Quantum information processing by nuclear magnetic resonance spectroscopy

Transcription

1 Quantum information processing by nuclear magnetic resonance spectroscopy T. F. Havel a) and D. G. Cory Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts S. Lloyd Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts N. Boulant, E. M. Fortunato, and M. A. Pravia Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts G. Teklemariam Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts Y. S. Weinstein Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts A. Bhattacharyya and J. Hou Research Science Institute, Center for Excellence in Education, 140 Park Street SE, Vienna, Virginia Received 26 August 2001; accepted 5 December 2001 Nuclear magnetic resonance NMR 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 see the last two authors. 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 American Association of Physics Teachers. DOI: / 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 NMR 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 intrinsic angular momenta 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 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 QIP. 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 molecules, wherein the states of their nuclear spins over the ensemble are a little short of 345 Am. J. Phys. 70 3, March American Association of Physics Teachers 345

2 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. This point was also stressed by Feynman et al. 12 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 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. For additional accounts of QIP by NMR, the reader is referred to Refs 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 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 1 or down 0. 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 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 usually just one or two, 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 see Sec. III. In keeping with their usage, the basis states for the twodimensional Hilbert space of a qubit are denoted in Dirac notation by 0 and 1. General states are superpositions of these basis states: where 0, 1 are complex numbers with The basis states of an array of qubits are variously denoted by 1 2 N 1 2 N 1 2 N, 1 where n 0,1 (n1,...,n) and denotes the tensor or Kronecker product: ) ) 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 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 346

3 The matrix N is clearly both Hermitian and unitary, that is, N N and NN N 2 1 the 22 identity, from which it follows that N10 as well. Indeed, the Hamiltonia H N that, when substituted into Schrödinger s equation, i/th N, 4 for a time t1, subjects the initial state t0 to a NOT, is H N Thus we may express N as the exponential of this Hamiltonian, that is, expih N / 1 ih N / 2H 1 N / 2 N, 6 where i 2 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 or even invertible. 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 1, but does nothing if the other is in the state 0. Specifically, the c-not gate the flips the first left-hand qubit conditional on the state of the second right-hand qubit is given by N 12, N , N , N , N This operation may also be viewed as replacing the first leftmost qubit by its XOR exclusive OR with the second. The unitary matrix for this c-not is simply N , which corresponds to the Hamiltonian H N Note also that (N 12 ) the 44 identity. 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: W0 1 & 01), W1 1 01). 10 & The unitary matrix for this operation is W & 1 1, and its generating Hamiltonian is H W & &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 k1 N k by k, where n k denotes the nth bit in the binary expansion of the integer k, n1,...,n, k0,...,2 N 1 for example, In addition, we will need to extend our previous definition of the tensor product of state vectors k to the tensor product of operators on them, which is also known as the Kronecker product of their representative matrices, that is, 11B a 12B A Ba a 21 B a 22 B ] ] a 11b 11 a 11b 12 a 11 b 21 a 11 b 22 ] ]. 13 Then it can be shown by a straightforward application of these definitions that W N kw Wk 2 N 1 l0 l0 N n11 n n k l 2 N/2 2 N 1 1 kl l 2 N/2 l, 14 N where kl n0 n k n l mod 2. Like the Fourier transform, 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 W 1 k ) W N N k ) n1 01 n k 1) W N k, 15 where denotes the Kronecker product of the factors ordered 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 N ) 1 W 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, 347 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 347

