Download Quantum information processing by nuclear magnetic resonance

Quantum information processing by nuclear magnetic resonance
spectroscopy
T. F. Havela) and D. G. Cory
Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02115
S. Lloyd
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02115
N. Boulant, E. M. Fortunato, and M. A. Pravia
Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02115
G. Teklemariam
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02115
Y. S. Weinstein
Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02115
A. Bhattacharyya and J. Hou
Research Science Institute, Center for Excellence in Education, 140 Park Street SE, Vienna, Virginia 22180
共Received 26 August 2001; accepted 5 December 2001兲
Nuclear magnetic resonance 共NMR兲 is a direct macroscopic manifestation of the quantum
mechanics of the intrinsic angular momentum of atomic nuclei. It is best known for its extraordinary
range of applications, which include molecular structure determination, medical imaging, and
measurements of flow and diffusion rates. Most recently, liquid-state NMR spectroscopy has been
found to provide a powerful experimental tool for the development and evaluation of the coherent
control techniques needed for quantum information processing. This burgeoning new
interdisciplinary field has the potential to achieve cryptographic, communications, and
computational feats far beyond what is possible with known classical physics. Indeed, NMR has
made the demonstration of many of these feats sufficiently simple to be carried out by high school
summer interns working in our laboratory 共see the last two authors兲. In this paper the basic
principles of quantum information processing by NMR spectroscopy are described, along with
several illustrative experiments suitable for incorporation into the undergraduate physics
curriculum. These experiments are spin–spin interferometry, an implementation of the quantum
Fourier transform, and the quantum simulation of a harmonic oscillator. © 2002 American Association
of Physics Teachers.
关DOI: 10.1119/1.1446857兴
I. INTRODUCTION
Quantum mechanics has a reputation for being not only
mysterious, but also far removed from everyday experience.1
Thus it may come as something of a surprise to learn that
one need go no further than the nearest chemistry laboratory
to witness it first-hand. There one will find a device, called a
nuclear magnetic resonance 共NMR兲 spectrometer, whose operation is described using language that is usually reserved
for elementary quantum systems. Nonetheless, this device
operates on macroscopic samples of rather ordinary liquids,
usually simple organic compounds in inert solvents. Of
course, a complete explanation of any spectroscopic experiment ultimately requires quantum mechanics, if only to justify the absorption and emission of radiation at distinct frequencies. But only in NMR are the lifetimes of multiparticle
quantum states often long enough to allow one to fully appreciate the complexity of the dynamics of such states.
Nuclear magnetic resonance uses radio-frequency radiation to observe and manipulate the magnetic dipoles associated with the spins 共intrinsic angular momenta兲 of the atomic
nuclei in molecules. It was first demonstrated by the groups
of Felix Bloch at Stanford and Edward Purcell at Harvard in
the late 1940s, for which they shared the 1952 Nobel prize in
345
Am. J. Phys. 70 共3兲, March 2002
http://ojps.aip.org/ajp/
physics. Because of the many applications that have since
been discovered, particularly to chemical analysis and biological imaging, it is now far more often encountered in
chemical and medical research laboratories than it is in physics. Nevertheless, NMR is currently making something of a
comeback in physics, because it recently has been used to
perform the first experimental demonstrations of quantum
information processing 共QIP兲.2– 4 These demonstrations are
the first step toward realizing the dream of a quantum computer: a device that harnesses the complexity of quantum
dynamics to solve computational problems that are, and will
remain forever beyond the reach of classical computers.5–9
Despite their name and enormous computational power,
quantum computers are actually much more than fast computers. When one is ultimately built, it will be capable of
creating and transforming quantum states arbitrarily and to
any desired precision.
The catch that prevented early pioneers in the field of QIP
from realizing that NMR provides an astonishingly easy
route to at least a limited form of quantum information processing is that NMR samples consist of a macroscopic ensemble on the order of 1020 molecules, wherein the states of
their nuclear spins over the ensemble are a little short of
© 2002 American Association of Physics Teachers
345
completely random. Such a system might well be able to
process information, but it seemed one would have little control over what information! This problem was circumvented
by taking advantage of the very ensemble nature of the system which gave rise to it in the first place. Namely, we define
a ‘‘pseudopure’’ ensemble to be one in which the sum of the
nuclear magnetic moments over all the molecules present in
the sample yields a system of moments proportional to that
which would be observed if the ensemble were pure, that is,
if the spin states of all its molecules were the same. It will be
shown below that the dynamics of the unique spin state associated with a pseudopure ensemble are identical to those of
a single molecule in that state, and that the ensemble average
magnetism observed is proportional to the corresponding
quantum expectation value.
Nevertheless, there are some important limitations associated with pseudopure ensembles. The first is that the preparation of pseudopure ensembles from the largely random
thermal equilibrium ensembles available in liquid-state NMR
entails an exponential loss of magnetization as the number of
spins in the molecule increases. Thus signal-to-noise considerations preclude the preparation of useful pseudopure ensembles for molecules containing more than about ten
spins—enough to demonstrate the principles of QIP, but certainly not enough to compete with a modern personal
computer.10 The second is that ensemble averaging entails a
loss of information about the microscopic states of the spins
in the individual molecules; it is even possible to devise
pseudopure ensembles where none of the molecules are in
the associated spin state at all.11 This lack of a well-defined
microscopic interpretation means that quantum information
processing based on liquid-state NMR cannot be used to
settle foundational issues in quantum mechanics such as the
existence of nonclassical correlations and nonlocality.
The purpose of this paper is to present some of the NMR
experiments that we have performed over the last few years
to demonstrate quantum information processing, in sufficient
detail that physicists with access to a reasonably good NMR
spectrometer could incorporate them into their curriculum.
These experiments, which form the beginnings of a ‘‘quantum cookbook,’’ illustrate many of the basic principles and
experimental subtleties not only of QIP, but also of quantum
physics more generally. It is our view that QIP provides an
excellent approach to teaching introductory quantum physics, because it deals mainly with finite-state quantum systems, which involve much simpler mathematics than that
needed to treat the spatial degrees of freedom of continuous
systems. 共This point was also stressed by Feynman et al.12兲
Indeed, the only mathematical prerequisites for the present
paper are calculus and linear algebra, and the experiments
are simple enough to be carried out by dedicated undergraduates or even high-school students 共as demonstrated by the
fact that a pair of high school summer interns in our laboratory, namely the last two authors on this paper, performed the
Mach–Zehnder interferometer experiments described in Sec.
V兲. For additional accounts of QIP by NMR, the reader is
referred to Refs. 13–16.
II. A PRIMER ON QUANTUM INFORMATION
PROCESSING
The new gedanken technology of quantum computation provides an unfamiliar perspective on
such vexing questions, by using the quantum
346
Am. J. Phys., Vol. 70, No. 3, March 2002
theory, not to expand our understanding and control of the physical world, but to exploit the
quantum behavior of the physical world as a
novel way to encode and process information.
The information is primary; the underlying
physical system only matters as a vehicle for that
information. Quantum computer scientists view a
set of n interacting spins-1/2 not for the insight it
offers into the nature of magnetic materials, but
as a way to represent and manipulate integers,
through their n-bit binary representations as orthogonal states in a ‘‘computational basis’’ that
specifies whether each individual spin is up 共1兲
or down 共0兲.
N. David Mermin, ‘‘The Contemplation of Quantum Computation,’’ Physics Today (July, 2000)
Information, however it may be conceived, exists only by
virtue of being stored in the state of some physical system.
The physics of its embodiment, in turn, determines what can
be done with the information.17 Quantum information processing is the study of the encoding, transmission, and dynamics of information contained within a quantum system,
which may also interact with a far more complex and largely
unknown environment in which it is embedded.18 –21 A
simple but important paradigm for QIP is a largely hypothetical device called a quantum computer, which stores binary
information in an array of distinguishable two-state quantum
systems, or ‘‘qubits.’’ 6,18,22 It operates on this information by
applying unitary transformations to small subsets of the qubits at a time 共usually just one or two兲, thereby building up
arbitrarily complex unitary transformations of all the qubits
together. We shall deal exclusively with systems that are, in
this sense, quantum computers, because such systems are the
easiest to understand and to implement via NMR spectroscopy 共see Sec. III兲.
In keeping with their usage, the basis states for the twodimensional Hilbert space of a qubit are denoted in Dirac
notation by 兩0典 and 兩1典. General states are superpositions of
these basis states: 兩 ␺ 典 ⫽ ␺ 0 兩 0 典 ⫹ ␺ 1 兩 1 典 共where ␺ 0 , ␺ 1 are
complex numbers with 兩 ␺ 0 兩 2 ⫹ 兩 ␺ 1 兩 2 ⫽1兲. The basis states of
an array of qubits are variously denoted by
兩 ␦ 1典 丢 兩 ␦ 2典 丢 ¯ 丢 兩 ␦ N典 ⫽ 兩 ␦ 1典兩 ␦ 2典 ¯ 兩 ␦ N典 ⫽ 兩 ␦ 1␦ 2¯ ␦ N典 ,
共1兲
where ␦ n 苸 兵 0,1其 (n⫽1,...,N) and ‘‘ 丢 ’’denotes the tensor 共or
Kronecker兲 product:
共 ␺ 0 兩 0 典 ⫹ ␺ 1 兩 1 典 ) 丢 共 ␺ 0⬘ 兩 0 典 ⫹ ␺ 1⬘ 兩 1 典 )
⫽
冋 册冋 册
␺0
␺1
丢
冋 册
␺ 0 ␺ 0⬘
␺ 0⬘
␺ 0 ␺ 1⬘
⫽
.
␺ ⬘1
␺ 1 ␺ ⬘0
␺ 1 ␺ 1⬘
共2兲
It should be noted that the dimension of the tensor product
space of N qubits grows exponentially as 2 N .
Logical operations on qubits are implemented by unitary
transformations, the simplest of which is the so-called NOT
gate:
N兩 0 典 ⫽
冋 册冋 册 冋 册
0
1
1 1
0
⫽ ⫽兩1典.
0
1
0
共3兲
Havel et al.
346
The matrix N is clearly both Hermitian and unitary, that is,
N† ⫽N and NN† ⫽N2 ⫽ ␴1 共the 2⫻2 identity兲, from which it
follows that N兩 1 典 ⫽ 兩 0 典 as well. Indeed, the Hamiltonia HN
that, when substituted into Schrödinger’s equation,
iប ⳵ / ⳵ t 兩 ␺ 典 ⫽HN兩 ␺ 典 ,
共4兲
for a time t⫽1, subjects the initial state 兩 ␺ 典 兩 t⫽0 to a NOT, is
HN ⫽
冋
⫺1
␲ប 1
2 ⫺1
1
册
共5兲
.
Thus we may express N as the exponential of this Hamiltonian, that is,
exp共 ⫺iHN /ប 兲 ⫽ ␴1 ⫺iHN /ប⫹ 21 共 HN /ប 兲 2 ⫹¯⫽N,
共6兲
where i ⫽⫺1.
A complete repertoire of quantum logic gates must also
include operations on one qubit conditional on the state of
another. Classical logic gates, such as the AND and OR, are
not possible because they are not unitary 共or even invertible兲.
The simplest conditional unitary operation is known as the
controlled NOT, or c-NOT gate. This operation takes the
NOT of one qubit providing that another is in the state 兩1典,
but does nothing if the other is in the state 兩0典. Specifically,
the c-NOT gate the flips the first 共left-hand兲 qubit conditional on the state of the second 共right-hand兲 qubit is given
by N1 兩 2 ,
2
N1 兩 2 兩 00典 ⫽ 兩 00典 ,
N1 兩 2 兩 01典 ⫽ 兩 11典 ,
N1 兩 2 兩 10典 ⫽ 兩 10典 ,
N1 兩 2 兩 11典 ⫽ 兩 01典 .
共7兲
This operation may also be viewed as replacing the first 共leftmost兲 qubit by its XOR 共exclusive OR兲 with the second. The
unitary matrix for this c-NOT is simply
冋 册
冋 册
1
N1 兩 2 ⫽
0
0
0
0
1
0
0
1
0
0
1
0
0
HN 1 兩 2 ⫽
0
0
0
␲ប 0
2 0
1
0
⫺1
0
0
0
⫺1
0
1
0
共9兲
.
Note also that (N1 兩 2 ) 2 ⫽ ␴1 丢 ␴1 共the 4⫻4 identity兲.
Thus far, we have considered only gates that can also be
implemented on purely classical bits, because they map basis
states to other basis states. A true quantum logic gate maps
basis states to superpositions thereof, a prime example of
which is the Hadamard gate:
W兩 0 典 ⫽
1
&
共 兩 0 典 ⫹ 兩 1 典 ),
W兩 1 典 ⫽
1
&
共 兩 0 典 ⫺ 兩 1 典 ).
共10兲
The unitary matrix for this operation is
W⫽
冋
1 1
1
& 1
⫺1
册
,
and its generating Hamiltonian is
347
Am. J. Phys., Vol. 70, No. 3, March 2002
共11兲
&⫹1
冋
冋
a 11B
a 12B
A 丢 B⫽ a 21B
]
a 22B
册
]
¯
¯
a 11b 11
a 11b 12
⫽ a 11b 21
]
a 11b 22
共12兲
.
