There is doubt, however, that solution NMR quantum computing will ever be useful. Ensembles of uncoupled two-level systems (magnetic resonance or any other form) have quite classical dynamics, as shown by Feynman (3). Thus the clock cycles for any nonclassical dynamics, including all of the computing operations in the report (2) and in any other conceivable treatment, require times on the order of the reciprocal of the spin-spin couplings (200 Hz for directly bonded atoms, 10 Hz for protons on nearest-neighbor carbons) per step. Many such steps would be needed for logic operations between two separated spins. Dipolar couplings (for example, in solids) can increase the couplings by another factor of 10, but then the eigenstates are not the simple spin product states, and each logical manipulation will be much more complex. The slowest limit of speed estimated by Gershenfeld and Chuang (2) (10 logic gates per second) is thus grossly overoptimistic for a reasonably sized molecule.

Speed is not an important problem for demonstration experiments; perhaps new quantum algorithms will be found that compensate for the enormous slowdown. However, NMR is the premier spectroscopic example, not of quantum mechanics, but of quantum statistical mechanics including ensemble averaging. For a macroscopic sample (say N  10²² spins) the evolution is essentially deterministic. For example, all modern spectrometers routinely measure I_x and I_y simultaneously, despite the Uncertainty Principle. Fluctuations from the expectation value scale as 1/, or about 10¹¹  (10¹¹ of the magnetization, but as I show below, this is still not good enough for solution NMR quantum computing). In addition, in NMR the energy difference between the two spin states of each atom is small, which implies that the possible signal in a quantum computing experiment suffers a severe degradation for systems that might be big enough, in principle, to contain a useful number of qubits (Fig. 1). For example, in a 100-spin system at room temperature, the expected signal for an ideal quantum computer is 28 orders of magnitude smaller than the room temperature magnetization.

34

Why is the scaling such a problem? Because the quantum computing signal relies on the fraction of the molecules starting in a single specific eigenstate in the equilibrium density matrix _eq. After the evolution through an assumed ideal set of quantum gates onto a target state, the computing signal has to be detected (converted into observable magnetization). Because the observable operators are traceless, this reduces the signal further: Only the largest population difference in _eq (the difference between the all- and all- states) can be made observable. It can be shown that the best way to do this is to overlap the population with the largest matrix element of I_z, so the ideal observable signal is

(1)

This upper limit is actually quite similar to doing N-quantum selective excitation (4) in a totally asymmetric N-spin system. This has been done for N  6 only in high symmetry cases such as benzene or solid adamantane. For the quantum computing problem, symmetry hurts instead of helps--it reduces the number of possible qubits.

A (classical) computer can evaluate equation (1) by an explicit sum over states for moderate values of N. It can also be simplified in the high temperature limit Nh/kT<<1:

(2)

This can be compared to the magnetization after one pulse in a normal NMR experiment:

(3)

It is apparent that the scaling to a useful number of spins is extremely unfavorable. To fully understand the scope of this problem, note that 99.99999999% of the time a generously sized room-temperature sample (10²² spins) contains no 100-spin molecules in the ground state ₁₂ ... _n, or in any other single one of its 2¹⁰⁰ quantum states. Furthermore, the all- state is only 1% less probable than the all- state in a 600-MHz spectrometer. Thus, for every 100 times one molecule accidentally gets in the "right" (all-) initial state, there will be 99 occurrences of the "wrong" (all-) initial state, giving exactly the negative of the desired signal. Finally, the "random" component of the magnetization (10¹¹ , as discussed earlier) is 10²² times larger than the expected signal and evolves at the same frequency.