4 2 N 1 2 N 1 U k0 k k k0 k Uk. 16 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 N U cl W N, where U cl is a unitary classical logic gate. Such nonlocal operations are what allow QIP to perform communication and computation feats far beyond what is possible classically III. STATES AND OPERATORS IN NMR SPECTROSCOPY 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 /2 spin 1/2. Such a nucleus, itself called a spin, constitutes an ideal qubit, because the component of its angular momenta measured along any given direction for example, by a Stern Gerlach apparatus is always /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 0 be the state obtained following a measurement giving /2 in the z direction, and 1 that following a measurement of /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 /2. To see how this works, let us write the spin s state vector in terms of the polar angles of n, 0 1 cos/2ei/2 sin/2e i/2. 17 It is readily verified that 1, so that this state vector is properly normalized, and that arg( 0 )arg( 1 )0 where arg(aib)arctan(b/a), so that the physically irrelevant total phase of the state vector is eliminated by this parametrization. In the 0, 1 basis specified above, the operators for the x, y, and z components of the spin s angular momentum are just /2 times the Pauli matrices: x , y i i, 18 i 0 z It follows that the expectation values of the angular momenta along the coordinate axes are /2 times: x cos/2sin/2e i e i sincos, y cos/2sin/2ie i e i sinsin, 19 z cos 2 /2sin 2 /2cos. 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 /2 times n sincos x sinsin y cos z. 20 It is readily verified that the expectation value of the angular momentum along n is (/2) n (/2)n 2 /2, and because measurements of the component of the angular momentum along any axis are quantized at /2, the outcome of such a measurement is always /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 1, meaning we measure angular momentum in multiples of and energy in radians per unit time. Consider the Hamiltonian for the Zeeman interaction, H Z (/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 te ih Z t 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 ih Z expib 0t/2 0 t 0 expib 0 t/2 1 cosb 0 t/2i z sinb 0 t/2. 22 This same formula can be shown to also hold for a field along x and y. Letting 0 B 0, it follows that t cosei 0 t/2 sine i 0 t/2, 23 meaning that the unit vector n precesses in a left-hand sense for 0 about the z axis at a rate of 0 radians per unit time, in accordance with our earlier claims. 348 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 348

5 For multiple spins, each with its own precession frequency 0 n due to their differing electronic environments within the molecule, this relation generalizes straightforwardly to e i 1 0 z t/2 1 ) e i N 0 z t/2 N ) e i 1 N 0 z t/2 e i 0 z t/2 1 N. 24 At this point it is convenient to introduce a notational shortcut, namely 25 When it is necessary to distinguish such a superscript from a power, parentheses will be used, for example, ( n z ) 2 1. Because of the general formula for all 1mn N the Zeeman propagator in Eq. 24 can now be written as e i 1 N 0 z t/2 e i 0 z t/2 e i z t/2 e i N N 0 z t/2 26 e i 1 1 N N 0 z 0 z t/2, 27 where in the last equality we have also used the fact that all the z n commute and operate on the same tensor product of Hilbert spaces. It follows that the multispin Zeeman Hamiltonian is H Z z 1 0 N z N. In the case of two spins 1/2, for example, we obtain 28 expih Z texp i 0 1 t/2 0 0 i 1 0 t/2 exp i 0 2 t/2 0 0 i 2 0 t/2 ei t/ e i t/ e i t/ e i t/2. 29 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 H Z z 1 0 N z N, 30 with n 0 n 0 0 for 1nN. The use of a rotating frame also makes it relatively easy to explain the action of a rf radio-frequency 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 magnetic dipole of the spin rotates at a constant velocity n 1, determined by the power of the pulse, around the y axis see Fig. 1. Thus after a time t /(2 n 1 ), 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 n 0 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 n 0 without ambiguity. 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 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:. 31 P k ekh Z k/k B T Z eq T 2 N 1 Z eq l0 e lh Z l/k B Here, we are using the ket k to denote an N-spin state in which spins parallel to B 0 along the z axis correspond to the bits k n 0, and antiparallel to k n 1, in accord with our intention to store binary information in these states. It is easily seen that these states are eigenvectors of H Z. 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 liquid-state NMR to a matter of seconds, 349 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 349

