Introduction to NMR Quantum Information Processing

R. Laflamme, E. Knill, D. G. Cory, E. M. Fortunato, T. Havel,
C. Miquel, R. Martinez, C. Negrevergne, G. Ortiz, M. A. Pravia, Y. Sharf,
S. Sinha, R. Somma and L. Viola

28 July 2002

Using quantum physics to represent and manipulate information makes possible surprising improvements in the efficiency with which some problems can be solved. But can these improvements be realized experimentally? If we consider the history of implementing theoretical ideas about classical information and computation, we find that initially, small numbers of simple devices were used to explore the advantages and the difficulties of information processing. For example, in 1933 Atanasoff and his colleagues at the Iowa State College were able to implement digital calculations using about 300 vacuum tubes (see [1], the entry for ``computing, modern history of''). Although the device was never practical because its error rate was too large, it was probably the first instance of a programmable computer using vacuum tubes and it opened the way for more stable and reliable devices. Progress toward implementing quantum information processors is also initially confined to limited capacity and error-prone devices.

There are numerous proposals for implementing quantum information processing (QIP) prototypes. To date (2002), only three of them have been used to successfully manipulate more than one qubit: cavity quantum electrodynamics (cavity QED), ion traps and nuclear magnetic resonance (NMR) with molecules in a liquid (liquid state NMR). The difficulty of realizing QIP devices can be attributed to an intrinsic conflict between two of the most important requirements: On the one hand, it is necessary for the device to be well isolated from, and therefore interact only weakly with, its environment; otherwise, the crucial quantum correlations on which the advantages of QIP are based are destroyed. On the other hand, it is necessary for the different parts of the device to interact strongly with each other and for some of them to be coupled strongly with the measuring device, which is needed to read out ``answers''. That few physical systems have these properties naturally is apparent from the absence of obvious quantum effects in the macroscopic world.

One system whose properties constitute a reasonable compromise between the two requirements consists of the nuclear spins in a molecule in the liquid state. The spins, particularly those with spin ${1\over 2}$, provide a natural representation of quantum bits. They interact weakly but reliably with each other and the effects of the environment are often small enough. The spins can be controlled with radio-frequency (RF) pulses and observed with measurements of the magnetic fields that they generate. Liquid state NMR has so far been used to demonstrate control of up to seven physical qubits.

It is important to remember that the idea of QIP is less than two decades old, and, with the notable exception of quantum cryptography, experimental proposals and efforts aimed at realizing modern QIP began only in the last five years of the 20'th century. Increasingly advanced experiments are being implemented. But from an information processing point of view, we are a long way from using quantum technology to solve an independently posed problem not solvable on a standard personal computer--a typical ``classical'' computer. In order to get to the point where such problems can be solved by QIP, current experimental efforts are devoted to understanding the behavior of and the methods for controlling various quantum systems, as well as ways of overcoming their limitations. The work on NMR QIP has focused on the control of quantum systems by algorithmically implementing quantum transformations as precisely as possible. Within the limitations of the device, this approach has been surprisingly successful, thanks to the many scientists and engineers who have perfected NMR spectrometers over the past 50 years.

After a general introduction to NMR, we give the basics of implementing quantum algorithms. We describe how qubits are realized and controlled with RF pulses, their internal interactions, and gradient fields. A peculiarity of NMR is that the internal interactions (given by the internal Hamiltonian) are always on. We discuss how they can be effectively turned off with the help of a standard NMR method called ``refocusing''. Liquid state NMR experiments are done at room temperature, leading to an extremely mixed (that is, nearly random) initial state. Despite this high degree of randomness, it is possible to investigate QIP because the relaxation time (the time scale over which useful signal from a computation is lost) is sufficiently long. We explain how this feature leads to the crucial ability of simulating a pure (non-random) state by using ``pseudopure'' states. We discuss how the ``answer'' provided by a computation is obtained by measurement and how this measurement differs from the ideal, projective measurement of QIP. We then give implementations of some simple quantum algorithms with a typical experimental result. We conclude with a discussion of what we have learned from NMR QIP so far and what the prospects for future NMR QIP experiments are. For an elementary, device-independent introduction to quantum information and definitions of the states and operators used here, see [2].