册
¯
册
¯ .
]
共13兲
Then it can be shown by a straightforward application of
these definitions that
W 丢 N 兩 k 典 ⫽ 共 W 丢 ¯ 丢 W兲 兩 k 典
2 N ⫺1
兺
l⫽0
2 N ⫺1
⫽
which corresponds to the Hamiltonian
⫺1
We leave it as an exercise to show that once again we have
WW† ⫽(W) 2 ⫽ ␴1 .
Applied to multiple qubits simultaneously, the Hadamard
gate becomes an analog of the Fourier transform, commonly
known as the Walsh–Hadamard transform, which maps information stored locally in the individual qubits into the
same information distributed globally over all of the qubits.
We may express this fact by abbreviating an arbitrary basis
state 兩 ␦ 1k ¯ ␦ Nk 典 by 兩 k 典 , where ␦ nk denotes the nth bit in the
binary expansion of the integer k, n⫽1,...,N, k⫽0,...,2 N ⫺1
共for example, 兩 5 典 ↔ 兩 101典 兲. In addition, we will need to extend our previous definition of the tensor product of state
vectors 兩k典 to the tensor product of operators on them, which
is also known as the Kronecker product of their representative matrices, that is,
共8兲
,
0
冋
␲ &⫺1
冑8 ⫺1
⫽
0
0
HW⫽
兺
l⫽0
冉
n n
N
兿 n⫽1
共 ⫺1 兲 ␦ k ␦ l
2 N/2
冊
兩l典
共 ⫺1 兲 k䉺l
兩l典,
2 N/2
共14兲
N
␦ nk ␦ nl mod 2. Like the Fourier transform,
where k䉺l⫽ 兺 n⫽0
computing this transform on a classical computer requires
time of order N2 N and memory 2 N . 23 The quantum version,
however, can be computed using time and memory merely
proportional to N—exponentially faster than a classical
computer—simply by applying a Hadamard gate to each qubit, and then letting the distributivity of the tensor product do
the rest, or
N
ជ n⫽1
共 W兩 ␦ 1k 典 ) 丢 ¯ 丢 共 W兩 ␦ Nk 典 )⫽ 丢
共 兩 0 典 ⫹ 共 ⫺1 兲 ␦ k 兩 1 典 )
n
⫽W 丢 N 兩 k 典 ,
共15兲
ជ denotes the 共Kronecker兲 product of the factors orwhere 丢
dered from left-to-right by increasing index.
Conversely, the Walsh–Hadamard transform is capable of
taking global information distributed across a superposition,
and mapping it back into local information that can be read
by determining the states of the individual qubits. This fact
follows simply by observing that it is self-inverse:
(W 丢 N ) ⫺1 ⫽W 丢 N . In addition, any quantum logic gate U,
applied to a superposition, effectively operates in parallel on
all basis states of the superposition, that is,
Havel et al.
347
2 N ⫺1
U
兺
k⫽0
2 N ⫺1
␺ k兩 k 典 ⫽
兺
k⫽0
␺ k U兩 k 典 .
共16兲
Together these facts allow for what Deutsch and Jozsa have
called ‘‘computation by quantum parallelism,’’ 24 which
means using nonlocal logic gates of the form W 丢 N UclW 丢 N ,
where Ucl is a 共unitary兲 classical logic gate. Such nonlocal
operations are what allow QIP to perform communication
and computation feats far beyond what is possible
classically.25–27
册
␺0
cos共 ␽ /2兲 e i ␸ /2
⫽
.
␺1
sin共 ␽ /2兲 e ⫺i ␸ /2
共17兲
It is readily verified that 具 ␺ 兩 ␺ 典 ⫽1, so that this state vector is
properly normalized, and that arg(␺0)⫹arg(␺1)⫽0 关where
arg(a⫹ib)⬅arctan(b/a)兴, so that the physically irrelevant total phase of the state vector is eliminated by this parametrization. In the 兩0典, 兩1典 basis specified above, the operators for
the x, y, and z components of the spin’s angular momentum
are just ប/2 times the Pauli matrices:
348
Am. J. Phys., Vol. 70, No. 3, March 2002
0
1
1
0
␴y ⫽i 共 兩 1 典具 0 兩 ⫺ 兩 0 典具 1 兩 兲 ⫽
␴z ⫽ 兩 0 典具 0 兩 ⫺ 兩 1 典具 1 兩 ⫽
,
0
⫺i
i
0
1
0
0
⫺1
共18兲
,
.
It follows that the expectation values of the angular momenta
along the coordinate axes are ប/2 times:
⫽sin共 ␽ 兲 cos共 ␸ 兲 ,
A major advantage of NMR in comparison with
other forms of spectroscopy is the possibility of
manipulating and modifying the nuclear spin
Hamiltonian at will, almost without any restriction, and to adapt it to the special needs of the
problem to be solved.
R. R. Ernst, G. Bodenhausen, and A. Wokaun,
Principles of Nuclear Magnetic Resonance in
One and Two Dimensions (Oxford University
Press, 1987)
Demonstrations of QIP by liquid-state NMR utilize an ensemble of identical molecules, each containing one or more
atomic nuclei with an intrinsic angular momentum of ប/2
共spin 1/2兲. Such a nucleus, itself called a ‘‘spin,’’constitutes
an ideal qubit, because the component of its angular momenta measured along any given direction 共for example, by a
Stern–Gerlach apparatus兲 is always ⫾ប/2. When placed in a
strong, uniform magnetic field, conventionally taken to be
along the z axis, the Zeeman interaction between the spins’
magnetic moments and the field causes them to precess
around the z axis, much like a gyroscope does in a gravitational field. The z component of the angular momentum of
the spins is quantized, meaning that it is not affected by the
Zeeman interaction and so constitutes a ‘‘good’’ quantum
number by which the state may be characterized. Therefore,
the natural basis in which to represent qubits by the states of
spins in a magnetic field parallel to the z axis is to let 兩0典 be
the state obtained following a measurement giving ⫹ប/2 in
the z direction, and 兩1典 that following a measurement of
⫺ប/2.
The physical nature of spins leads naturally to a geometric
interpretation of the quantum state of a single qubit as a real,
three-dimensional unit vector n. This vector, variously
known as the Bloch vector or polarization vector, defines the
unique direction in space along which a measurement of the
spin’s angular momentum always has the deterministic outcome ⫹ប/2. To see how this works, let us write the spin’s
state vector in terms of the polar angles of n,
冋 册冋
冋 册
冋 册
冋 册
具 ␺ 兩 ␴x 兩 ␺ 典 ⫽cos共 ␽ /2兲 sin共 ␽ /2兲共 e ⫺i ␸ ⫹e i ␸ 兲
III. STATES AND OPERATORS IN NMR
SPECTROSCOPY
兩␺典⫽
␴x ⫽ 兩 1 典具 0 兩 ⫹ 兩 0 典具 1 兩 ⫽
具 ␺ 兩 ␴y 兩 ␺ 典 ⫽cos共 ␽ /2兲 sin共 ␽ /2兲 i 共 e ⫺i ␸ ⫺e i ␸ 兲
⫽sin共 ␽ 兲 sin共 ␸ 兲 ,
共19兲
具 ␺ 兩 ␴z 兩 ␺ 典 ⫽cos 共 ␽ /2兲 ⫺sin 共 ␽ /2兲 ⫽cos共 ␽ 兲 .
2
2
These are just the Cartesian coordinates of the unit vector n
in terms of its polar coordinates. Because the components of
the angular momentum transform as a vector under rotations,
the operator for the angular momentum along n is ប/2 times
␴n ⫽sin共 ␽ 兲 cos共 ␸ 兲 ␴x ⫹sin共 ␽ 兲 sin共 ␸ 兲 ␴y ⫹cos共 ␽ 兲 ␴z .
共20兲
It is readily verified that the expectation value of the angular
momentum along n is (ប/2) 具 ␺ 兩 ␴n 兩 ␺ 典 ⫽(ប/2) 储 n储 2 ⫽⫹ប/2,
and because measurements of the component of the angular
momentum along any axis are quantized at ⫾ប/2, the outcome of such a measurement is always ប/2, as claimed.
This geometric interpretation of a spin 1/2 state as a vector
in physical space rather than as a ‘‘ket’’ in an abstract Hilbert
space is of great utility in understanding both NMR and QIP
more generally. To simplify what follows, we shall henceforth use units in which ប⫽1, meaning we measure angular
momentum in multiples of ប and energy in radians per unit
time. Consider the Hamiltonian for the Zeeman interaction,
HZ⫽⫺( ␥ /2)B 0 ␴z , where B 0 is the strength of the static
magnetic field along z, and ␥ is a conversion factor called the
gyromagnetic ratio, which gives the magnetic dipole moment
of the nucleus from its angular momentum. Schrödinger’s
equation thus implies that the evolution of a spin in the magnetic field is
兩 ␺ 共 t 兲 典 ⫽e ⫺iHZ t 兩 ␺ 0 典 .
共21兲
The matrix exponential is easily evaluated because ␴z is diagonal, leading to the following analog of de Moivre’s formula for the complex exponential:
e ⫺iHZ t ⫽
冋
exp共 i ␥ B 0 t/2兲
0
0
exp共 ⫺i ␥ B 0 t/2兲
册
⫽ ␴1 cos共 ␥ B 0 t/2兲 ⫹i ␴z sin共 ␥ B 0 t/2兲 .
共22兲
This same formula can be shown to also hold for a field
along ␴x and ␴y . Letting ␻ 0 ⫽ ␥ B 0 , it follows that
兩␺共 t 兲典⫽
冋
册
cos共 ␽ 兲 e i 共 ␸ ⫹ ␻ 0 t 兲 /2
,
sin共 ␽ 兲 e ⫺i 共 ␸ ⫹ ␻ 0 t 兲 /2
共23兲
meaning that the unit vector n precesses in a left-hand sense
共for ␥ ⬎0兲 about the z axis at a rate of 兩 ␻ 0 兩 radians per unit
time, in accordance with our earlier claims.
Havel et al.
348
For multiple spins, each with its own precession frequency
␻ n0 共due to their differing electronic environments within the
molecule兲, this relation generalizes straightforwardly to
共e
1
兩␺ 典) 丢 ¯ 丢 共 e
i ␻ 0 ␴z t/2
⫽共 e
1
i ␻ 0 ␴z t/2
丢
N
i ␻ 0 ␴z t/2
i ␻ 0 ␴z t/2
1
¯丢e
N
兩␺ 典)
共26兲
N
兲兩 ␺ 1¯ ␺ N典 .
共24兲
the Zeeman propagator in Eq. 共24兲 can now be written as
1
共25兲
关When it is necessary to distinguish such a superscript from
a power, parentheses will be used, for example, ( ␴zn ) 2
⫽ ␴1 .兴 Because of the general formula 共for all 1⭐m⬍n
⭐N兲
exp共 ⫺iHZ t 兲 ⫽exp
⫽
冋
冉冋
i ␻ 10 t/2
0
0
⫺i ␻ 10 t/2
1
2
e i 共 ␻ 0 ⫹ ␻ 0 兲 t/2
0
册冊 冉冋
0
e
1
2
i 共 ␻ 0 ⫺ ␻ 0 兲 t/2
丢 exp
0
0
0
0
册冊
0
e ⫺i 共 ␻ 0 ⫺ ␻ 0 兲 t/2
0
0
0
0
e ⫺i 共 ␻ 0 ⫹ ␻ 0 兲 t/2
1
with ⌬ ␻ n0 ⫽ ␻ n0 ⫺ ␻ 0 for 1⭐n⭐N. The use of a rotating
frame also makes it relatively easy to explain the action of a
rf 共radio-frequency兲 pulse on the spins in the sample, where
the direction of propagation coincides with the B 0 field’s
axis. When the frequency of the pulse equals that of the nth
spin and the phase of the pulse puts its magnetic component
along the y ⬘ axis, the 共magnetic dipole of the兲 spin rotates at
a constant velocity ␻ n1 , determined by the power of the
pulse, around the y ⬘ axis 共see Fig. 1兲. Thus after a time t
⫽␲/(2␻n1), the spin points along the x ⬘ axis, where its precessing magnetic dipole produces the maximum signal back
in the lab frame. If the transmitter and receiver frequencies
are the same, the real part of the Fourier transform of the
signal then contains a resonance at the frequency ␻ n0 ⫺ ␻ 0 .
Henceforth, unless otherwise stated, all our observables and
propagators will be referenced to such a common rotating
frame, so that we may drop the primes from the coordinate
labels x, y and the ⌬’s from the precession rates ␻ n0 without
ambiguity.
Am. J. Phys., Vol. 70, No. 3, March 2002
共27兲
共28兲
In the case of two spins 1/2, for example, we obtain
⫺i ␻ 20 t/2
共30兲
N N
HZ ⫽⫺ 21 共 ␻ 10 ␴z1 ⫹¯⫹ ␻ N0 ␴zN 兲 .