6 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 vertical 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 primes, 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 important issue will not be further discussed here. It is nevertheless essential to understand how the accessible measurable 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 k0 P k kk, and more generally, given any probability distribution p k 0( k p k 1) over an ensemble of quantum states k, itis k p k k k. 32 It follows easily that the ensemble average of the expectation value of any observable M is trm k p k tr k k M k p k k M k, 33 and that the time evolution of this ensemble average, and hence the density operator itself, under any Hamiltonian H is given by Mt k p k k e iht Me iht k tre iht e iht MtrtM. 34 n At the temperatures characteristic of liquid samples, 0 k B T, so that the Boltzmann factors P k are essentially equal to their linear approximations, 2 N 1 1 eq Z eq l0 1lH Z l/k B Tll Z 1 eq IH Z /k B T, 35 where I 1 1 and the partition function Z eq tr(i H Z /(k B T))2 N because the zero of energy is chosen so that tr(h Z )0. Given that the observables for the transverse Fig. 2. Energy level diagram for two coupled spins in a magnetic field above, 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 EE 1 E 2. Assuming a high-temperature Boltzmann distribution of energy level populations, the spectrum, obtained by rotating the spins into the transverse xy plane with a /2 rf pulse and Fourier transforming the resulting signal below, 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 0 or antiparallel 1 to the field, respectively. 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 0 n /(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 homonuclear equilibrium density operator: eq 1 2 z 1 z N. 36 In the case of two spins, for example, the equilibrium density operator becomes eq , 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 cf. Fig. 2. Together with operator- 350 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 350

7 valued expansions of the propagators, as in Eq. 22, 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 It can in fact be viewed as a special case of the geometric or Clifford algebras that have been shown to provide a unified mathematical language for much of physics. 31,32 IV. HOW TO DEMONSTRATE QIP BY LIQUID- STATE 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 or y axis of the rotating frame, thereby converting 0 1 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 n 0, 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 n 0 /2, so that only spins with precession frequencies in this range are significantly 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. 12 is represented in product operator notation by H W 8 & 1 x z. 38 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 t1 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: e ih W 1 & x z. 39 Rotations about such oblique axes are most conveniently implemented by a sequence of pulses affecting rotations about the transverse x and y axes, for example, e i y /8 e i x /2 e i y /8 expe i y /8 i x /2e i y /8 e i x z /&/2. 40 Implementation of the c-not gate requires an interaction between spins, but as mentioned above 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 coupling constant for this interaction is J mn m 0 n 0 /(2), the scalar coupling Hamiltonian for a pair of spins m, n assumes the following simple form: H mn J J mn m z n z /2. 41 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 0, and the other peak due to those for which the other spin is antiparallel 1. The time has come to present our first NMR pulse sequence, this being one that implements the c-not gate N 12 in a simple two-spin system: 4 1. x y 4 z 1 z 2 In this notation, (angle/2) spin axis indicates a rf pulse implementing the rotation by the given angle about the subscript axis and selective for the superscript spins. A bracketed expression of the form (angle/2) 1 z 2 z, on the other hand, refers to a time delay waiting period 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 H z 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 (/4) 1 z 2 z by the subsequence 8 z 1 z 2 2 x 1 2 x 2 x 1 2 x, 43 8 z 1 z 2 where the sum x 1 x 2 indicates a rotation of both spins. The easiest way to understand the overall sequence is via a vector diagram Fig. 3, in which the magnetization vectors of the first spin, in each of the two subensembles defined by the orientation 0 or 1 of the second, are plotted in a 351 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 351

8 Fig. 3. Vector diagrams showing the magnetization components of spin 1 in those molecules wherein spin 2 is parallel 0 and antiparallel 1 to the static magnetic field B 0 after each pulse or delay in the sequence which implements the c-not gate N 12 see text. frame rotating at the first spin s Zeeman precession rate at each step along the way. The initial /2 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 /2 pulse rotates one back to the z axis and the other to the z axis. The accompanying pulse with no delay 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. 9, that is, H N 1/2X 1 1 Z 1 2, 44 where Z ( 1 (1) z ) and X ( 1 (1) x ) ( 0,1). Because these operators are easily seen to be idempotent meaning that each squares to itself and Z 1 2 commutes with X 1 1, up to an irrelevant overall phase factor, the Taylor series for the exponential of this Hamiltonian collapses to the following simple form: e ih N 1/2 Z 2 0 e ix 1 1 Z 2 1 Z x Z Alternatively, we can fully expand the generator and use the commutivity of all the terms to obtain up to a phase factor as before e ih N 1/2 e i 2 z /4 e i 1 x /4 e i 1 2 x z /4 e i 2 z /4 e i 1 x /4 e i 1 y /4 e i 1 2 z z /4 e i 1 y /4 e i 1 2 z z /4 e i 1 x /4 e i 1 2 z z /4 e i 1 y /4. 46 Reading this product of propagators in temporal order from right-to-left 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 that is, unitary process. Instead, an incoherent process is needed, by which the distribution of eigenvalues can be evened out that is, its von Neumann entropy increased 34. 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 providing the molecules have not moved a significant distance during the time between the two gradients. 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 z 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 cf. Eq. 29: 352 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 352

