Ensemble quantum computing by NMR spectroscopy

Abstract

A quantum computer (QC) can operate in parallel on all its possible inputs at once, but the amount of information that can be extracted from the result is limited by the phenomenon of wave function collapse. We present a new computational model, which differs from a QC only in that the result of a measurement is the expectation value of the observable, rather than a random eigenvalue thereof. Such an expectation value QC can solve nondeterministic polynomial-time complete problems in polynomial time. This observation is significant precisely because the computational model can be realized, to a certain extent, by NMR spectroscopy on macroscopic ensembles of quantum spins, namely molecules in a test tube. This is made possible by identifying a manifold of statistical spin states, called pseudo-pure states, the mathematical description of which is isomorphic to that of an isolated spin system. The result is a novel NMR computer that can be programmed much like a QC, but in other respects more closely resembles a DNA computer. Most notably, when applied to intractable combinatorial problems, an NMR computer can use an amount of sample, rather than time, which grows exponentially with the size of the problem. Although NMR computers will be limited by current technology to exhaustive searches over only 15 to 20 bits, searches over as much as 50 bits are in principle possible, and more advanced algorithms could greatly extend the range of applicability of such machines.

Figure 1

The four energy levels associated with two spins α and β, whose resonance frequencies are ν_α and ν_β (in Hz). The energy levels when there is no coupling between the spins are shown on the left, and those with a coupling of J_αβ (in Hz) on the right. The allowed transitions between the energy levels are indicated with dashed double-headed arrows.

Figure 2

The simulated NMR spectrum of a two-spin system, with ν_α = 100 Hz, ν_β = 400 Hz, and J_αβ = 40 Hz. From left to right, the four peaks in this spectrum correspond to the transitions 0 ↔ 1, 2 ↔ 3, 0 ↔ 2, and 1 ↔ 3 indicated in Fig. 1 (be forewarned, however, that NMR spectroscopists traditionally plot their spectra with frequency increasing from right to left). In practice, this spectrum would be obtained by applying a nonselective π/2 pulse to the equilibrium state and Fourier transforming the resulting signal.