0
HZ⬘ ⫽⫺ 21 共 ⌬ ␻ 10 ␴z1 ⫹¯⫹⌬ ␻ N0 ␴zN 兲 ,
N N
where in the last equality we have also used the fact that all
the ␴zn commute and operate on the same 共tensor product of兲
Hilbert space共s兲. It follows that the multispin Zeeman Hamiltonian is
0
In NMR spectrometers, the voltage induced in a coil by
the precession of the spins’ magnetic dipoles is measured
relative to a carrier signal, the frequency of which is close to
those of the spins. This process too can be viewed geometrically as a change of coordinate system from a laboratory
frame to a frame that rotates at the carrier frequency ␻ 0 , in
which the Zeeman Hamiltonian becomes
349
1 1
0
2
1 1
⫽e i 共 ␻ 0 ␴z ⫹¯⫹ ␻ 0 ␴z 兲 t/2,
i ␻ 20 t/2
1
N
e i ␻ 0 ␴z t/2 丢 ¯ 丢 e i ␻ 0 ␴z t/2⫽e i ␻ 0 ␴z t/2¯e i ␻ 0 ␴z t/2
At this point it is convenient to introduce a notational shortcut, namely
2
册
共29兲
.
The final aspect of liquid-state NMR spectroscopy that
must be dealt with here lies in the ensemble nature of its
samples, which consist of ⬃1020 identical molecules each
containing N spins. The use of such macroscopic samples is
an absolute necessity, because the signal due to a single spin
is orders of magnitude too small to detect. Due to the rapid
thermal motions of the molecules in liquids, the forces between spins in different molecules are averaged to zero so
that, to an excellent approximation, they are noninteracting.
As a result, the presence of multiple copies of the molecule
in the sample simply amplifies their spins’ signals, with no
effect on their evolution. In addition, the Zeeman energy of
the spins in the strongest available fields B 0 is small compared to the mean thermal energies k B T in liquid samples, so
that the spins are in a statistical mixture of Zeeman eigenstates with probabilities given by the Boltzmann distribution:
e ⫺ 具 k 兩 HZ 兩 k 典 / 共 k B T 兲
P k⫽
Z eq
冉
冊
2 N ⫺1
Z eq⫽
兺
l⫽0
e ⫺ 具 l 兩 HZ 兩 l 典 / 共 k B T 兲 . 共31兲
Here, we are using the ket 兩k典 to denote an N-spin state in
which spins parallel to B 0 along the z axis correspond to the
bits ␦ nk ⫽0, and antiparallel to ␦ nk ⫽1, in accord with our intention to store binary information in these states. It is easily
seen that these states are eigenvectors of HZ .
The process by which an arbitrary ensemble with density
operator ␳ evolves toward equilibrium is known as relaxation. This fundamentally limits the time available for performing QIP by 共liquid-state兲 NMR to a matter of seconds,
Havel et al.
349
Fig. 1. The left-hand diagram shows a trajectory for the polarization vector
of a nuclear spin in a static magnetic field along the z axis 共vertical兲 combined with resonant electromagnetic radiation which exerts a constant torque
upon it. The right-hand diagram shows the same trajectory in a frame that
rotates along with the radiation 共primes兲, where it appears as a constant
angular velocity rotation about y ⬘ axis.
and hence the number of gates that can be applied to a hundred or so. All of the experiments described in this paper can
of course be performed before significant relaxation occurs,
and hence this 共important兲 issue will not be further discussed
here.
It is nevertheless essential to understand how the accessible 共measurable兲 quantum information contained in an ensemble is represented. This is done by reducing the full probability distribution over the spins’ states to the so-called
‘‘density operator’’ description of the ensemble’s statistics.
At thermal equilibrium, this operator is given by ␳eq
2 N ⫺1
⫽ 兺 k⫽0
P k 兩 k 典具 k 兩 , and more generally, given any probability
distribution p k ⭓0 ( 兺 k p k ⫽1) over an ensemble of quantum
states 兵 兩 ␺ k 典 其 , it is
␳⫽
兺k p k兩 ␺ k 典具 ␺ k兩 .
共32兲
It follows easily that the ensemble average of the expectation
value of any observable M is
tr共 ␳M兲 ⫽
兺k p k tr共 兩 ␺ k 典具 ␺ k兩 M兲 ⫽ 兺k p k 具 ␺ k兩 M兩 ␺ k 典 ,
共33兲
and that the time evolution of this ensemble average, and
hence the density operator itself, under any Hamiltonian H is
given by
M 共 t 兲⫽
共34兲
At the temperatures characteristic of liquid samples, ␻ n0
Ⰶk B T, so that the Boltzmann factors P k are essentially equal
to their linear approximations,
2 N ⫺1
⫺1
␳eq⫽Z eq
兺
l⫽0
共35兲
where I⫽ ␴1 丢 ¯ 丢 ␴1 and the partition function Z eq⫽tr(I
⫺HZ /(k B T))⫽2 N 关because the zero of energy is chosen so
that tr(HZ )⫽0兴. Given that the observables for the transverse
350
Am. J. Phys., Vol. 70, No. 3, March 2002
共36兲
In the case of two spins, for example, the equilibrium density
operator becomes
␳eq⫽
⫽
共 1⫺ 具 l 兩 HZ 兩 l 典 / 共 k B T 兲兲 兩 l 典具 l 兩
⫺1
⫽Z eq
共 I⫺HZ / 共 k B T 兲兲 ,
magnetization ␴x , ␴y are also traceless, the identity component I is not detectable in NMR experiments and so is usually dropped when writing the density operator. Similarly,
because the factors ␻ n0 /(2 N k B T) are essentially equal for nuclei of the same atomic isotope, and are also multiplied by
nonintrinsic factors depending on the spectrometer setup,
they are usually set to unity in analyzing NMR experiments.
These sins of omission lead to the following simple expression for the 共homonuclear兲 equilibrium density operator:
␳eq⫽ 21 共 ␴z1 ⫹¯⫹ ␴zN 兲 .
兺k p k 具 ␺ k兩 e iHt Me ⫺iHt兩 ␺ k 典
⫽tr共 e ⫺iHt ␳e iHt M兲 ⫽tr共 ␳共 t 兲 M兲 .
Fig. 2. Energy level diagram for two coupled spins in a magnetic field
共above兲, where the single spin–flip transitions are indicated by two-headed
arrows. The Zeeman energies of the spins are E 1 and E 2 , and their difference is ⌬E⫽ 兩 E 1 ⫺E 2 兩 . Assuming a high-temperature Boltzmann distribution of energy level populations, the spectrum, obtained by rotating the spins
into the transverse 共xy兲 plane with a ␲/2 rf pulse and Fourier transforming
the resulting signal 共below兲, exhibits a peak for each of the single spin–flip
transitions, where each pair of peaks separated by the coupling constant J
corresponds to the flips of one spin in those molecules for which the other
spin is either parallel 共兩0典兲 or antiparallel 共兩1典兲 to the field, respectively.
冉冋
1
0
冋 册
1
0
0
册冋 册冋 册冋
1
1
2
⫺1
丢
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⫺1
⫹
1
0
0
1
丢
1
0
0
⫺1
册冊
共37兲
,
indicating that the populations of the molecules in the lowest
and the highest of the four energy levels are increased and
decreased by the same amount from the totally random state
of all equal populations 共cf. Fig. 2兲. Together with operatorHavel et al.
350
valued expansions of the propagators, as in Eq. 共22兲, these
notations are sufficient to describe the vast majority of
liquid-state NMR experiments, including all those to be discussed in this paper. The notation itself is commonly known
in NMR as the product operator formalism.28 –30 It can in
fact be viewed as a special case of the geometric 共or Clifford兲 algebras that have been shown to provide a unified
mathematical language for much of physics.31,32
e ⫺iHW ⫽
1
&
共 ␴x ⫹ ␴z 兲 .
共39兲
Rotations about such oblique axes are most conveniently
implemented by a sequence of pulses affecting rotations
about the transverse 共x and y兲 axes, for example,
e i ␴y ␲ /8e ⫺i ␴x ␲ /2e ⫺i ␴y ␲ /8
⫽exp共 e i ␴y ␲ /8共 ⫺i ␴x 共 ␲ /2兲兲 e ⫺i ␴y ␲ /8兲
⫽e ⫺i 共 ␴x ⫹ ␴z 兲 /& 共 ␲ /2兲 .
IV. HOW TO DEMONSTRATE QIP BY LIQUIDSTATE NMR
... it is pretty clear why present quantum theory
not only does not use—it does not even dare to
mention—the notion of a ‘‘real physical situation.’’ Defenders of the theory say that this notion is philosophically naive, a throwback to outmoded ways of thinking, and that recognition of
this constitutes deep new wisdom about the nature of human knowledge. I say that it constitutes
a violent irrationality, that somewhere in this
theory the distinction between reality and our
knowledge of reality has become lost, and that
the result has more the character of a medieval
necromancy than of science.
E. T. Jaynes, ‘‘Quantum Beats,’’ in Foundations
of Radiation Theory and Quantum Electrodynamics, edited by A. O. Barut (Plenum, 1980)
Given the foregoing background in NMR, we are now in a
position to explain how to implement quantum logic gates on
a NMR spectrometer. The quantum NOT gate, for example,
is simply a rf pulse with its total power and phase set to
rotate the spin by ␲ about the x 共or y兲 axis of the rotating
frame, thereby converting 兩 0 典 ↔ 兩 1 典 in every molecule of the
equilibrium ensemble. Such a pulse can be made selective
for a single spin by setting its frequency to the spin’s precession rate ␻ n0 , and giving the pulse an envelope determined
by sin(t⌬␻/2)/(t⌬ ␻ /2). Because the Fourier transform of
this envelope is a square wave, this modulation results in an
approximately uniform distribution of power across all frequencies in the range ␻ n0 ⫾⌬ ␻ /2, so that only spins with
precession frequencies in this range are 共significantly兲 affected by the pulse. This technique also illustrates the first of
many practical tradeoffs that must be made in designing a
NMR experiment, because the narrower the range of frequencies required, the longer the duration of the pulse must
be, during which the other spins will evolve under other
interactions, or lose phase coherence through their natural
relaxation processes.
Implementing the Hadamard gate is also straightforward,
because the generator given in Eq. 共12兲 is represented in
product operator notation by
HW ⫽
␲
冑8
共 & ␴1 ⫺ ␴x ⫺ ␴z 兲 .
共38兲
The ␴1 part generates a simple phase shift, while the remainder generates a rotation about an axis halfway between x and
z, in analogy with the Zeeman Hamiltonian. At t⫽1 the
angle of rotation is ␲, and because ( ␴x ⫹ ␴z ) 2 ⫽2 ␴1 , the
Taylor series for the exponential collapses to the Hadamard
gate in product operator notation:
351
Am. J. Phys., Vol. 70, No. 3, March 2002
共40兲
Implementation of the c-NOT gate requires an interaction
between spins, but 共as mentioned above兲 the through-space
interaction between the magnetic dipoles of the spins is averaged to zero by the rapid diffusional motions of the molecules in liquids. Fortunately, a through-chemical-bond interaction between pairs of spins in the same molecule, known
as ‘‘scalar coupling,’’ remains active. Assuming that the coun
pling constant for this interaction is J mn Ⰶ 兩 ␻ m
0 ⫺ ␻ 0 兩 /(2 ␲ ),
the scalar coupling Hamiltonian for a pair of spins m, n assumes the following simple form:
mn m n
Hmn
J ⫽ ␲ J ␴z ␴z /2.
共41兲
This Hamiltonian can be viewed as increasing or decreasing
the field at one spin, depending on whether the other spin is
parallel or antiparallel to the z axis, and thereby changing the
first spin’s precession rate. Thus the peak in the NMR spectrum due to the first spin splits into two peaks separated by a
frequency of J mn , one peak due to those molecules for which
the other spin is parallel to the field 共兩0典兲, and the other peak
due to those for which the other spin is antiparallel 共兩1典兲.
The time has come to present our first NMR pulse sequence, this being one that implements the c-NOT gate N1 兩 2
in a simple two-spin system:
冋 册 冋
册 冋 册
␲ 1
␲
␲
␴y → ␴z1 ␴z2 → ␴1x .
4
4
4
共42兲
spin
In this notation, 关 (angle/2) ␴axis
兴 indicates a rf pulse implementing the rotation by the given angle about the subscript
axis and selective for the superscript spin共s兲. A bracketed
expression of the form 关 (angle/2) ␴z1 ␴z2 兴 , on the other hand,
refers to a time delay 共waiting period兲 of duration
angle/( ␲ J), during which the system evolves freely under
the internal scalar coupling, but not the Zeeman Hamiltonian. Of course one cannot actually ‘‘turn off’’ Hz at will,
but it is possible to do something almost as good, which is to
refocus the Zeeman Hamiltonian while allowing the scalar
coupling to evolve. The refocusing is done by replacing
关 ( ␲ /4) ␴z1 ␴z2 兴 by the subsequence
冋
册 冋
冋
册
册 冋
␲ 1 2
␲
␴z ␴z → 共 ␴1x ⫹ ␴2x 兲
8
2
→
册
␲ 1 2
␲
␴ ␴ → ⫺ 共 ␴1x ⫹ ␴2x 兲 ,
8 z z
2
共43兲
where the sum ␴1x ⫹ ␴2x indicates a rotation of both spins.