Gershenfeld and Chuang state that the signal grows exponentially with decreasing temperature, but exponential growth does not start until h/kT>>1 (<<1K even for ¹H in large magnets). The sample then will not be a liquid; lines will be broadened and intermolecular couplings will complicate logic gates enormously. It is possible to polarize nuclear spins from electronic spins using laser excitation, but doing this efficiently requires isolated atoms with sharp electronic transitions (for example, ¹²⁹Xe or ³He in contact with Rb atoms in the gas phase (5). Spin-polarized ¹²⁹Xe can polarize room-temperature ¹H in solution (6), but the fractional polarization is fundamentally limited by the nature of the interaction. Solid ³He at mK temperatures has sharp resonance lines due to spin diffusion (the linewidths are on the order of 1 Hz, similar to liquids), but in this case there are no scalar couplings. Finally, perhaps someday we will have 100-kT magnets with the required 10-nT inhomogeneity, but in that case the field itself will surely align the solute and reintroduce dipolar couplings (as happens now in proteins).

These problems are not found with other potential implementations of quantum computing. For transitions with h>>kT the initial state can be prepared essentially without loss, no matter how many systems are coupled. This means, for example, that electron spin resonance (ESR) spectroscopy in modern superconducting magnets (resonance frequencies around 300 GHz) can get into the right regime at liquid helium temperatures; one could conceive of quantum computing with multiple-radical molecules in an inert matrix, using dipolar couplings plus g value differences that are far larger than J couplings and chemical shifts, respectively, in NMR. It seems more likely, however, that if quantum computing will ever be practical, it will be with "designer materials" such as precisely spaced quantum dots or free radicals positioned on a surface by force microscopy.

In summary, quantum computing might well turn out to be capable someday of solving certain problems better than conventional techniques; but if so, bulk NMR is not likely to play any role in a practical implementation.

Warren S. Warren
Department of Chemistry,
Princeton University,
Princeton, NJ 08544-1009, USA

REFERENCES AND NOTES

1. P. Shor, Proceedings of the 35th Annual Symposium on Foundations of Computer Science (IEEE Computer Society Press, New York, 1994); A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996) .
2. N. A. Gershenfeld and I. L. Chuang, Science 275, 350 (1997) [Abstract/Free Full Text] .
3. R. P. Feynman, F. L. Vernon Jr., R. W. Hellwarth, Appl. Phys. 49, 28 (1957).
4. W. S. Warren, S. Sinton, D. P. Weitekamp, A. Pines, Phys. Rev. Lett. 43, 1791 (1979) [CrossRef]; W. S. Warren and A. Pines, J. Chem. Phys. 74, 2808 (1981) [CrossRef] .
5. See, for example, B. Driehuys et al., Appl. Phys. Lett. 69, 1668 (1996) [CrossRef]; A. S. Barton, G. D. Cates, W. Happer, H. Middleton, Phys. Rev. A48, 4441 (1993) , and references therein.
6. G. Navon et al., Science 257, 1848 (1996) ; K. L. Sauer, R. J. Fitzgerald, W. Happer, Chem. Phys. Lett., in press.
7. Supported by the National Science Foundation under grant CHE-9412948. I thank H. Rabitz and K. K. Lehmann for useful discussions.

Response: Few researchers can match Warren's innovative contributions to NMR techniques, but he makes a number of restrictive assumptions about areas of active research that led him to draw unduly pessimistic conclusions about the usefulness of bulk spin resonance quantum computing.

First, our conclusion (which he agrees with) was that realizing a 10-qubit quantum computer is within reach of existing NMR spectrometers (1). While there are grounds to think that it will be possible to scale beyond that, even 10 qubits is remarkable, because just a year ago there were confident predictions that building any nontrivial quantum computer would be impossible. In the short time since its introduction, this new technique has already led to the first experimental realization of quantum gates connected into circuits (2) that have been used to run programs to experimentally test theoretical predictions of unusual quantum dynamics (3). Even in these first simple examples, the introduction of quantum computation to NMR has already proved itself to provide fruitful new ways to explore the creation and loss of coherence in complex systems.

