The Use of NMR as a Quantum Computer
By: Timothy K McIntosh
University of California, Davis, Davis California Physics 121
March 17, 2000
In 1982, Feinman suggested that quantum computing would be exceptionally better at simulating quantum particles than classical computers. This speculation was developed into a full set of theories by the start of the 1990's, and Computational algorithms were developed that would utilize the unique processing power of a quantum computers. Chasing an exponential speed up of classical computers, scientists began developing methods for carrying out he quantum algorithms.
The development of experiments that could test these algorithms were severely limited by the lack of devices that could manipulate quantum particles, and maintain the coherence of the quantum states. Advances in laser technology, leading to quantum cooling, and advances in precision detectors allowed for initial steps toward controllable quantum experiments. Only in the last three years have groups been able to generate experiments where theorized algorithms can be tested.
In attempts to realize a quantum computer capable of limited computational algorithms, researches have utilized trapped ions, quantum dots, and cavity quantum electrodynamics. The most successful attemp to build a quantum bit or “qubit” computer, has been the use of a single cooled beryllium ion. Unfortunatly, the immense effort needed to cool the ion and restrict the quantum states to only a few, have demonstrated the limited accessibility and limited use of such systems.
The search for a viable quantum system, the nuclear spin has been an attractive alternative to other more cumbersome quantum systems. Nuclear spins are particularly isolated form electronic and vibrational mechanisms that can lead to decoherance. In modern spectrographic nuclear magnetic resonance ( NMR ) machines, which utilize the nuclear spins of protons, relaxation times are routinely measured in the hundreds of seconds. In addition to the lengthy coherence times, the use of hundreds of rf pulses are used to manipulate proton spins. Meeting these requirements of quantum computing points to the NMR as a realistic approach to the development of qubit computers.
I will discuss the underlying concept of quantum computing and the requirements for implementing qubit algorithms. I will then describe the methods of modern NMR and how they may be used to generate qubits for use in calculations. Finally, I will bring up the short comings of the NMR as a pure state quantum computer and suggestions of NMR as an intermediate between classical and quantum computing.
II. Quantum Computing
Quantum systems offer two features that a classical computer can not; superposition and entanglement. A classical bit exists in one of two states; 1 or 0. Superposition allows for a qubit to exist in two or more states at once. Therefore a system of N qubits may represent 2N different states simultaneously. Entanglement combines the states of many particles, shifting the determination of a quantum state away from individual qubits to their combined states.
Entanglement and superposition have been thought of as necessary for quantum computing since it was first proposed. For many qubit computers, entanglement allows for calculations on multiple qubits at the same time. Unlike classical computers, quantum computers may operate on a single entangled state, which by its nature encompasses many individual qubits, carrying the calculation to all the involved qubits. Superposition allows unitary operations to represent many calculations at once. While a qubit resides in a superposition of states, an operation will make appropriate, yet different changes to all the superimposed states of the qubit at the same time. In this way, many different calculations may be represented by a single operation. This is referred to as quantum parallelism.
With careful development, the parallelism that quantum computers offer may lead to algorithms that solve problems faster than any classical computer. Examples of these clever algorithms include; factoring, database searching, quantum system simulation, and error correction. It is noted that in a quantum computer, the logic circuitry and time steps would be the same as for classical computers but the memory would be composed of qubits . To carry one of these quantum algorithms, an initial pure state must be set up, a set of unitary operations preformed, and a measurement made.
Meeting the requirements of a quantum computer poses several problems. First, to generate a pure quantum state, the system must be isolated from its environment and must be controllable. The coherence length of quantum states is proportional to how much influence its surroundings can impart on it. Any interaction may alter the wavfunction, and prevent the system from maintaining the initial and desired state. Second, to preform the needed unitary operations, the system must be manipulatable by the environment. This contradicts our first problem of state isolation. A system is needed that is isolated enough to have long coherence lengths yet capable of undergoing an intended state change. Finally, measuring a state wavefunction necessarily collapses it. In order to readout information the wavefunction must be destroyed, along with any unmeasured information. Here, a system must be large enough to ensure the probability of detection. Through wave-particle duality of quantum particles, often all that is needed for measurement, is the probability of detection. In this way, not all particles need to be measured, just a small portion, so that the remaining qubits will not be lost.
Modern NMR possesses traits that have thrust it to the forefront of quantum computer research. NMR has been used for chemical analysis for many years, so the technology is thoroughly advanced, and readily available. Since it is dependent on the manipulation of quantum states, namely nuclear spins, NMR is already attractive to developer of quantum computers.
For a spin system, the application of a magnetic field causes Zeeman energy level splitting. The alignment of the spins to the magnetic field dictates the interaction between the two. A preferred lower energy orientation exists in the splitting, so the populations of the spins aligned with the field and opposite the field will be different. This population difference results in a net magnetization which may be detected.
Detection of the magnetization employs pickup coils which receive signals from energy transitions in the spins. An applied magnetic field B0, causes a torque on the spins in the system, The torque is perpendicular to the plane of the spin and B0 field, causing a precession. Now, a radio frequency field Brf is applied in the xy plane. The x and y components are put 90° out of phase with each other and oscillated, so that Brf rotates around B0 similar to the spin. If resonance is reached, the rotation of Brf matches the precession of the spin and a torque is created on the spin ( figure 1 ). The torque flips the spin direction causing an energy level transition. If the spin flips back, a photon is released, which is detected by the pickup coils.