9 zexpi z H Z z expi z H Z z 00 e i z 2 0 z 01 e i z 1 0 z 02 e i z z 03 e i z 2 0 z 01 * 11 e i z z 12 e z 1 0 z 13 e i z 1 0 z 02 * e i z z 12 * 22 e i z 2 0 z e i z z 03 * e i z 1 0 z 13 * e i z 2 0 z 23 * 33 To a good approximation, the rate at which the coherences off-diagonal elements 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 in units of /2 of the states j, k 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 the rate of phase variation increases, save for the zeroquantum coherences 12 in a homonuclear two-spin system, which dephase at a rate z ( 0 j 0 k ) z 0 j, z 0 k. In the following rf and gradient pulse sequence, each arrow connects the preceding density operator in product operator notation 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: 48 For the case of a heteronuclear system, wherein the gyromagnetic ratios of the two spins will differ substantially, the second gradient pulse z and its bracketing /2 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) Fig. 4. a 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. b 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 (01)/& into which the H qubit is put by its first Hadamard. This state is shown without normalization by 1/& in a dashed box that points at that place in the circuit. The conditional rotation exp(i(/2)z 1 H x C ) transforms it into the generally mixed state with density operator z cos(/2) y sin(/2) z, which is likewise shown in a dashed box pointing to the corresponding place in the circuit with the identity component omitted. Here, z tr( z )/2isthez 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 decoupling all further interactions with 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 see the text. 353 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 353

10 A. Background and theory Figure 4a 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. 4b, wherein: 1 the H qubit top line representing the photon s path is placed in a known quantum state 0; 2 a Hadamard gate is applied to this qubit, creating a superposition of states (01)/& just as a beam splitter creates a superposition of photon paths; 3 the C qubit bottom, representing the photon s possibly unknown polarization, is subjected to a phase shift exp(i C z ) conditional on the state of the first; because z W x W, this phase shift may be accomplished by sandwiching a c-not gate see below between a pair of Hadamards; 4 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 5 its state 0 or 1, corresponding to the path taken by the photon, is measured. In terms of equations, this circuit does the following. Because X H 1 W H Z H 1 W H, the unitary transformation implemented by the overall circuit can be written compactly using idempotents cf. Eq. 44 as U MZ e i/2x H C 1 z H X1 e i/2 C z H X0. 49 Thus for an arbitrary input state C of the C qubit, we obtain HC out e i/2 C z H X1 X H 0 Z H 0 C e i/2 C z H X1 X H 0 1 2X H 1 e i/2 C z C e i/2 C z H z X H 0 e i/2 C z C X H 0 H z C e i/2 C z H X0 C. 50 If we now perform a z measurement on the H qubit tracing over the C qubit as indicated in Fig. 4, we find that tr H z HC out 2trX 1 H 0 e i/2 C z C X H 1 C e i/2 C z realtre i/2 C z C. In a similar fashion, measuring y H gives tr H y HC out 2triX 1 H 0 e i/2 C z C ix H 1 C e i/2 C z i imagtre /2 C z C, whereas measuring x H gives tr H x C out 2trX 1 H 0 C X H 1 e i/2 C z C e i/2 C z We can thus use our knowledge of C to determine the rotation angle and hence the strength of the interaction scattering, or we can use our knowledge of to determine the prior state of the C qubit tomography. 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 X C 0. The overall density operator Z H 0 X C 0 is converted by the first two Hadamards to X H 0 Z C 0, which in turn is transformed by the c-not gate W H W C U MZ W C W H as follows: Z H 1 e i C x /2 Z H 0 X H 0 Z C 0 Z H 1 e i C x /2 Z H 0 Z H 0 X H 0 Z H 0 Z C 0 Z H 1 X H 0 Z H 1 e i C x /2 Z C 0 e i C x /2 Z H 0 X H 0 Z H 1 Z C 0 cos/2 i x C sin/2z 1 H X 0 H Z 0 H cos/2i x C sin/2z 0 C 1 2Z 0 H Z 0 C 1 4Z 1 H 1cos z C sin y C i 2Z 0 H x H Z 0 C x C x H Z 0 H x C Z 0 C sin/2 1 2 x H Z 0 C cos/ Z 0 H Z 1 H cos z C 1 4Z 1 H y C sin 1 4 x H y C y H x C sin/2 1 2 x H Z 0 C cos/2. 54 Clearly when 0 or4, the result is equal to the c-not input X 0 H Z 0 C, so that the overall circuit does nothing. In the case 2, however, we obtain Finally, if we set, we simply apply the usual c-not gate to X 0 H Z 0 C, obtaining C z 1 4 H x 1 4 H x C z X H 1 Z C 0, x H y C y H x C z H z C which is transformed by the last pair of Hadamards to Z 1 H X 0 C. 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 that is, has periodicity 4 rather than 2 as in spatial rotations. e i H C z z /8 e i H C z z /8, 56 where (0011)/& is one of the maximally entangled Bell states. It is easily verified that is unaffected by the final pair of Hadamards, and hence 354 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 354