Warren makes two fundamental points, both of which were clearly identified in our Research Article (1). First, the slowest operations come from coupling terms, which usefully range down to on the order of 10 Hz. Warren states that this rate severely limits the feasible molecule size, but we gave two reasons why this conclusion is incorrect. The first is that a quantum computer does exponential work per time step. At the speed of the fastest (publically known) classical factoring to date, finding the factors of a 1000-digit number would take on the order of the age of the universe. A quantum computer with a 1-Hz clock cycle time could reduce this time to a matter of days. Further, a polynomial decrease in coupling strength with increasing molecule size will still lead to an exponential increase in computation speed. The second reason why the coupling frequency does not limit the molecule size is that, as with classical computation, quantum computation is possible using only local (and hence strong) interactions. Quantum cellular automata are computationally universal, and have favorable scaling properties (4).

Warren than reproduces our conclusion that, in the high-temperature limit, the signal strength falls off roughly exponentially in the number of qubits. This limits conventional spectrometers to roughly 10 qubits. But, the special features of quantum computation suggest many optimizations not possible in a general-purpose spectrometer. Starting by transferring electron polarization gives an improvement by a factor of 10³ in signal strength (the ratio of the gyromagnetic ratios); reading out with transfer back to electrons gives another factor of 10³ in magnetization and another factor of 10³ in the time rate of change of flux in a pick-up coil. Because the system can be designed around a large sample of a known fluid, increasing the sample radius by a factor of 30 gives another factor of 10³, and because the computation can be designed to read out on a single line, a high-Q resonantor can provide another factor of 10³. Each factor of 10³ provides sensitivity for 10 more qubits. Realizing these potential improvements presents a significant experimental challenge, but taken together they suggest that it might be possible to reach sizes larger than the biggest classical computers (2⁴⁰ ~ 10¹² bits).

These refinements remain in the high-temperature limit and thus cannot be expected to scale further still. As we have noted, the underlying problem arises from the small Boltzmann factors. But Boltzmann factors of order unity are routinely observed in a number of systems, including optically pumped vapors and cryogenically cooled materials. While this work to date has focused on simple systems without the nonlinear interactions required for computation, recent experimental studies have shown that it is possible to transfer large polarizations to molecules with multiple spins (5). This is now an active research area. Given the possibility of such large Boltzmann factors, the remaining scaling limit is the decoherence time, but NMR approaches the range of the decoherence per gate required for steady-state error correction (6).

Warren concludes by stating that alternatives such as quantum dots have much more promise. This unexpected conclusion neglects the experimental reality that bulk spin resonance quantum computing is the only experimental approach to date that has implemented nontrivial quantum circuits, while quantum dots have decoherence times below nanoseconds and have yet to demonstrate a coherent quantum gate. This does not mean that NMR is the final solution; a useful quantum computer will almost surely look nothing at all like a conventional NMR spectrometer, and will draw on the best features of all of the alternative approaches. It may not necessarily even involve nuclear spins, but we are confident that it will take advantage of the desirable features, introduced by NMR, of representing quantum information in ensembles using naturally occurring nonlinear interactions and performing quantum readout by a weak ensemble measurement. Scaling beyond roughly 10 spins poses daunting experimental problems for NMR, but we have found nothing in his arguments or our experiments to dissuade us from this exciting challenge.

Neil Gershenfeld
Physics & Media Group,
Massachusetts Institute of Technology
Media Laboratory,
E15-495, 20 Ames Street,
Cambridge, MA 02139, USA
E-mail: neilg{at}media.mit.edu
Isaac Chuang
Los Alamos National Laboratory,
Los Alamos, NM 87545, USA

REFERENCES

1. N. A. Gershenfeld and I. L. Chuang, Science 275, 350 (1997) .
2. I. L. Chuang, N. A. Gershenfeld, M. G. Kubinec, D. W. Leung, in preparation.
3. I. L. Chuang, in preparation.
4. S. Lloyd, Science 261, 1569 (1993) .
5. G. Navon et al., ibid. 271, 1848 (1996) [Abstract].
6. R. Laflamme, E. Knill, W. Zurek, Los Alamos preprint quant-ph/961011 (1997).