The spin systems used in NMR are typically Hydrogen or carbon atoms. Hydrogen atoms are simple because they consist of a single proton. The protons in water have been used in NMR for many years, and the protons in any liquid are preferred. Aside from the lack of need for cooling, the thermal energy will cause the bulk of spins to have random orientations. All the signals of the randomly oriented molecules will cancel each other out, and the resulting signals will only originate from the population of spins that contribute to the net magnetization. Also, the thermal energy in the liquid will tend to wash out any coupling to the environment that might be present.
IV. NMR as a quantum computer
The fundamental techniques and design of NMR machines lend themselves to their use in quantum computing. Coherence lengths for spin states in water and other molecular liquids regularly exceed many seconds, which allows for the time needed in carrying out quantum algorithms. Since large samples are be used with these techniques, many molecules contribute to a high signal-to-noise. NMR’s may have on the order of 1022 molecules participating in calculations. Along with the features that benefit quantum computing there are also pitfalls. The same thermal energy that decouples the molecules in liquid state NMR from the environment also prevents the individual spins from maintaining coherence in pure quantum states. Also, since an NMR is made up of a bulk of molecules, they will be in a distribution of states, where individual states can not be isolated. Measurements and operations on this distribution will result in different outcomes for each molecule.
A solution to the pure state problem in NMR was provided by Gershenfeld et. al  in 1997. His group proposed a way to generate a psudo-pure state in bulk quantum computing. A proposed thermal ensemble of (2,3)-dibromothiophene ( figure 2 ) molecules in a 4-7 Tesla field offers a deviation from the equilibrium density matrix that acts as a small pure state. For a molecule such as this, the chemical environment will cause a shift in the resonant frequencies of the two separate hydrogen atoms. Although small, this shift of the resonant frequency allows for the individual manipulation of the spins. The spins are coupled in a limited way through the intra-molecule forces, so operations on the spins provide a means for logical qubit operations.
The state of an N-spin system is well described by the density matrix ( see  for derivation ):
The two terms represent an equilibrium part, proportional to the unitary matrix I and a deviation matrix . Under a unitary operation, utilized in quantum computing, the I term will experience no change, while the second, deviation term, will undergo the wanted operation. Therefore, the bulk dynamics of the ensemble may be represented by the deviation matrix alone. Furthermore, for the two spin system such as the molecule shown above, the equilibrium configurations of the 4 possible spin states are not equal. The state where both spins are aligned opposite the field is more probable than the other 3 ( for derivation, see  ). The populations of the lower probability states will generate signals that will cancel under an operation. The result is that the molecules in the higher probability configuration will produce the only detectable signal. Thus, at equilibrium, the deviation matrix will produce a population of pure state spin down molecules that all act like a single psudo-pure quantum state. With the existence of a psudo-pure state resulting from the population differences in the deviation matrix, quantum calculation can proceed.
Implementation of NMR as a quantum computer has succeeded in opening the doors to testing the many quantum algorithms under development. Chuang et al have experimented in the implementation of fast Quantum searching . Cory et al  have also conducted work on the realization of quantum error correction. Other groups, such as Jones et al  have started quantum counting work. Although many of the requirement needed for pure state quantum computing have not been met, some success in developing quantum computers has been achieved using NMR.
Opponents of bulk quantum computing argue that due to the lack of true pure states and the lack of entanglement between the qubits, the NMR is not utilizing the features of quantum mechanics that make it so appealing. Scientist like Gershenfeld  admit that NMR does not rely on the quantum parallelism thought to be the underlying power of quantum computing, but NMR is based in the quantum world, and classical interpretation of their work have not been fount. Proponents of NMR quantum computation propose that NMR lies somewhere in between the classical and quantum computer . It provides a speed up of classical calculations , but does not match the power of a pure quantum computer. One point made is the ability of the NMR to enact unitary operations; a purely quantum mechanical phenomena.
Few groups expect NMR or any of the current methods of quantum computation to lead to a useful computer. To scale up the quantum computer for the proposed exponential speed up would require an exponential increase in instrumentation and resources as well. Scientist expect that as system sizes increase, and qubit numbers grow, entanglement will follow closely behind. Work is now shifting to the possibilities of mixed states and bulk quantum computers.
 Lov K. Grover. Phys. Rev. Lett. 79, (no. 2) 325 (1997).
 Neil A. Gershenfeld, Isaac L. Chuang. Science 275, 350 (1997).
 Isaac L. Chuang, Neil Gershenfeld, Mark Kubinec. Phys. Rev. Lett. 80,
(no. 15) 3408 (1998).
 D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F.
Havel, S. S. Somaroo. Phys. Rev. Lett. 81, (no. 10) 2152 (1998).
 J. A. Jones, M. Mosca. Phys. Rev. Lett. 83, (no. 5) 1050 (1999).
 S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden, S. Popescu, R.
Schack. Phys. Rev. Lett. 83, (no. 5) 1054 (1999).
 Richard Fitzgerald. Physics Today 53, ( no. 1 ) 20 (2000).
 E. Knill, I. Chuang, R. Laflamme. Phys. Rev. A 57, (no. 5) 3348 (1998).
 S. Somaroo, C. H. Tseng, T. F. Havel, R. Laflamme, D. G. Cory. Phys.
Rev. Lett. 82, (no. 26) 5381(1999).
 Isaac L. Chuang, Yoshihisa Yamamoto. Preprint ERATO Quantum
Fluctuation Project, ( March 28, 1995).