The easiest way to understand the overall sequence is via a
vector diagram 共Fig. 3兲, in which the magnetization vectors
of the first spin, in each of the two subensembles defined by
the orientation 共兩0典 or 兩1典兲 of the second, are plotted in a
Havel et al.
351
Fig. 3. Vector diagrams showing the magnetization components of spin 1 in those molecules wherein spin 2 is parallel 共兩0典兲 and antiparallel 共兩1典兲 to the static
magnetic field B 0 after each pulse or delay in the sequence which implements the c-NOT gate N1 兩 2 共see text兲.
frame rotating at the first spin’s Zeeman precession rate at
each step along the way. The initial 共␲/2兲 pulse rotates the
magnetization due to the first spin into the transverse plane,
where it precesses freely in the applied magnetic field. In
those molecules in which the second spin is parallel to the
field, the first precesses J/2 Hz faster, whereas in those in
which the second spin is antiparallel, it precesses J/2 Hz
slower, so that after a 1/(4J) delay there is a ␲/2 difference
in the accumulated phase of the first spin between these two
subensembles of molecules. The ␲ pulse after the first such
delay negates the accumulated phase evolution due to both
the coupling as well as the Zeeman Hamiltonian on both
spins. As a result, the net accumulated phase due to the Zeeman Hamiltonian after the second delay vanishes, so that it
has no net effect on the system. Because the ␲ pulse affects
both spins, however, the direction of phase evolution due to
scalar coupling also changes its sign at each spin, so that the
scalar coupling phase evolution accumulates. Thus after the
second delay, the phase of the first spin differs by ␲ between
the two subensembles, so that a properly phased 共␲/2兲 pulse
rotates one back to the ⫹z axis and the other to the ⫺z axis.
The accompanying ⫺␲ pulse 共with no delay兲 reverses the
effect of the first save for its ‘‘refocusing’’ of the Zeeman
evolution.
A more analytical means of deriving the same pulse sequence is obtained from the product operator form of the
generating Hamiltonian for the c-NOT in Eq. 共9兲, that is,
HN1/2⫽ ␲ X11 Z21 ,
共44兲
where Z␦ ⫽( ␴1 ⫹(⫺1) ␦ ␴z ) and X␦ ⫽( ␴1 ⫹(⫺1) ␦ ␴x ) ( ␦
苸 兵 0,1其 ). Because these operators are easily seen to be idempotent 共meaning that each squares to itself兲 and Z21 commutes
with X11 , up to an irrelevant overall phase factor, the Taylor
series for the exponential of this Hamiltonian collapses to the
following simple form:
1
e ⫺iHN1/2⫽Z20 ⫹e ⫺iX1 ␲ Z21 ⫽Z20 ⫹ ␴1x Z21 .
共45兲
Alternatively, we can fully expand the generator and use the
commutivity of all the terms to obtain 共up to a phase factor
as before兲
352
Am. J. Phys., Vol. 70, No. 3, March 2002
1/2
2
1
1 2
2
1
1
e ⫺iHN ⫽e i ␴z 共 ␲ /4兲 e i ␴x 共 ␲ /4兲 e ⫺i ␴x ␴z 共 ␲ /4兲
1 2
1
⫽e i ␴z 共 ␲ /4兲 e i ␴x 共 ␲ /4兲 e ⫺i ␴y 共 ␲ /4兲 e ⫺i ␴z ␴z 共 ␲ /4兲 e i ␴y 共 ␲ /4兲
1
2
1
1 2
1
⫽e i 共 ␴z ⫹ ␴z 兲共 ␲ /4兲 e i ␴x 共 ␲ /4兲 e ⫺i ␴z ␴z 共 ␲ /4兲 e i ␴y 共 ␲ /4兲 .
共46兲
Reading this product of propagators in temporal order 共from
right-to-left兲 immediately gives the previous pulse sequence
for the c-NOT gate, up to a ␲/2 phase shift on each qubit.
These are of little consequence and can be implemented, if
need be, by correcting the accumulated phase prior to Fourier transformation.
Similar strategies for implementing a wide variety of
quantum logic gates by NMR may be found for example, in
Refs. 29 and 33. We now turn to methods of preparing
pseudopure ensembles, which are a prerequisite for most
demonstrations of QIP by NMR. Because the eigenvalues of
the homonuclear equilibrium density matrix form an equally
spaced sequence with a binomial distribution of degeneracies, whereas all but one of the eigenvalues of a pseudopure
density matrix are degenerate, it is immediately clear that the
former cannot be transformed to the latter by any coherent
共that is, unitary兲 process. Instead, an ‘‘incoherent’’ process is
needed, by which the distribution of eigenvalues can be
‘‘evened out’’ 共that is, its von Neumann entropy increased34兲.
In our work as well as in NMR spectroscopy more generally,
the experimental technique most widely used to implement
incoherent processes is a magnetic field gradient. In this
technique, a variation in the strength of the field across the
sample is introduced, so that the spins precess at differing
rates depending on their location within it. The rapid loss of
net phase coherence resembles the process of decoherence in
many respects, in which information on the state of the system irreversibly leaks away into the environment,35 save that
in NMR the phase coherence can be restored simply by reversing the direction of the field gradient 共providing the molecules have not moved a significant distance during the time
between the two gradients兲.
In particular, the effect of a gradient along the z axis is to
correlate the phases of the density operator’s entries in the
Zeeman basis with the z coordinates of the molecules in the
ensemble. For example, if we let ⵜz ␻ be the rate at which the
phase of a spin varies along the z axis following such a
gradient pulse, then an arbitrary two-spin density matrix ␳
becomes a periodic function of z, specifically 关cf. Eq. 共29兲兴:
Havel et al.
352
␳共 z 兲 ⫽exp共 ⫺iⵜz HZ z 兲 ␳ exp共 iⵜz HZ z 兲 ⫽
冋
2
1
1
2
␳ 00
e ⫺iⵜz ␻ 0 z ␳ 01
e ⫺iⵜz ␻ 0 z ␳ 02
*
e iⵜz ␻ 0 z ␳ 01
2
␳ 11
e ⫺iⵜz 共 ␻ 0 ⫺ ␻ 0 兲 z ␳ 12
e ⫺ⵜz ␻ 0 z ␳ 13
1
*
e iⵜz 共 ␻ 0 ⫺ ␻ 0 兲 z ␳ 12
␳ 22
e ⫺iⵜz ␻ 0 z ␳ 23
1
*
e iⵜz ␻ 0 z ␳ 23
2
␳ 33
*
e iⵜz ␻ 0 z ␳ 02
1
2
*
e iⵜz 共 ␻ 0 ⫹ ␻ 0 兲 z ␳ 03
To a good approximation, the rate at which the coherences
共off-diagonal elements兲 of this density matrix dephase is proportional to their coherence order, that is, the difference in
the total z angular momentum along the z axis 共in units of
ប/2兲 of the states 兩j典, 兩k典 connected by the coherence ␳ jk .
Because a NMR spectrometer normally detects only the total
signal from the entire sample volume, the density matrix that
gives the experimentally observable signal is the integral of
␳(z) over z. Thus all the coherences rapidly decay to zero as
1
1
2
*
e iⵜz ␻ 0 z ␳ 13
2
e ⫺iⵜz 共 ␻ 0 ⫹ ␻ 0 兲 z ␳ 03
1
2
册
.
共47兲
the rate of phase variation increases, save for the zeroquantum coherences 共␳12 in a homonuclear two-spin system兲,
which dephase at a rate 兩 ⵜz ( ␻ 0j ⫺ ␻ k0 ) 兩 Ⰶ 兩 ⵜz ␻ 0j 兩 , 兩 ⵜz ␻ k0 兩 .
In the following rf and gradient pulse sequence, each arrow connects the preceding density operator 共in product operator notation兲 to the density operator it is converted to by
the pulse given above the arrow. The sequence as a whole
transforms the equilibrium ensemble of a homonuclear twospin system into a pseudopure ensemble:
共48兲
For the case of a heteronuclear system, wherein the gyromagnetic ratios of the two spins will differ substantially, the
second gradient pulse 关 ⵜz 兴 and its bracketing 共␲/2兲 pulses
can be omitted, because in this case the zero-quantum coherences will also be decohered by the first gradient.
Although this sequence may appear somewhat formidable,
it is straightforward to implement on a NMR spectrometer
and can be verified by straightforward trigonometry. A general method of preparing pseudopure states on any number of
spins may be found in Ref. 36.
V. SPIN–SPIN INTERFEROMETRY AND SPINOR
BEHAVIOR
We choose to examine a phenomenon which is
impossible, absolutely impossible, to explain in
any classical way, and which has in it the heart of
quantum mechanics. In reality, it contains the
only mystery. We cannot make the mystery go
away by explaining how it works. We will just
tell you how it works. In telling you how it
works we will have told you about the basic peculiarities of all quantum mechanics.
R. P. Feynman, R. B. Leighton, and M. Sands,
The Feynman Lectures on Physics, Vol. III, Sec.
1-1 (Addison–Wesley, 1965)
353
Am. J. Phys., Vol. 70, No. 3, March 2002
Fig. 4. 共a兲 Mach–Zehnder interferometer, with beam splitters drawn as grey
rectangles, mirrors as white rectangles, detectors as half circles, and the
phase shifter as a square box with the phase shift ␾ inside it. 共b兲 Corresponding logic circuit, with Hadamard gates indicated by W and the conditional rotation by circles containing the corresponding operators on each
qubit with the rotation angle ␾ in the box connecting them. The two paths
taken by the photon in the interferometer correspond to the two components
of the superposition ( 兩 0 典 ⫹ 兩 1 典 )/& into which the H qubit is put by its first
Hadamard. This state is shown 共without normalization by 1/&兲 in a dashed
box that points at that place in the circuit. The conditional rotation
exp(i(␾/2)ZH1 ␴xC) transforms it into the generally mixed state with density
1
operator 2 ␴1 ⫹ ␴z cos(␾/2)⫹ ␴y sin(␾/2) ␳ z , which is likewise shown in a
dashed box pointing to the corresponding place in the circuit 共with the
1
identity component 2 ␴1 omitted兲. Here, ␳ z ⫽tr( ␳␴z )/2 is the z component of
the C-qubit’s input state ␳ which represents the polarization of the photons
entering the interferometer. Finally, the ␴y or ␴z polarization of the H qubit
is measured while ‘‘tracing over’’ 共decoupling all further interactions with兲
the C qubit. This polarization varies with the half angle ␾/2 of the rotation
angle, just as does the ‘‘which path’’ information obtained from the measurements made in the interferometer 共see the text兲.
Havel et al.
353
C
A. Background and theory
Figure 4共a兲 shows an optical device for measuring phase
shifts in polarized light, known as a Mach–Zehnder interferometer. This device provides a simple and direct illustration
of the quantum nature of photons, the ‘‘spinor’’ nature of
phase shifts, and the destructive nature of quantum measurements. A Mach–Zehnder interferometer is logically equivalent to the quantum circuit shown in Fig. 4共b兲, wherein:
共1兲 the H qubit 共top line兲 representing the photon’s path is
placed in a known quantum state 兩0典;
共2兲 a Hadamard gate is applied to this qubit, creating a superposition of states ( 兩 0 典 ⫹ 兩 1 典 )/& just as a beam splitter creates a superposition of photon paths;
共3兲 the C qubit 共bottom兲, representing the photon’s 共possibly
unknown兲 polarization, is subjected to a phase shift
exp(⫺i␾␴zC) conditional on the state of the first; because
␴z ⫽W␴x W, this phase shift may be accomplished by
sandwiching a c-NOT共␾兲 gate 共see below兲 between a
pair of Hadamards;
共4兲 the reciprocal phase shift on the H qubit is transformed
into a rotation about the x axis by another Hadamard,
which corresponds to rejoining the paths with a second
beam splitter; and
共5兲 its state 兩0典 or 兩1典, corresponding to the path taken by the
photon, is measured.
In terms of equations, this circuit does the following. BeH H H
cause XH
1 ⫽W Z1 W , the unitary transformation implemented by the overall circuit can be written compactly using
idempotents 关cf. Eq. 共44兲兴 as
H C
C
⫺i 共 ␾ /2兲 ␴z
UM Z ⫽e ⫺i 共 ␾ /2兲 X1 ␴z ⫽XH
⫹XH
1e
0 .