11 HC out We i H C z z /8 WW WWe i H C z z /8 W e i H C x x /8 e i H C x x / H z C y H y C z H x C x. 57 It follows that tr( HC out H y )tr( HC out H z )0, meaning that no 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 single spin 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 13 C-labeled chloroform ( 1 H 13 C 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 pseudopure ground state Z 0 H Z 0 C 0000, as described in Eq. 48, 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. 4b are so simple, we shall omit them in what follows, and just input X 0 H Z 0 C to the c-not gate directly. A pulse sequence to implement the remaining Hadamard gates has been described already Eq. 40, as has one for the c-not gate Eq. 46. The latter s extension to general angles is easily obtained by expanding its exponential, as follows: exp i 2 x C Z 1 H exp C i 4 x exp i 4 x C z H exp C i 4 x exp C i 4 y exp i 4 z C z H exp C i 4 y. 58 Fig. 5. Pulse sequence for x rotation of C by an angle conditional on the polarization of H, with an optional 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 /2 pulses, while the last pulse to the C is labeled by the variable and outlined by a dotted line. The third factor on the last line corresponds to delay of length t/(2j), where J is the coupling constant, leading to the pulse sequence shown in Fig. 5 cf. Eq. 46. The trace operation indicated in Fig. 4b is implemented 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 or has been traced over. 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 magnitudes of H x and H y only, whereas if coupling is allowed during acquisition, the antiphase components of the density operator, H x C z and H y C z, will evolve under scalar coupling into observable components whose signals will be /2 out-of-phase with those due to H x and H y at the beginning of the data acquisition. By not including the final Hadamard W H, one also obtains the amplitudes of the product operators H z and H z C z, and if W C is then included, one also obtains H z C x and H y C x. Finally if both W C and W H are included, one obtains H x C x as well. With a bit more work, one can actually read out all 15 product operators H 1 C 1 is, of course, not detectable, 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 simply swap the input states of the spins, X H 0 Z C 0, 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 H y C y. This, however, can also be obtained simply by replacing the final Hadamards in the swapped experiments with their phase-shifted analog Vi y z /&, Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 355

12 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 1. 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 implementation 39 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 where N is the number of qubits, as opposed to N2 N for the classical fast Fourier transform FFT. The efficiency with which this quantum Fourier transform QFT 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 algorithm 41 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, 2 N 1 U QFT l2 N/2 k0 e 2ikl/2N k, 60 where 2 N/2 exp(2ikl/2 N ) k,l is just the usual unitary matrix of the discrete Fourier transform. We can rewrite this matrix as a product by substituting the binary expansion k N m1 k m 2 Nm with k m m k into Eq. 60 and splitting the sum over all integers into a sum over all combinations of their N bits 0 or 1, obtaining Fig. 6. Plots of matrices of product operator coefficients obtained via tomography for the spin spin interferometry experimental results see the text. The columns of each matrix have been ordered according to the longitudinal (L1,z) and transverse (Tx,y) components as follows: LL TLLTTT. Each column, in turn, has been ordered in the same way with 1z and xy only the labels on the LL columns are shown. The three matrices plotted are a the results of the 4 conditional rotation, which returns essentially the input state X 0 H Z 0 C ; b the results of the 2 conditional rotation, which shows the change of sign X 1 H Z 0 C duetothe spinor nature of qubit rotations; c and the results of the conditional rotation, which yields a rotated Bell state with no observable magnetization see the text. 356 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 356

