Overview of Contributions to QIP

Important issues in current experimental efforts toward realizing QIP are to find ways of achieving the necessary quantum control and to determine whether sufficiently low error-rates are possible. Liquid-state NMR is the only extant system (as of 2002) with the ability to realize relatively universal manipulations on more than two qubits (restricted control has been demonstrated in four ions [30]). For this reason, NMR serves as a useful platform for developing and experimentally verifying techniques for QIP and for establishing simple procedures for benchmarking information processing tasks. The ``cat-state'' and the various error-correction benchmarks [23,25] consist of a set of quantum control steps and measurement procedures that can be used with any general-purpose QIP system to determine, in a device independent way, the degree of control achieved. The demonstration of error rates in the few percent per non-trivial operation is encouraging. For existing and proposed experimental systems other than NMR, achieving such error rates is still a great challenge.

Prior research in NMR, independent of quantum information, has proved to be a rich source of basic quantum control techniques useful for physically realizing quantum information in other settings. We mention four examples. The first is the development of sophisticated shaped-pulse techniques that can selectively control transitions or spins while being robust against typical errors. These techniques are finding applications to quantum control involving laser pulses [31] and are likely to be very useful when using coherent light to accurately control transitions in atoms or quantum dots, for example. The second is the recognition that there are simple ways in which imperfect pulses can be combined to eliminate systematic errors such as those associated with miscalibration of power or side-effects on off-resonant nuclear spins. Although many of these techniques were originally developed for such problems as accurate inversion of spins, they are readily generalized to other quantum gates [32,33]. The third example is decoupling used to reduce unwanted external interactions. For example, a common problem in NMR is to eliminate the interactions between proton and labeled carbon nuclear spins to observe ``decoupled'' carbon spins. In this case, the protons constitute an external system with an unwanted interaction. To eliminate the interaction, it is sufficient to invert the protons frequently. Sophisticated techniques for ensuring that the interactions are effectively turned off independent of pulse errors have been developed (See, for example, [5]). These techniques have been greatly generalized and shown to be useful for actively creating protected qubit subsystems in any situation in which the interaction has relatively long correlation times [34,35]. Refocusing to undo unwanted internal interactions is our fourth example. The technique for ``turning off'' the coupling between spins that is so important for realizing QIP in liquid-state NMR is a special case of much more general methods of turning off or refocusing Hamiltonians. For example, a famous technique in solid state NMR is to reverse the dipolar coupling Hamiltonian using a clever sequence of $180^\circ$ pulses at different phases (see, for example, [5], page 48). Many other proposed QIP systems suffer from such internal interactions while having similar control opportunities.

The contributions of NMR QIP research extend beyond those directly applicable to experimental QIP systems. It is due to NMR that the idea of ensemble quantum computation with weak measurement was introduced and recognized as being, for true pure initial states, as powerful for solving algorithmic problems as the standard model of quantum computation. (It cannot be used in settings involving quantum communication.) One implication is that to a large extent, the usual assumption of projective measurement can be replaced by any measurement that can statistically distinguish between the two states of a qubit. Scalability still requires the ability to ``reset'' qubits during the computation, which is not possible in liquid-state NMR. Another interesting concept emerging from NMR QIP is that of ``computational cooling'' [36], which can be used to efficiently extract initialized qubits from a large number of noisy qubits in initial states that are only partially biased toward $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. This is a very useful tool for better exploiting otherwise noisy physical systems.

The last example of interesting ideas arising from NMR studies is the ``one-qubit'' model of quantum computation [37]. This is a useful abstraction of the capabilities of liquid-state NMR. In this model, it is assumed that initially, one 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$}$ and all the others are in random states. Standard unitary quantum gates can be applied and the final measurement is destructive. Without loss of generality, one can assume that all qubits are re-initialized after the measurement. This model can perform interesting physics simulations with no known efficient classical algorithms. On the other hand, with respect to oracles, it is strictly weaker than quantum computation. It is also known that it cannot ``faithfully'' simulate quantum computers [38].