共49兲
Thus for an arbitrary input state ␳ of the C qubit, we obtain
C
C
C
HC
H
H C
i 共 ␾ /2兲 ␴z H
␳out
⫽ 共 e ⫺i 共 ␾ /2兲 ␴z XH
X1 ⫹XH
1 ⫹X0 兲共 Z0 ␳ 兲共 e
0兲
C
C
C
⫺i 共 ␾ /2兲 ␴z C i 共 ␾ /2兲 ␴z
⫺i 共 ␾ /2兲 ␴z C
⫽ 12 共 XH
␳ e
⫹ ␴zHXH
␳
1e
0e
C
H C i 共 ␾ /2兲 ␴z
C
⫹XH
⫹XH
0 ␴z ␳ e
0 ␳ 兲.
共50兲
If we now perform a ␴z measurement on the H qubit 共tracing
over the C qubit as indicated in Fig. 4兲, we find that
C
C
HC
⫺i 共 ␾ /2兲 ␴z C
␳ ⫹XH1 ␳Ce ⫺i 共 ␾ /2兲 ␴z 兲
tr共 ␴zH␳out
兲 ⫽ 21 tr共 XH
0e
C
⫽real共 tr共 e ⫺i 共 ␾ /2兲 ␴z ␳C兲兲 .
共51兲
In a similar fashion, measuring ␴Hy gives
C
C
HC
⫺i 共 ␾ /2兲 ␴z C
␳ ⫺iXH1 ␳Ce i 共 ␾ /2 兲 ␴z 兲
tr共 ␴Hy ␳out
兲 ⫽ 21 tr共 iXH
0e
C
⫽i imag共 tr共 e ⫺ 共 ␾ /2兲 ␴z ␳C兲兲 ,
共52兲
whereas measuring ␴H
x gives
C
C
1
C
H C
H ⫺i 共 ␾ /2兲 ␴z C i 共 ␾ /2兲 ␴z
tr共 ␴H
␳ e
兲 ⫽0.
x ␳out 兲 ⫽ 2 tr共 X0 ␳ ⫺X1 e
共53兲
We can thus use our knowledge of ␳C to determine the rotation angle ␾ and hence the strength of the interaction 共‘‘scattering’’兲, or we can use our knowledge of ␾ to determine the
prior state of the C qubit 共‘‘tomography’’兲. The Mach–
Zehnder interferometer circuit also illustrates the essential
logic behind most of the quantum algorithms currently believed to be more efficient than their classical counterparts.37
The most interesting results are obtained with an input of
␳C⫽XC0 . The overall density operator ZH0 XC0 is converted
C
by the first two Hadamards to XH
0 Z0 , which in turn is
transformed by the c-NOT共␾兲 gate WHWCUM Z WCWH as
follows:
C
H C
H i ␴x ␾ /2
⫺i ␴x ␾ /2
⫹ZH
⫹ZH
共 ZH
1e
0 兲共 X0 Z0 兲共 Z1 e
0兲
C
C
H H C
H H H ⫺i ␴x ␾ /2 C i ␴x ␾ /2
H H C
⫽ZH
Z0 e
⫹ZH
0 X0 Z0 Z0 ⫹Z1 X0 Z1 e
0 X0 Z1 Z0 共 cos共 ␾ /2 兲
H H H
C
C
⫹i ␴C
x sin共 ␾ /2 兲兲 ⫹Z1 X0 Z0 共 cos共 ␾ /2 兲 ⫺i ␴x sin共 ␾ /2 兲兲 Z0
i
1 H C
C 1 H
C
C
H H C C
H H C C
⫽ 12 ZH
0 Z0 ⫹ 4 Z1 共 1⫹cos共 ␾ 兲 ␴z ⫺sin共 ␾ 兲 ␴y 兲 ⫹ 2 共 Z0 ␴x Z0 ␴x ⫺ ␴x Z0 ␴x Z0 兲 sin共 ␾ /2 兲 ⫹ 2 ␴x Z0 cos共 ␾ /2 兲
1
1 H C
H
C 1 H C
H C
H C
⫽ 14 ⫹ 14 共 ZH
0 ⫹Z1 cos共 ␾ 兲兲 ␴z ⫺ 4 Z1 ␴y sin共 ␾ 兲 ⫺ 4 共 ␴x ␴y ⫹ ␴y ␴x 兲 sin共 ␾ /2 兲 ⫹ 2 ␴x Z0 cos共 ␾ /2 兲 .
Clearly when ␾⫽0 or ␾⫽4␲, the result is equal to the
C
c-NOT共␾兲 input XH
0 Z0 , so that the overall circuit does nothing. In the case ␾ ⫽2 ␲ , however, we obtain
1
4
1 H C
H C
⫹ 14 ␴zC⫺ 41 ␴H
x ⫺ 4 ␴x ␴z ⫽X1 Z0 ,
共55兲
Finally, if we set ␾⫽␲, we simply apply the usual c-NOT
C
gate to XH
0 Z0 , obtaining
1
4
C
H C
H C
共 1⫺ ␴H
x ␴y ⫺ ␴y ␴x ⫹ ␴z ␴z 兲
H
which is transformed by the last pair of Hadamards to
C
ZH
1 X0 . The polarization of the hydrogen has been negated
by what, applied unconditionally, would be a rotation by
2␲, demonstrating that the underlying dynamics is ‘‘spinorial’’ 共that is, has periodicity 4␲ rather than 2␲ as in spatial
rotations兲.
354
Am. J. Phys., Vol. 70, No. 3, March 2002
共54兲
C
H
C
⫽e i 共 ␴z ⫹ ␴z 兲 ␲ /8兩 ␾ ⫹ 典具 ␾ ⫹ 兩 e ⫺i 共 ␴z ⫹ ␴z 兲 ␲ /8,
共56兲
where 兩 ␾ ⫹ 典 ⫽( 兩 00典 ⫹ 兩 11典 )/& is one of the maximally entangled Bell states. It is easily verified that 兩 ␾ ⫹ 典 is unaffected by the final pair of Hadamards, and hence
Havel et al.
354
HC
⬘ 典具 ␾ ⫹
⬘兩
␳out
⫽兩␾⫹
H
C
⫽We i 共 ␴z ⫹ ␴z 兲 ␲ /8WW兩 ␾ ⫹ 典
H
C
⫻具 ␾ ⫹ 兩 WWe ⫺i 共 ␴ z ⫹ ␴z 兲 ␲ /8W
H
C
H
C
⫽e i 共 ␴x ⫹ ␴x 兲 ␲ /8兩 ␾ ⫹ 典具 ␾ ⫹ 兩 e ⫺i 共 ␴x ⫹ ␴x 兲 ␲ /8
C
⫽ 14 共 1⫹ ␴zH␴Cy ⫹ ␴Hy ␴zC⫹ ␴H
x ␴x 兲 .
共57兲
HC H
HC H
␴y )⫽tr( ␳out
␴z )⫽0,
tr( ␳out
meaning that no
It follows that
signal will be seen in the corresponding NMR spectrum, or
equivalently, that the outcomes of measurements of these
observables, if performed on the individual molecules in the
sample, would be completely random. This randomness is a
key characteristic of maximally entangled states: no information about them can be obtained by local 共single spin兲 measurements. Indeed, the apparent disappearance of order when
one qubit becomes entangled with another unobserved qubit
is widely believed to be a paradigm for decoherence: the loss
of observable quantum mechanical effects through interaction of a quantum system with its surroundings.19
B. NMR implementation
The NMR experiments described here use the hydrogen
and carbon nuclei in a sample of 13C-labeled chloroform
( 1 H – 13C – Cl 3 ). This molecule is convenient not only because it justifies our use of the labels H and C for the qubits
above, but also because separate rf channels can be used for
each nucleus, allowing each spin to be placed in a separate
frame rotating at its precession frequency so that its Zeeman
evolution has no effect on the results. To imitate the Mach–
Zehnder interferometer by NMR, we first create the two-spin
C
pseudopure ground state ZH
0 Z0 ↔ 兩 00典具 00兩 , as described in
Eq. 共48兲, save that in this heteronuclear system it is necessary
to reduce the magnetization on the hydrogen to that of the
carbon before beginning the experiment. Because ␥ H is almost exactly 4 ␥ C , this reduction is easily done by applying
an arccos(1/4)⫽75.5° pulse to the hydrogen, followed by a
strong gradient pulse to destroy all the resulting transverse
magnetization.
Because the results of applying the first two Hadamards in
the circuit of Fig. 4共b兲 are so simple, we shall omit them in
C
what follows, and just input XH
0 Z0 to the c-NOT共␾兲 gate
directly. A pulse sequence to implement the remaining Hadamard gates has been described already 关Eq. 共40兲兴, as has one
for the c-NOT共␲兲 gate 关Eq. 共46兲兴. The latter’s extension to
general angles ␾ is easily obtained by expanding its exponential, as follows:
冉
exp ⫺i
冊 冉
冉
冊 冉
冊
冊 冉 冊
冊 冉 冊
␾ C H
␾
␾
␴x Z1 ⫽exp ⫺i ␴Cx exp ⫺i ␴Cx ␴zH
2
4
4
⫽exp ⫺i
冉
␾ C
␲
␴x exp ⫺i ␴Cy
4
4
⫻exp ⫺i
␾ C H
␲
␴z ␴z exp i ␴Cy . 共58兲
4
4
The third factor on the last line corresponds to delay of
length t⫽ ␾ /(2 ␲ J), where J is the coupling constant, leading to the pulse sequence shown in Fig. 5 关cf. Eq. 共46兲兴.
The ‘‘trace’’ operation indicated in Fig. 4共b兲 is imple355
Am. J. Phys., Vol. 70, No. 3, March 2002
Fig. 5. Pulse sequence for x rotation of C by an angle ␾ conditional on the
polarization of H, with an 共optional兲 Hadamard gate applied to the H spin.
Left-upper-to-right-lower hatched pulses are y pulses, while left-lower-toright-upper hatched pulses are about x. The ␲ pulse of the Hadamard gate
has twice the width of the 共␲/2兲 pulses, while the last ␾ pulse to the C is
labeled by the variable ␾ and outlined by a dotted line.
mented by decoupling the carbon from the hydrogen while
the data for the spectrum is collected. This may be done by
applying a sequence of ␲ pulses to the carbon between the
time points at which the signal is measured, thereby refocusing the scalar coupling so that the carbon has no effect upon
the hydrogen spectrum 共or has been ‘‘traced’’ over兲. Although decoupling is necessary if one wishes to implement
exactly the Mach–Zehnder circuit of Fig. 4, we can actually
obtain more information about the state of the system as a
whole by not decoupling during acquisition. Thus, if decoupling is used, the spectrum allows one to determine the magH
nitudes of ␴H
x and ␴y only, whereas if coupling is allowed
during acquisition, the ‘‘antiphase’’ components of the denC
H C
sity operator, ␴H
x ␴z and ␴y ␴z , will evolve under scalar
coupling into observable components whose signals will be
␲/2 out-of-phase with those due to ␴Hx and ␴Hy at the beginning of the data acquisition. By not including the final Hadamard WH, one also obtains the amplitudes of the product
operators ␴zH and ␴zH␴zC , and if WC is then included, one
H C
C
H
also obtains ␴zH␴C
x and ␴y ␴x . Finally if both W and W
H C
are included, one obtains ␴x ␴x as well.
With a bit more work, one can actually read out all 15
C
product operators 共␴H
1 ␴1 is, of course, not detectable兲, a
procedure usually referred to as tomography. One could,
for example, also collect spectra from the carbon channel,
although this channel suffers from lower sensitivity due to
13
C’s smaller gyromagnetic ratio. Alternatively, one can
C
simply swap the input states of the spins, XH
0 Z0 , along
with the pulses applied to the H and C channels. Together
with the foregoing, this procedure yields all 15 observable
product operators with the exception of ␴Hy ␴Cy . This, however, can also be obtained simply by replacing the final Hadamards in the swapped experiments with their phase-shifted
analog
V⫽i 共 ␴y ⫹ ␴z 兲 /&,
共59兲
Havel et al.
355
which can be implemented just by replacing ␴x by ␴y in the
usual pulse sequence for the Hadamard gate. Figure 6 shows
the results of tomography following gates with ␾ ⫽ ␲ , 2␲,
and 4␲, illustrating spinor behavior as well as the loss of
observable magnetization in an entangled state.
VI. QUANTUM FOURIER TRANSFORM ON THREE
QUBITS
It always seems odd to me that the fundamental
laws of physics, when discovered, can appear in
so many different forms that are not apparently
identical at first, but with a little mathematical
fiddling you can show the relationship... . There
is always another way to say the same thing that
doesn’t look at all like the way you said it before... . Perhaps a thing is simple if you can describe it fully in several different ways without
immediately knowing that you are describing the
same thing.
