InterJournal Complex Systems, 33
Status: Accepted
Manuscript Number: [33]
Submission Date: 963011
Nuclear magnetic resonance spectroscopy: an experimentally accessible paradigm for quantum computing
Author(s): David Cory , Amr Fahmy , Timothy Havel

Subject(s): CX.09

Category: Brief Article

Abstract:

We describe a new physical mechanism which is capable of computation, along with experimental results that demonstrate its potential. The mechanism is Nuclear Magnetic Resonance spectroscopy, or NMR. This approach is based on the fact that the spin 1/2 nuclei of each molecule in a liquid sample are largely isolated from the spins in all the other molecules. This makes it possible to describe the state of the sample by a reduced density matrix of size 2^n, where n is the number of spins in one molecule, rather than 2^N where N ~ 10^23 is the total number of spins in the sample. It also effectively makes each molecule into an independent quantum computer, with a decoherence time on the order of seconds. We call a computer based on these principles an Ensemble Quantum Computer, or EQC.

The main problem that had to be overcome, in order to use a macroscopic NMR sample as an EQC, lies in the fact that even with superconducting magnets, the net polarization of the spins is only about one part in a million. We have solved this problem by introducing a new concept into NMR spectroscopy, which we call pseudo-pure states. Whenever the reduced density matrix has (2^n)-1 degenerate eigenvalues, it can be shifted by the addition of an appropriate multiple of the unit matrix to a rank one density matrix, which in turn factorizes into the dyadic product of a spinor with its conjugate, just as the density matrix of a true pure state does. This mapping between pseudo-pure density matrices and "pseudo-spinors" is covariant, in that when the density matrix is subjected to a two-sided unitary transform- ation, the corresponding pseudo-spinor is transformed by the same unitary matrix. In addition, the ensemble-average expectation value of any observable versus a pseudo-pure state is a simple linear function of the ordinary expectation value of the observable versus the corresponding pseudo-spinor. Thus pseudo- pure states in NMR enable us to implement an EQC using only weakly polarized samples at room temperature and pressure.

In this article we show how this approach enables us to easily implement the usual controlled-NOT or quantum XOR gate, on single states as well as on "coherent superpositions", using a commercially available NMR spectrometer. We also show how standard techniques in experimental NMR spectroscopy enable us to implement the universal Toffoli gate. Scaling these experiments to molecules with as many as ten inequivalent spins in them is straightforward. A variety of experimental difficulties, in particular the limited signal-to-noise of the receivers used to collect NMR spectra, make scaling much past ten spins challenging. Nevertheless, the availability of this simple paradigm for quantum computing should be of considerable utility in further developing the theory and algorithms that will be needed to solve significant problems by either quantum computing, or by ensemble quantum computing.

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