one can implement an experiment just once with a series
of extra pulses at the end that will rotate all possible terms to be observable at some
point after the completion of the experiment proper. To be able to measure each
term it would be necessary to acquire data after each of these extra pulses.
This
method has the advantage of being able to collect all the data on one go which avoids
42
_ _ _
the problem of the experiments being run under different conditions. However, this
method entails refocusing all spins due to the time of the acquiring periods. For three
qubits this method requires 7 pulses to be appended to the end of the experiment in
the following order,
2 X
-
2 -X
2 X
-
()l2
Y
_
2 Y
-
2 Y
-
2 Y'
(4.2)
This method has not been implemented experimentally and, though it requires
less readout pulses, it is not necessarily recommended due to the added complexity
of refocusing.
In between the two extremes there are many possible combinations of numbers of
experiments and appended pulses required to read out all terms of the density matrix.
For example,
S1
2(4.3)
requires only three experiments and a total of only 8 pulses.
Given a series of spectra the terms of the density matrix can be calculated by
determining how much each of the observable product operators (Ix, Iy...) contributes
to spectra. This is done here by integrating over every peak and solving a series of
equations for the coefficients of the product operators. Repeating this for each spectra
taken during tomography results in complete data for the coefficients of the product
operators. These can be added to produce the complete deviation density matrix.
43
Chapter 5
Quantum Chaos and the Quantum
Baker's Map
5.1
Classical Chaos and Quantum Chaos
Classically, chaotic dynamics can be defined by an exponential divergence of arbitrarily close points in phase space. In quantum mechanics chaos defined in these terms
cannot exist. This is due to the impossibility of arbitrarily close points in quantum
phase space. Furthermore, the Schroedinger wave equation is linear and does not
allow for such divergence of state vectors in the quantum Hilbert space. This seeming non-correspondence between classical and quantum dynamics forces the question,
why do some quantum systems become chaotic in the classical realm. This has led to
a search for the characteristic or 'fingerprint' of certain quantum systems that in the
classical realm manifest themselves as chaos [39] [40]. This characteristic has been
termed "quantum chaos" [41] [42] [43].
A number of characteristics about quantum systems have been advanced in the
attempt to define quantum chaos. Schack and Caves [45] [46] [47], building on work
by Peres [48], define a quantum chaotic system as one that shows hypersensitivity to
perturbations. A quantum system that requires an exponential growth of information
to track it when it is disturbed by small random perturbations. Baranger and Latora
suggest the linear rate of increase in coarse grained entropy as a signature of quantum
44
Figure 5-1: The action of the baker's map on phase space. The unit square, which
we have divided in two just to show the action of the map, is first stretched in one
direction and squeezed in the other, keeping the total area constant. Then it is
cut vertically down the middle and one side is placed on top of the other, similar
to the way a baker kneads dough. Two initial conditions arbitrarily close together
that are split by the cut on the map, will end up very far from each other. Continual
applications of the map will cause the whole of phase space to act in a chaotic manner.
chaos. Lloyd has noted that quantum chaotic systems can be identified with those
systems whose decoherent histories produce classical information.
5.2
The Quantum Baker's Map
To help the search for quantum chaos, Balazs and Voros [49] (also see Saraceno [50])
devised a quantum version of the classical baker's map [44]. The baker's map is one
of the most often used maps to study classical chaos. Acting on the unit square in
phase space, the baker's map first stretches the phase space to twice its length, while
squeezing it to half its height, as shown in figure 1. Then, the map cuts phase space
vertically in half and stacks the right portion on top of the left portion, similar to
the way a baker kneads dough. Due to this cut, initial conditions that started close
together, are sent far apart by application of the map. Balazs and Voros reasoned
that if the classical baker's map helps study classical chaos perhaps the quantized
baker's map will help in the search for quantum chaos.
5.3
The Baker's Map on a Quantum Computer
Balazs and Voros' quantization of the baker's map leads to a simple unitary operator
which consists of a QFT on half of the Hilbert space followed by an inverse QFT on
the whole of Hilbert space. For a quantum computer of L bits this would mean a
45
QFT on L - 1 bits, followed by an inverse QFT on all L bits. Schack [51] pointed
out that this map can be implemented on a quantum computer.
Because the quantum baker's map is made up of QFTs, it can be implemented by
a series of Aj and Bjk gates. For example, the three spin version of the map is made
up of the gates
SO
2 AoB' 1 Bt 2 A 1 B12 A 2 SO1 AOBO1 A 1
(5.1)
where Sjk is a swap gate, and Bt is the inverse of the B k gate. The Btk gate is
implemented by the following pulse sequence
Bk-
5.4
=
2
(
,,jk
)
-
)
O6~
_
_' (7'jk
-
-).
(5.2)
Other Quantum Chaotic Maps
Other chaotic maps may also be implemented on a quantum computer. Among these
are the quantum lazy baker's maps [52]. Lazy bakers do not "knead" their dough fully.
Phase space is stretched, squeezed, and cut much like the baker's map, but a portion
in the middle of phase space is not affected by the stretching and squeezing. These
maps are chaotic in some areas of phase space while some areas remains non-chaotic.
This allows for easy comparison of chaotic and non-chaotic dynamics.
Other quantum chaotic maps implementable on a quantum computer, include the
quantum kicked top [53] [54] and the quantum cat map [55] [56].
46
Chapter 6
Conclusions and Future Work
6.1
Conclusions
A number of experiments have been conducted to show the viability and benefit of
quantum information processing. These experiments showed that quantum systems
can be controlled and have taken the first step to the implementation of powerful
quantum algorithms. A metric to measure the accuracy of a QIP experiment has
been introduced and a scheme for quantum tomography of NMR systems has been
developed. Though there is much more work to be done before QIP on small systems can be used to show the basic framework of full-scale QIP and to study some
fundamental physics via simulation and experimental realization of quantum chaos.
6.2
Future Work
Even on the level of a small number of qubits there is much to be done in the area of
QIP. This includes implementations of the various quantum chaotic maps which can
be used as a test bed for theories of quantum chaos and decoherence. Other simulations of quantum systems may also prove interesting. Implementations of quantum
algorithms on these small qubit systems, while not necessarily easier or faster than
using classical computers, are important to show the basic abilities of QIP and quantum algorithms. A small scale implementation of Shor's algorithm should be possible
47
with 5-6 bits.
48
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53