R. P. Feynman, ‘‘Nobel Lecture,’’ Physics Today
(August, 1966)
A. Background and theory
The Fourier transform and its extensions play a pivotal
role throughout science and engineering, and no less so in
quantum mechanics, where for example it converts between
the position and momentum representations of operators on
Hilbert space.34 We have already made good use of the
Walsh–Hadamard transform, which can be regarded as a
Fourier transform over the multiplicative two-element group
兵⫾1其. Such generalizations of the Fourier transform to Abelian groups have been proposed as a unifying foundation for
quantum algorithms.26,38 In this section we present a quantum implementation39 of the discrete Fourier transform
which, like the Walsh–Hadamard transform, maps basis
states into superpositions but can be implemented in time
proportional to N 2 共where N is the number of qubits兲, as
opposed to N2 N for the classical fast Fourier transform
共FFT兲. The efficiency with which this ‘‘quantum Fourier
transform’’ 共QFT兲 can be implemented is the key to Shor’s
algorithm for fast integer factorization.40,41 It also plays an
important role in certain examples of quantum chaos, specifically the baker’s map.42,43
The original QFT algorithm41 decomposes the QFT into a
product of Walsh–Hadamard transforms and conditional
phase rotations, in analogy to the classical FFT. This algorithm may be derived by considering how the FT acts on the
computational basis, that is,
UQFT兩 l 典 ⫽2
2 N ⫺1
⫺N/2
兺
k⫽0
N
e 2 ␲ ikl/2 兩 k 典 ,
共60兲
where 关 2 ⫺N/2 exp(2␲ikl/2N ) 兴 k,l is just the usual 共unitary兲 matrix of the discrete Fourier transform. We can rewrite this
matrix as a product by substituting the binary expansion k
N
k m 2 N⫺m 共with k m ⫽ ␦ m
⫽ 兺 m⫽1
k 兲 into Eq. 共60兲 and splitting
the sum over all integers into a sum over all combinations of
their N bits 0 or 1, obtaining
356
Am. J. Phys., Vol. 70, No. 3, March 2002
Fig. 6. Plots of matrices of product operator coefficients obtained via tomography for the spin–spin interferometry experimental results 共see the
text兲. The columns of each matrix have been ordered according to the longitudinal (L⫽1,z) and transverse (T⫽x,y) components as follows: LL
⬍TL⬍LT⬍TT. Each column, in turn, has been ordered in the same way
with 1⬍z and x⬍y 共only the labels on the LL columns are shown兲. The
three matrices plotted are 共a兲 the results of the ␾ ⫽4 ␲ conditional rotation,
which returns essentially the input state XH0 ZC0 ; 共b兲 the results of the ␾
⫽2 ␲ conditional rotation, which shows the change of sign XH1 ZC0 due to the
spinor nature of qubit rotations; 共c兲 and the results of the ␾ ⫽ ␲ conditional
⬘ 典具 ␾ ⫹
⬘ 兩 with no observable
rotation, which yields a rotated Bell state 兩 ␾ ⫹
magnetization 共see the text兲.
Havel et al.
356
1
UQFT兩 l 典 ⫽2
冉
1
N
兺 ¯ k兺⫽0 m⫽1
兿 e 2 ␲ ik
k ⫽0
⫺N/2
1
N
N
ជ m⫽1
⫽丢
共 2 ⫺1/2共 兩 0 典 ⫹e 2 ␲ il/2
m⫺1
m l/2
m⫺1
冊
兩 k 1典 ¯ 兩 k N典
兩 1 典 ))
共61兲
关see Eq. 共15兲 for the notation兴. On also substituting for l
N
⫽ 兺 n⫽1
l N⫺n⫹1 2 n⫺1 , we obtain
冉
冊
m
N
ជ m⫽1
UQFT兩 l 典 ⫽2 ⫺N/2 丢
兩0典⫹
兿
n⫽1
共 e i ␲ l n /2
m⫺n
兲兩 1 典 ,
共62兲
where we have used the fact that exp(i␲/2m⫺n )⫽1 for all
integers n⬎m. It follows that we can write the QFT as a
product UQFT⫽U1 ¯UN , where each Um is a factor of the
ជ . Each Um in turn can be further expanded as
product 丢
Um ⫽Wm Vm,m⫹1 ¯Vm,N ,
共63兲
which consists of the two-qubit conditional phase shifts
n
Vm,n ⫽exp共 ⫺i ␻ m,n Zm
1 Z1 兲 ,
共64兲
, followed by a Hadamard W to the mth
with ␻ mn ⫽ ␲ /2
qubit. This algorithm yields the Fourier transform of l with
its qubits reversed, so opposite pairs of qubits must be
swapped in order to complete it: l n ↔l N⫺n⫹1 . Finally, it is
easily shown that the QFT of any superposition
n⫺m
m
2 N ⫺1
UQFT兩 f 典 ⫽UQFT
兺
k⫽0
f 共 k 兲兩 k 典
2 N ⫺1
⫽
兺
k⫽0
兺
l⫽0
F共 l 兲兩 l 典 ⫽ 兩 F 典 ,
共65兲
where F is the Fourier transform of the function f.
The above QFT can be rearranged to an equivalent form
that is somewhat easier to implement by NMR.29,44 This rearrangement is done by inserting the identity I⫽(Wm ) 2 on
the right of each factor Um and bringing the result into the
combined exponential of the conditional phase shifts:
冉
冉
m 2
冊
N
U 共 W 兲 ⫽exp ⫺i
m
兺
␻ m,n Wm Zm1 Wm Zn1 Wm
n⫽m⫹1
N
⫽exp ⫺iXm
1
兺
n⫽m⫹1
冊
␻ m,n Zn1 Wm .
共66兲
Because Wm commutes with Un for all n⬎m, we may slide
all the Hadamards over to the right, obtaining
冉
N
ជ m⫽1
exp ⫺iXm
UQFT⫽ 丢
1
N
兺
n⫽m⫹1
冊
␻ m,n Zn1 W,
共67兲
where W⫽W1 ¯WN . That is to say, the QFT can be implemented by applying Hadamard gates to all the qubits, followed by a sequence of rotations of each qubit m⫽N
⫺1,...,1, conditional on all the qubits n⫽m⫹1,...,N, by the
angles ␻ m,n .
In many cases this rearranged form will be easier and
faster to implement, because the Hadamards can be implemented all together using ‘‘hard’’ pulses 共with a frequency
range that spans all the spins兲. Although the frequency range
of the spins in the molecule used below was too large to
achieve this, given the available rf transmitter power, it is a
description of this rearranged form that is given.
357
B. NMR implementation
The NMR implementation described here uses the three
carbon-13 atoms in alanine 共a common amino acid widely
available with 99% 13C incorporated into it for biomolecular
NMR purposes兲 as its qubits 共see Fig. 7兲.39 The data described here were collected a Bruker AVANCE-300 NMR
spectrometer with a magnetic field strength of about 7.0 T.
The differences in the precession frequencies of the spins
and the scalar coupling constants, all in hertz, were
兩 ␯ 1 ⫺ ␯ 2 兩 ⫽9456.1,
Am. J. Phys., Vol. 70, No. 3, March 2002
兩 ␯ 2 ⫺ ␯ 3 兩 ⫽2594.6,
兩 ␯ 1 ⫺ ␯ 3 兩 ⫽12 050.7,
J 12⫽54.1,
2 N ⫺1
f 共 k 兲 UQFT兩 k 典 ⫽
Fig. 7. Chemical bonds among atoms of alanine in a neutral aqueous solution, with the numbering of the three carbon-13 spins used as the qubits
indicated by superscripts; the hydrogen spins were decoupled throughout the
experiments, and so had no effect on the carbons.
共68兲
J 23⫽35.0,
J 13⫽⫺1.3.
共69兲
Because 兩 J kl Ⰶ 兩 ␯ k ⫺ ␯ l 兩 for all k, l, the weak coupling approximation holds extremely well.
For three qubits Eq. 共67兲 simplifies to
1
2
3
2
2
UQFT⫽e⫺iX1 共 ␲ /2兲共 Z1 ⫹Z1 /2兲 e ⫺iX1 共 ␲ /2兲 Z1 W1 W2 W3 .
共70兲
The Hadamards at the beginning are implemented in the
usual way, while the c-NOT 共␾兲 gates are implemented by
the previously described pulse sequence 关Eq. 共58兲兴. The complete pulse sequence will not be presented here, because it is
largely a concatenation of those already given. It should be
noted, however, that considerable time can be saved by executing each block of c-NOT共␾兲 gates in parallel, that is, by
allowing multiple couplings to evolve simultaneously.44
Also, because J 13 is much smaller than J 12 and J 23, the
c-NOT共␲/2兲 between spins 1 and 3 is more rapidly implemented as a ‘‘relayed gate,’’ which does not use J 13 at all.
This is done by sandwiching a coupling evolution in between
a rf pulse sequence and its inverse sequence, as shown below
共with only the innermost pair of inverse transformations displayed on any one line兲:
2
1 2
2
¯ 共 e ⫺i 共 ␲ /4兲 ␴y e ⫺i 共 ␲ /4兲 ␴z ␴z e i 共 ␲ /4兲 ␴y 兲 ¯
2 3
1 2
2 3
⫽¯ 共 e ⫺i 共 ␲ /4兲 ␴z ␴z e ⫺i 共 ␲ /4兲 ␴z ␴x e i 共 ␲ /4兲 ␴z ␴z 兲 ¯
1
2
1 2 3
1
2
⫽¯ 共 e ⫺i 共 ␲ /4兲共 ␴x ⫹ ␴x 兲 e ⫺i 共 ␲ /4兲 ␴z ␴y ␴z e i 共 ␲ /4兲共 ␴x ⫹ ␴x 兲 兲 ¯
1 2
1 2 3
1 2
1 3
⫽e ⫺i 共 ␲ /4兲 ␴z ␴z e i 共 ␲ /4兲 ␴y ␴z ␴z e i 共 ␲ /4兲 ␴z ␴z ⫽e ⫺i 共 ␲ /4兲 ␴x ␴z .
共71兲
Putting all the pieces together on a single line now shows
that the pulse sequence for this component of the
c-NOT共␲/2兲 between 1 and 3 is
Havel et al.
357
冋
⫺
册 冋
冋
冋
册 冋
册
册 冋 册
册 冋 册
␲ 1 2
␲
␲
␴z ␴z → ⫺ 共 ␴1x ⫹ ␴2x 兲 → ⫺ ␴z2 ␴z3
4
4
4
→ ⫺
→
册 冋
册 冋
␲ 1
␲
␲
␴ → ␴z1 ␴z2 → ␴1y
4 y
4
4
␲ 2 3
␲
␲
␴ ␴ → 共 ␴1x ⫹ ␴2x 兲 → ␴z1 ␴z2 .
4 z z
4
4
共72兲
The negative coupling constants (rotations⬎ ␲ ) are most efficiently implemented by applying an x or y ␲ pulse to one of
the spins involved immediately before and after the coupling
delay 共while refocusing the other two couplings as usual with
a ␲ pulse to the other spin halfway through the coupling
period and at the end兲.
Finally, although one can certainly make sense of the results without so doing, in order to fully implement the QFT it
is necessary to reverse the order of the 共states of the兲 qubits
at the end. In the present case this means swapping qubits 1
and 3, once again without using the small J 13 coupling. It is
easily seen that the Swap13 gate is equal to the composition
Swap12 Swap23 Swap12, and that Swapkl is obtained from
the pulse sequence:
冋
册 冋
冋
冋
册 冋
册 冋
册 冋
␲ k
␲
␲
共 ␴y ⫹ ␴ly 兲 → ␴zk ␴zl → ⫺ 共 ␴ky ⫹ ␴ly 兲
4
4
4
册
册
→
␲ k l
␲
␴z ␴z → 共 ␴kx ⫹ ␴lx 兲
4
4
→
␲ k l
␲
␴z ␴z → ⫺ 共 ␴kx ⫹ ␴lx 兲 .
4
4
册
共73兲
With this last gate, the reader has all the pieces needed to
implement the QFT, except for tomography on the input and
output states. This procedure is a straightforward extension
of that previously given for two spins, and is omitted for the
sake of brevity.
The results of our implementation are shown in Fig. 8,
using the three-dimensional bar graphs of the product operator components introduced previously. The three-spin
pseudopure ensemble was prepared from the thermal equilibrium ensemble by first applying the rf and gradient pulse
sequence
Fig. 8. Plots of the product operator coefficients in the QFT input 共top兲 and
output 共bottom兲 pseudopure ensembles, as reconstructed by tomography.
The labels on the coordinate axes have the same meaning as in Fig. 6. The
input state was chosen to have a periodicity of two, that is, 兩 000典 ⫹ 兩 010典
⫹ 兩 100典 ⫹ 兩 110典 )/2, with theoretical output state 兩 00典 ( 兩 0 典 ⫹ 兩 1 典 )/& following the Swap 共Ref. 13兲. These correspond to the pure state ensembles
X10 X20 Z30 and Z10 Z20 X30 , respectively. The precision of the QFT implementation can be seen in the correlation between product operator amplitudes
obtained by expanding these theoretical expressions and the amplitudes plotted here.
VII. QUANTUM SIMULATION OF A HARMONIC
OSCILLATOR
共74兲
swapping spins 1 and 2 via the above pulse sequence, and
finally applying the two-spin pulse sequence given in Eq.
共48兲 to spins 2 and 3.