13 1 U QFT l2 N/2 k1 0 1 kn l/2m1 N e 2ik m k 1 k N 0 m1 N m1 2 1/2 0e 2il/2m1 1)) 61 see Eq. 15 for the notation. On also substituting for l N n1 l Nn1 2 n1, we obtain U QFT l2 N/2 N m1 m 0 n1 e il n /2mn 1, 62 where we have used the fact that exp(i/2 mn )1 for all integers nm. It follows that we can write the QFT as a product U QFT U 1 UN, where each U m is a factor of the product. Each U m in turn can be further expanded as U m W m V m,m1 Vm,N, 63 which consists of the two-qubit conditional phase shifts V m,n expi m,n Z m 1 Z n 1, 64 with mn /2 nm, followed by a Hadamard W m to the mth 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 Nn1. Finally, it is easily shown that the QFT of any superposition 2 N 1 U QFT f U QFT k0 2 N 1 f kk 2 N 1 k0 f ku QFT k l0 FllF, 65 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(W m ) 2 on the right of each factor U m and bringing the result into the combined exponential of the conditional phase shifts: U m W m exp 2 i exp ix 1m N nm1 N nm1 m,n W m Z 1 m W m Z 1 n m,n Z 1 n W m. W m 66 Because W m commutes with U n for all nm, we may slide all the Hadamards over to the right, obtaining N N U QFT m1 exp ix 1m m,n Z 1 n W, 67 nm1 where WW 1 W N. 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 mn 1,...,1, conditional on all the qubits nm1,...,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 with a frequency range that spans all the spins. 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. 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. B. NMR implementation The NMR implementation described here uses the three carbon-13 atoms in alanine a common amino acid widely available with 99% 13 C incorporated into it for biomolecular NMR purposes as its qubits see Fig 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 , , , 68 J , J , J Because J kl k l for all k, l, the weak coupling approximation holds extremely well. For three qubits Eq. 67 simplifies to U QFT e ix /2Z1Z1/2 e ix /2Z1W 1 W 2 W The Hadamards at the beginning are implemented in the usual way, while the c-not gates are implemented by the previously described pulse sequence Eq. 58. 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 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/2 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 with only the innermost pair of inverse transformations displayed on any one line: e i/ y e i/4 z ze i/4 y e i/ z ze i/4 z xe i/4 z z e i/4 1 2 x x e i/ z yze i/4 xx e i/ z ze i/4 yz ze i/4 z ze i/4 xz. 71 Putting all the pieces together on a single line now shows that the pulse sequence for this component of the c-not/2 between 1 and 3 is 357 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 357

14 4 z 1 z 2 4 x 1 2 x 4 z 2 z y 4 z 1 z y 4 z 2 z 3 4 x 1 2 x 4 z 1 z 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 while refocusing the other two couplings as usual with a pulse to the other spin halfway through the coupling period and at the end. 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 states of the 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 Swap 13 gate is equal to the composition Swap 12 Swap 23 Swap 12, and that Swap kl is obtained from the pulse sequence: 4 k y l y 4 z k z l 4 k y l y 4 z k z l 4 x k l x 4 x k l x. 4 z k z l 73 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 top and output bottom 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, )/2, with theoretical output state 00(01)/& following the Swap Ref. 13. These correspond to the pure state ensembles X 0 1 X 0 2 Z 0 3 and Z 0 1 Z 0 2 X 0 3, 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 74 swapping spins 1 and 2 via the above pulse sequence, and finally applying the two-spin pulse sequence given in Eq. 48 to spins 2 and 3....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 Bose- 358 Am. J. Phys., Vol. 70, No. 3, March 2002 Havel et al. 358