Research Update: December '96 - September '97

Research Update: November '97 - June '98

Overview:

In quantum computation, the traditional Ising-like bits of a classical computer are promoted to Heisenberg-like spin-1/2 systems. Computation in an N-bit system then takes place via unitary transformations of a 2^N-dimensional Hilbert space generated by pairwise spin-spin interactions. A quantum computer can thus be treated as an interacting multi-spin system familiar to many-body physicists, albeit with peculiar (time-dependent and spatially non-local) interactions.

More than sixty years of work in numerous physical contexts has led to much intuition about quantum many-body systems, and there are a variety of analytical and numerical techniques for predicting the behavior of such systems. We propose to explore the applicability of this body of knowledge to quantum computation. Among the questions to be considered are: "What are the computational imports of such familiar notions as the large-spin limit or phase transitions?" or "What is the connection between entanglement and multi-spin correlation functions?" Answers to questions like these will allow deeper insights into the nature, limitations, and potential of quantum computation.

Some of the more interesting questions along these lines concern the applicability of many-body approximation methods to quantum computation. The essential properties of many physical systems can be discerned through a hierarchy of simple approximations (mean-field, Random Phase Approximation, variational method, ...) that avoid the complexity of dealing with the entire Hilbert space. Such methods have been shown to be successful for describing the many-body S-matrix for interacting spin-systems subject to time-dependent one-body perturbations. The relevance of these methods to quantum computation will be explored.

Monte Carlo simulations are a powerful tool for determining the exact ground state and thermal properties of quantum spin systems and hence are a natural tool to apply to quantum computers. While Monte Carlo is not a panacea, for many systems (such as spins on lattices, Bose fluids, the nuclear shell model) and for many observables of physical interest, simulations avoid the necessity of manipulating the 2^N-dimensional matrices that are the naive description of the quantum mechanics, and so allow polynomial time calculations. A successful simulation of a quantum computer along these lines would result in computations on a classical computer that have all of the advantages of a quantum computer (a "quantum -inspired" classical algorithm).

Two postdoctoral researchers working in the group of Prof. S. E. Koonin, N. J. Cerf and C. Adami, have been investigating Quantum Information Theory (QIT). This subject is concerned with the storage, manipulation, and retrieval of information in qubits such as spin-1/2 particles or polarized photons. In recent papers [h], they have explored the information content of the quantum density matrix and it’s relation to classical information theory; negative entropies in the quantum system are an essential feature of their results. Apart from implications for quantum teleportation and quantum measurement theory, these results open the possibility for using reduced-spin density matrices to describe quantum coherence and entanglement in quantum computation. A common paradigm for describing many-body systems is to truncate the heirarchy of equations of motion for the N-body density matrices at some few-body level, and employ a phenomenological closure to arrive at a tractable description. Such an approach to quantum computation can lead to a compact description and understanding of the essential features of quantum circuits.

References

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b. S. Lloyd, Phys. Rev. Lett. 75, 346(1995).

c. S. Lloyd, "Quantum Analog Computers," submitted to Science (1996).

d. J. Preskill, "Can Quantum Errors Be Corrected?," unpublished notes (November, 1994).

e. A. Despain et al., JASON Report JSR-95-115, Mitre Corporation, McLean, VA.

f. D. Beckman, A. Chari, S. Devabhaktuni, and J. Preskill, Caltech Preprint CALT-68-2021 (1995).

g. H. Mabuchi and P. Zoller, Phys. Rev. Lett.(1996).

h. N. J. Cerf and C. Adami, "Negative Entropy and Information in Quantum Mechanics," Caltech Preprint (1995), quant-phy/9512022.

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5. J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).

6. A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, Phys. Rev. Lett. 74, 4083 (1995); T. Sleator and H. Weinfurter, ibid., 4087; J. I. Cirac and P. Zoller, ibid., 4091.

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11. R. Landauer, "Is quantum mechanically coherent computation useful?," in Proceedings of the Drexel-4 Symposium on Quantum Nonintegrability -- Quantum-Classical Correspondence, edited by D. H. Feng and B.-L. Hu (International Press, to appear).

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13. For an elementary review see C. H. Bennett, G. Brassard, and A. K. Ekert, Sci. Am. Oct. 1992, 50.

14. C. H. Bennett et al., "Purification of noisy entanglement and faithful teleportation via noisy channels," (1995, unpublished); D. Deutsch et al., "Entanglement-based quantum cryptography is unconditionally secure," (1995, unpublished).

15. V. Vedral, A. Barenco, and A. Ekert, "Quantum networks for elementary arithmetic operations," Oxford preprint (1995), quantu-ph/9511018.

16. C. Miquel, J. P. Paz, R. Perazzo, "Factoring in a dissipative quantum computer," Technical report 9601021 from Los Alamos National Laboratory repository, (1996).

17. C. H. Bennett et al., "Purification of noisy entanglement and faithful teleportation via noisy channels," (1995, unpublished).

18. D. Deutsch et al., "Entanglement-based quantum cryptography is unconditionally secure," (1995, unpublished).

19. H.-K. Lo and H. F. Chau, "Quantum cryptography in noisy channels," (1995, unpublished), quant-ph/9511025.