358
Am. J. Phys., Vol. 70, No. 3, March 2002
...it does seem to be true that all the various field
theories have the same kind of behavior, and can
be simulated in every way, apparently, with little
latticeworks of spins and other things. It’s been
noted time and time again that all the phenomena
of field theory are well imitated by many phenomena in solid state theory. For example, the
spin waves in a spin lattice imitating BoseHavel et al.
358
⬁
HS⫽
兺
S k 兩 k 典具 k 兩 ,
k⫽0
共76兲
it is clear that we can simulate at most the first 2 N energy
levels with N qubits. The obvious, though by no means only,
mapping to use is
⌽
Fig. 9. Diagram illustrating relations between the simulated and physical
Hamiltonians induced by the encoding mapping ⌽ 共see the text兲.
N
k
共78兲
k
where ␦ nk 苸 兵 0,1其 (n⫽1,...,N) are the bits in the binary expansion of the integer k. On expanding this latter product and
regrouping, we obtain
2 N ⫺1
兺
k⫽0
H̄P⫽
Much of the interest in quantum information processing
stems from the discovery of quantum algorithms, for which
the time required to solve certain types of discrete problems
共for example, integer factorization41兲 scales better with their
input size than any known classical algorithm. The fact remains that any quantum information processor, even one
based on a finite-dimensional Hilbert space, evolves continuously under Schrödinger’s equation and so can also be
viewed as an analog computer. Analog encoding of physical
systems, that is, direct mappings of their degrees of freedom
onto those of the processor, generally require far fewer qubits than discrete encodings over a grid in their parameter
space. Thus the first practical applications of QIP are likely
to be based on such direct analog encodings.
Quantum information processing based on an analog encoding is known as quantum simulation. Feynman’s early
insight regarding the possibility of a ‘‘universal quantum
simulator,’’ able to simulate any other quantum system far
more efficiently than is possible even on an analog classical
computer 共see the above quote45兲, was later validated, in general terms, by showing that a quantum computer would also
have this capability.46 The first concrete realizations of specific quantum simulations were performed using NMR a few
years later.47 In the following, we present a protocol that
allows one to simulate a general quantum harmonic oscillator by NMR, and hence any bosonic quantum field as well.
The general idea is illustrated by the diagram in Fig. 9.
The state of the simulated system is denoted by 兩S典 and is to
be calculated after the system has evolved under the simulated Hamiltonian HS for a time T. A linear mapping ⌽ is
chosen, which carries this system onto an actual physical
system used, 兩P典. It is assumed that this latter system can be
subjected to a limited set of unitary transformations Vk at
will, which are alternated with periods during which the system evolves under its natural internal Hamiltonian HP . The
goal is to find a sequence 兵 t k ;Vk 其 such that the resulting
‘‘average’’ Hamiltonian H̄P and simulation time t T , when
mapped back onto the simulated system, gives the product
THS , that is,
共75兲
Assuming that the Hamiltonian of the simulated system is
given in diagonal form,
Am. J. Phys., Vol. 70, No. 3, March 2002
1
兩 k 典具 k 兩 → Z␦ 1 ¯Z␦ N ⬅Zk ,
A. Background and theory
359
共77兲
or, in terms of density operators
⌽
particles in field theory. I therefore believe it’s
true that with a suitable class of quantum machines you could imitate any quantum system,
including the physical world.
R. P. Feynman, ‘‘Simulating Physics with Computers,’’ Int. J. Theor. Phys. 21, 467–488 (1982)
THS⫽t T ⌽ ⫺1 HP⌽.
兩 k 典 → 兩 ␦ 1k 典 ¯ 兩 ␦ Nk 典 ,
2 N ⫺1
S l Zk ⫽
兺
l⫽0
N
1
s l 共 ␴z1 兲 ␦ l ¯ 共 ␴zN 兲 ␦ l ,
共79兲
where the s l are coefficients to be determined and the zeroth
power ( ␴zn ) 0 ⫽I, the identity.
N
2 ⫺1
Zk into the
On inserting the 2 N ⫻2 N identity I⫽ 兺 k⫽0
n
right-hand side of Eq. 共79兲 and using the relation ␴zn Z␦ n
n
k
⫽(⫺1) ␦ k Z␦ n , we obtain
n
k
2 N⫺1
H̄P⫽H̄P
兺
k⫽0
Zk
2 N ⫺1 2 N ⫺1
⫽
兺 兺
k⫽0 l⫽0
1 N
N 1
s l 共 ⫺1 兲 ␦ k ␦ l ¯ 共 ⫺1 兲 ␦ k ␦ l Zk .
共80兲
On comparing this result with Eq. 共79兲, we see that
2 N ⫺1
S k⫽
兺
l⫽0
s l 共 ⫺1 兲
k䉺˜l
2 N ⫺1
⫽
兺
l⫽0
s˜l 共 ⫺1 兲 k䉺l ,
共81兲
where k䉺l is the bitwise inner product of the integers k and
N
␦ nl 2 n⫺1 is the result of reversing the order of
l and l̃ ⫽ 兺 n⫽1
the bits in l. It follows that the coefficients s l of the ␴z
product operators in HP are 共up to order and a factor of 2 N/2兲
the Walsh–Hadamard transform of the energy levels S k in
HS 关see Eq. 共14兲兴. In the case of a harmonic oscillator truncated after the first 2 N energy levels, we have
S k ⫽ 12 ⍀ 共 2k⫹1 兲
共 k⫽0,...,2 N ⫺1 兲
共82兲
from which it follows on taking the Walsh–Hadamard transform that the Hamiltonian can be written as
HS⫽ 21 ⍀ 共 2 N ⫺ ␴z1 ⫺2 ␴z2 ⫺2 2 ␴z3 ⫺¯⫺2 N⫺1 ␴zN 兲 .
共83兲
This encoding is only logarithmic in the number of qubits
required per energy level, although the range of their Zeeman
energies scale linearly. It has the significant advantage of
involving no multi-qubit product operators, which are relatively complicated to implement by NMR and require the use
of scalar couplings 共these are generally much smaller, and
hence slower, than the Zeeman energy differences兲. Nevertheless, in the following simulation of the first four levels of
a harmonic oscillator on a two-spin system, it proved more
Havel et al.
359
Because this propagator commutes with the propagator of
the last delay,
exp共 ⫺i 共 ␶ 1 /2⫹ ␶ 2 兲共 KI⫹J ␴z1 兲 ␴z2 兲 ,
共90兲
the net result of the pulse sequence is
exp共 ⫺i 共 ␶ 2 KI⫹ 共 ␶ 1 ⫹ ␶ 2 兲 ␴z1 兲 ␴z2 兲 .
Fig. 10. Chemical diagram of 2,3-dibromo-thiophene, with the numbering
of hydrogen used in the harmonic oscillator simulations indicated by superscripts. As is often done in such diagrams, the carbon atoms have not been
labeled.
convenient to use a Grey, rather than a binary, encoding, as
will now be described.
B. NMR implementation
The NMR implementation described here makes use of
the two hydrogen atoms in 2, 3-dibromo-thiophene 共see Fig.
10兲.47 The Grey code ⌽ used maps the first four energy levels of the oscillator onto the Zeeman energy levels of this
spin system as follows:
⌽
⌽
兩 0 典 ↔ 兩 00典
兩 2 典 ↔ 兩 11典
⌽
⌽
兩 1 典 ↔ 兩 10典
共84兲
兩 3 典 ↔ 兩 01典 .
The Walsh–Hadamard transform maps this ordering of the
energy levels to the following effective Hamiltonian:
H̄S⫽T⍀ 共 I⫺ 共 I⫹ ␴z1 /2兲 ␴z2 兲 .
HP⫽ ␲ 共 KI⫹J ␴z1 /2兲 ␴z2 ⬇ ␲ 共 226.0I⫹5.7␴z1 /2兲 ␴z2 ,
册
冋
␲
␲
⫺ 共 ␴1y ⫹ ␴2y 兲 → 关共 ␶ 1 /2兲 HP兴 → 共 ␴1y ⫹ ␴2y 兲
2
2
册
→ 关共 ␶ 1 /2⫹ ␶ 2 兲 HP兴 , 共87兲
where the times ␶ 1 , ␶ 2 are determined below. To see how
this pulse sequence works, we first note that the propagators
for ␲ rotations about the y axis are simply
冉
exp ⫾i
冊
␲ 1
共 ␴ ⫹ ␴2y 兲 ⫽⫾i ␴1y ␴2y .
2 y
共88兲
Thus the two ␲ pulses and intervening delay of the sequence
correspond to the propagator
⫺i ␴1y ␴2y exp共 ⫺i 共 ␶ 1 /2兲 HP兲 i ␴1y ␴2y
⫽exp共 ␴1y ␴2y 共 ⫺i 共 ␶ 1 /2兲共 KI⫹J ␴z1 兲 ␴z2 兲 ␴1y ␴2y 兲
⫽exp共 i 共 ␶ 1 /2兲共 KI⫺J ␴z1 兲 ␴z2 兲 .
360
冋
册 冋
Am. J. Phys., Vol. 70, No. 3, March 2002
共89兲
册 冋
␲ 1
␲
␲
共 ␴ ⫹ ␴2x 兲 → ␴z1 ␴z2 → ⫺ 共 ␴1x ⫹ ␴2x 兲
4 x
4
4
册
共92兲
into the superposition ( 兩 00典 ⫹i 兩 11典 )/&↔( 兩 0 典 ⫹i 兩 2 典 )/&, as
the reader can readily verify. Up to an overall phase factor,
this superposition should evolve as 兩 0 典 ⫹ie ⫺i⍀T 兩 2 典 . Finally,
the full superposition state
1
2
共 兩 00典 ⫹ 兩 01典 ⫹ 兩 11典 ⫹ 兩 10典 )↔ 21 共 兩 0 典 ⫹ 兩 1 典 ⫹ 兩 2 典 ⫹ 兩 3 典 )
共93兲
is prepared from 兩00典 by a simple ␲/2 y pulse, and should
evolve as ( 兩 0 典 ⫹e ⫺i⍀T/2兩 1 典 ⫹e ⫺i⍀T 兩 2 典 ⫹e ⫺i⍀3T/2兩 3 典 )/2.
These three states will henceforth be denoted as A, B, and C,
respectively.
The traceless part of the time-dependent two-spin density
matrices for each of these states is proportional to
␳ˆ A⫽
␳ˆ B⫽
␳ˆ C⫽
冋
冋
冋
3
1 0
4 0
共86兲
where J⫽5.7 Hz is the scalar coupling constant and K/2
⫽( ␻ 20 ⫺ ␻ 10 )/(2 ␲ )⫽113 Hz in the 9.6 T magnetic field used
to obtain the results below.
The pulse sequence used to convert the physical Hamiltonian HP into the desired average Hamiltonian H̄P is
冋
The argument of the exponential equals the desired average
Hamiltonian H̄P when ␶ 2 ⫽T⍀/K and ␶ 1 ⫽T⍀/(2J)⫺ ␶ 2 .
The experimental validation of this simulation protocol is
most convincingly carried out on pseudopure ensembles. A
pseudopure ensemble for the state 兩00典 may be prepared as
described in Eq. 共48兲. This state corresponds to the ground
state 兩0典 of the harmonic oscillator versus the encoding ⌽,
and so should not evolve under the simulated Hamiltonian. It
will be converted by the pulse sequence
共85兲
On choosing a frame that rotates at the Zeeman precession
rate of the first spin, the physical Hamiltonian of this system
becomes
共91兲
1
4
0
0
0
⫺1
0
0
0
⫺1
0
0
0
⫺1
0
册
,
1
0
0
2ie i⍀T
0
⫺1
0
0
0
0
⫺1
0
⫺2ie ⫺i⍀T
0
0
1
0
册
册
共94兲
,
e 3i⍀T
e i⍀T
e 2i⍀T
1 e ⫺3i⍀T
4 e ⫺i⍀T
0
e ⫺2i⍀T
e ⫺i⍀T
e 2i⍀T
0
e i⍀T
e ⫺2i⍀T
e i⍀T
e ⫺i⍀T
0
.
The single quantum coherences in these matrices 关␳ i j with
(i, j)⫽(0,1),(0,2),(1,3),(2,3)兴 correspond to the four peaks
in the spectrum. For ␳C , one of these four peaks will oscillate with T at three times the rate of the other three peaks.
For ␳B , the spectrum will contain no peaks, so that a ␲/2
pulse selective for one of the two spins will be needed in
order to convert the e i⍀T oscillation into an observable single
quantum coherence. For ␳A , of course, the spectra seen on
applying selective ␲/2 pulses will exhibit no oscillations in
their peaks.
The results of these experiments are shown in Fig. 11,
which exhibits the predicted oscillations quite nicely. It is
Havel et al.
360
The three examples given in this paper are only a sample
of many similar exercises that have been carried out in our
laboratory in recent years. These include also a variety of
quantum logic gates,33,49 quantum error correction,50,51 quantum simulation,52,53 and quantum eraser demonstrations.54 In
due course we expect a quantum cookbook to evolve from
this work, which can be used in an undergraduate laboratory
course on quantum mechanics. Regardless of scalability
issues,10 we believe that the educational potential of QIP by
NMR guarantees it a place in every physics department.
