A Brief Survey of NMR QIP

Concrete and workable proposals for using liquid-state NMR for quantum information were first given in 1996/7 by D. Cory, A. Fahmy and T. Havel [10] and by N. Gershenfeld and I. Chuang [11]. Three difficulties had to be overcome for NMR QIP to become possible. The first was that the standard definitions of quantum information and computation require that quantum information be stored in a single physical system. In NMR, an obvious such system consists of some of the nuclear spins in a single molecule. But it is not possible to detect single molecules with available NMR technology. The solution that makes NMR QIP possible can be applied to other QIP technologies: Consider the large collection of available molecules as an ensemble of identical systems. As long as they all perform the same task, the desired answers can be read out collectively. The second difficulty was that the standard definitions require that read-out take place by a projective quantum measurements of the qubits. From such a measurement, one learns whether a qubit is in the state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ or $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. The two measurement outcomes have probabilities determined by the initial state of the qubits being used, and after the measurement the state ``collapses'' to a state consistent with the outcome. The measurement in NMR is much too weak to determine the outcome and cause the state's collapse for each molecule. But because of the additive effects of the ensemble, one can observe a (noisy) signal that represents the average, over all the molecules of the probability that $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ would be the outcome of a projective measurement. It turns out that this so-called ``weak measurement'' suffices for realizing most quantum algorithms, in particular those whose ultimate answer is deterministic. Shor's factoring and Grover's search algorithm can be modified to satisfy this property. The final and most severe difficulty was that, even though in equilibrium there is a tendency for the spins to align with the magnetic field, the energy associated with this tendency is very small compared to room temperature. Therefore, the equilibrium states of the molecules' nuclear spins are nearly random, with only a small fraction pointing in the right direction. This difficulty was overcome by methods for singling out the small fraction of the observable signal that represents the desired initial state. These methods were anticipated in 1977 [12].

Soon after these difficulties were shown to be overcome or circumventable, two groups were able to experimentally implement short quantum algorithms using NMR with small molecules [13,14]. At present it is considered unlikely that liquid-state NMR algorithms will solve problems not easily solvable with available classical computing resources. Nevertheless, experiments in liquid-state NMR QIP are remarkable for demonstrating that one can control the unitary evolution of physical qubits sufficiently well to implement simple QIP tasks. The control methods borrowed from NMR and developed for the more complex experiments in NMR QIP are applicable to other device technologies, enabling better control in general.