ACKNOWLEDGMENTS
This work was made possible by NSF Grant No. EEC0085557 and ARO Grant Nos. DAAD19-01-1-0678 and
DAAD19-01-1-0519. We thank L. Viola, R. Laflamme, and
J.-P. Paz for useful discussions.
Fig. 11. Signed heights of peaks in NMR spectra revealing oscillations in
coherences obtained by simulation of a quantum harmonic oscillator for
varying times T. 共A兲 Simulation of ground state followed by nonselective
␲/2 pulse. 共B兲 Simulation of superposition of ground and second excited
state, following selective ␲/2 pulse to transfer coherence to a singlequantum antiphase state. 共C1兲 Simulation of uniform superposition over first
four energy levels, recording heights of peaks exhibiting single-quantum
oscillation. 共C2兲 Simulation of uniform superposition over first four energy
levels, recording height of peak exhibiting triple-quantum oscillation 共reproduced from Ref. 47 with permission from the American Physical Society兲.
noteworthy that, in a two-spin system like that used here, the
natural oscillations of the coherences go only as high as
twice the base frequency. Thus the observation of a term at
three times the lowest frequency is entirely a consequence of
our artificial Hamiltonian H̄P . Nevertheless, even the triple
quantum term is oscillating at less than the base frequency of
the natural Hamiltonian HP . This slowing down is part of
the overhead associated with averaging the natural Hamiltonian with those of the rf pulses. Although Eq. 共83兲 in principle leads to a recipe for generating a pulse sequence for an
N-spin system which simulates with first 2 N energy levels of
a quantum harmonic oscillator, it is not efficient because the
ratio of the largest to the smallest frequencies in H̄P decreases exponentially with the number of qubits N. Even
without this slowing down, however, before one could claim
that such a recipe broke an exponential-time computational
barrier, one would also have to show that the pulses required
did not have to become exponentially more precise, that is,
that quantum error correction extends to such analog
computations.48
VIII. CLOSING REMARKS
The above examples should be sufficient to show that:
• quantum information processing provides an excellent
milieu in which to teach the basic principles of quantum
mechanics, without being distracted by the mathematical
challenges of infinite dimensional spaces.
• NMR spectroscopy provides a concrete, convenient, and
highly complementary illustration of these principles, in that
the design and execution of the corresponding NMR experiments require the students to think things through and verify
them for themselves.
361
Am. J. Phys., Vol. 70, No. 3, March 2002
a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
1
D. Z. Albert, Quantum Mechanics and Experience 共Harvard U.P., Cambridge, MA, 1992兲.
2
I. L. Chuang, N. Gershenfeld, M. G. Kubinec, and D. W. Leung, ‘‘Bulk
quantum computation with nuclear magnetic resonance: Theory and experiment,’’ Proc. R. Soc. London, Ser. A 454, 447– 467 共1998兲.
3
D. G. Cory, M. D. Price, and T. F. Havel, ‘‘Nuclear magnetic resonance
spectroscopy: An experimentally accessible paradigm for quantum computing,’’ Physica D 120, 82–101 共1998兲.
4
J. A. Jones, M. Mosca, and R. H. Hansen, ‘‘Implementation of a quantum
search algorithm on a quantum computer,’’ Nature 共London兲 393, 344 –
346 共1998兲.
5
Phys. World, special issue on quantum information 共March, 1998兲; available on-line at http://physicsweb.org/article/world/11/3/8.
6
C. H. Bennett and D. P. DiVincenzo, ‘‘Quantum information and computation,’’ Nature 共London兲 404, 247–255 共2000兲.
7
S. Haroche and J.-M. Raimond, ‘‘Quantum computing: Dream or nightmare?,’’ Phys. Today 49 共8兲, 51–52 共1996兲.
8
S. Lloyd, ‘‘Quantum-mechanical computers,’’ Sci. Am. 273, 140–145 共October, 1995兲.
9
C. P. Williams and S. H. Clearwater, Ultimate Zero and One: Computing
at the Quantum Frontier 共Copernicus Books, Springer-Verlag, New York,
1999兲.
10
W. S. Warren, ‘‘The usefulness of NMR quantum computing,’’ Science
277, 1688 –1689 共1997兲; see also response by N. Gershenfeld and I.
Chuang, ibid. 277, 1689–1690 共1997兲.
11
S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden, S. Popescu, and R.
Schack, ‘‘Separability of very noisy mixed states and implications for
NMR quantum computing,’’ Phys. Rev. Lett. 83, 1054 –1057 共1999兲.
12
R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on
Physics 共Addison–Wesley, Reading, MA, 1965兲, Vols. I–III.
13
D. G. Cory et al., ‘‘NMR based quantum information processing,’’ Prog.
Phys. 48, 875–907 共2000兲.
14
H. K. Cummins and J. A. Jones, ‘‘Nuclear magnetic resonance: A quantum
technology for computation and spectroscopy,’’ Contemp. Phys. 41, 383–
399 共2000兲.
15
N. A. Gershenfeld and I. L. Chuang, ‘‘Quantum computing with molecules,’’ Sci. Am. 278, 66 –71 共June, 1998兲.
16
T. F. Havel, S. S. Somaroo, C.-H. Tseng, and D. G. Cory, ‘‘Principles and
demonstrations of quantum information processing by NMR spectroscopy,’’ in Applicable Algebra in Engineering, Communications and Computing, edited by T. Beth and M. Grassl 共Springer-Verlag, Berlin, 2000兲,
Vol. 10, pp. 339–374. See also quant-ph/9812086.
17
C. H. Bennett and R. Landauer, ‘‘The fundamental physical limits of computation,’’ Sci. Am. 253, 38 – 46 共July, 1985兲.
18
D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum
Information 共Springer-Verlag, Berlin, 2001兲.
19
D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H. D. Zeh,
Decoherence and the Appearance of a Classical World in Quantum Theory
共Springer-Verlag, Berlin, 1996兲.
20
G. Mahler and V. A. Weberruss, Quantum Networks 共Springer-Verlag, Berlin, 1998兲, 2nd ed.
Havel et al.
361
A. Peres, Quantum Theory: Concepts and Methods 共Kluwer Academic,
Amsterdam, 1993兲.
22
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum
Information 共Cambridge U.P., Cambridge, 2000兲.
23
K. Beauchamp, Applications of Walsh and Related Functions: With an
Introduction to Sequency Theory 共Academic, Englewood Cliffs, NJ, 1984兲.
24
D. Deutsch and R. Jozsa, ‘‘Rapid solution of problems by quantum computation,’’ Proc. R. Soc. London, Ser. A 439, 553–558 共1992兲.
25
A. Ekert and R. Jozsa, ‘‘Quantum algorithms: Entanglement enhanced
information processing,’’ Philos. Trans. R. Soc. London, Ser. A 356, 1769–
1782 共1998兲.
26
R. Jozsa, ‘‘Quantum algorithms and the Fourier transform,’’ Proc. R. Soc.
London, Ser. A 454, 323–337 共1998兲.
27
R. Jozsa, ‘‘Quantum factoring, discrete logarithms, and the hidden subgroup problem,’’ Comput. Sci. Eng. 3, 34 – 43 共2001兲.
28
R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions 共Oxford U.P., Oxford, 1987兲.
29
S. S. Somaroo, D. G. Cory, and T. F. Havel, ‘‘Expressing the operations of
quantum computing in multiparticle geometric algebra,’’ Phys. Lett. A 240,
1–7 共1998兲.
30
O. W. Sörensen, G. W. Eich, M. H. Levitt, G. Bodenhausen, and R. R.
Ernst, ‘‘Product operator formalism for the description of NMR pulse experiments,’’ Prog. Nucl. Magn. Reson. Spectrosc. 16, 163–192 共1983兲.
31
T. F. Havel, D. G. Cory, S. S. Somaroo, and C.-H. Tseng, ‘‘Geometric
algebra methods in quantum information processing by NMR spectroscopy,’’ in Advances in Geometric Algebra with Applications, edited by E.
Corrochano and G. Sobczyk 共Birkhauser, Boston, 2000兲.
32
T. F. Havel and C. Doran, ‘‘Geometric algebra in quantum information
processing,’’ Contemporary Math. 共in press兲 共see LANL preprint
quant-ph/0004031兲.
33
M. D. Price, S. S. Somaroo, A. E. Dunlop, T. F. Havel, and D. G. Cory,
‘‘Generalized (controlled) n -NOT quantum logic gates,’’ Phys. Rev. A 60,
2777–2780 共1999兲.
34
J. J. Sakurai, Modern Quantum Mechanics 共Addison–Wesley, Reading,
MA, 1994兲, revised edition.
35
W. H. Zurek, ‘‘Decoherence and the transition from quantum to classical,’’
Phys. Today 44 共10兲, 36 – 44 共1991兲.
36
Y. Sharf, T. F. Havel, and D. G. Cory, ‘‘Spatially encoded psuedo-pure
states for NMR quantum information processing,’’ Phys. Rev. A 62,
052314 共2000兲.
37
R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, ‘‘Quantum algorithms
revisited,’’ Proc. R. Soc. London, Ser. A 454, 339–354 共1998兲.
38
M. Mosca and A. Ekert, ‘‘The hidden subgroup problem and eigenvalue
estimation on a quantum computer,’’ in Proceedings of the NASA International Conference on Quantum Computers and Quantum Communications
(QCQS ’98), edited by C. P. Williams, Lecture Notes in Computer Science
Vol. 1509 共Springer-Verlag, Berlin, 1999兲.
21
362
Am. J. Phys., Vol. 70, No. 3, March 2002
39
Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, S. Lloyd, and D. G. Cory,
‘‘Implementation of the quantum Fourier transform,’’ Phys. Rev. Lett. 86,
1889–1891 共2001兲.
40
N. D. Mermin, ‘‘Notes for physicists of the theory of quantum computation,’’ 1999, available from http://www.lassp.cornell.edu/lassp – data/
NMermin.html.
41
P. W. Shor, ‘‘Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,’’ SIAM J. Comput. 26, 1484 –
1509 共1997兲.
42
R. Schack, ‘‘Using a quantum computer to investigate quantum chaos,’’
Phys. Rev. A 57, 1634 –1635 共1998兲.
43
Y. S. Weinstein, S. Lloyd, J. V. Emerson, and D. G. Cory, ‘‘Experimental
implementation of the quantum baker’s map,’’ Phys. Rev. Lett. 共submitted兲.
44
M. D. Price, T. F. Havel, and D. G. Cory, ‘‘Multiqubit logic gates in NMR
quantum computing,’’ New J. Phys. 2, 10 共2000兲, or http://njp.org.
45
R. P. Feynman, ‘‘Simulating physics with computers,’’ Int. J. Theor. Phys.
21, 467– 488 共1982兲.
46
S. Lloyd, ‘‘Universal quantum simulator,’’ Science 273, 1073–1078
共1996兲.
47
S. S. Somaroo, C.-H. Tseng, T. F. Havel, R. Laflamme, and D. G. Cory,
‘‘Quantum simulations on a quantum computer,’’ Phys. Rev. Lett. 82,
5381–5384 共1999兲.
48
A. M. Steane, ‘‘Introduction to quantum error correction,’’ Philos. Trans.
R. Soc. London, Ser. A 456, 1739–1756 共1998兲.
49
M. D. Price, S. S. Somaroo, C.-H. Tseng, J. C. Gore, A. F. Fahmy, T. F.
Havel, and D. G. Cory, ‘‘Construction and implementation of NMR quantum logic gates for two-spin systems,’’ J. Magn. Reson. 140, 371–378
共1999兲.
50
Y. Sharf, D. G. Cory, T. F. Havel, S. S. Somaroo, E. Knill, R. Laflamme,
and W. H. Zurek, ‘‘A study of quantum error correction by geometric
algebra and liquid-state NMR spectroscopy,’’ Mol. Phys. 98, 1347–1363
共2000兲.
51
Y. Sharf, T. F. Havel, and D. G. Cory, ‘‘Quantum codes for controlling
coherent evolution,’’ J. Chem. Phys. 113, 10878 –10885 共2000兲.
52
C.-H. Tseng, S. Somaroo, Y. Sharf, E. Knill, R. Laflamme, T. F. Havel, and
D. G. Cory, ‘‘Quantum simulation with natural decoherence,’’ Phys. Rev.
A 62, 032309 共2000兲.
53
C. H. Tseng, S. Somaroo, Y. Sharf, E. Knill, R. Laflamme, T. F. Havel, and
D. G. Cory, ‘‘Quantum simulation of a three-body interaction Hamiltonian
on an NMR quantum computer,’’ Phys. Rev. A 61, 012302 共2000兲.
54
G. Teklemariam, E. M. Fortunato, M. A. Pravia, T. F. Havel, and D. G.
Cory, ‘‘NMR analog of the quantum disentanglement eraser,’’ Phys. Rev.
Lett. 86, 5845–5849 共2001兲.
Havel et al.
362