On the identification of the ground state based on occupation probabilities: An investigation of Smith's apparent counterexamples (original) (raw)
2006
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Abstract
We study a set of truncated matrices, given by Smith~\cite{Smith2005}, in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert's tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable choice of some parameter it can always be removed. We also argue that
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A quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. This quantum adiabatic behavior is the basis of a new class of algorithms for quantum computing. We tested one such algorithm by applying it to randomly generated hard instances of an NP-complete problem. For the small examples that we could simulate, the quantum adiabatic algorithm worked well, providing evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.
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References (8)
- T.D. Kieu. Computing the non-computable. Contemporary Physics, 44:51-77, 2003.
- T.D. Kieu. Quantum adiabatic algorithm for Hilbert's tenth problem: I. The algorithm. ArXiv:quant-ph/0310052, 2003.
- T.D. Kieu. Quantum algorithms for Hilbert's tenth problem. Int. J. Theor. Phys., 42:1451-1468, 2003.
- T.D. Kieu. A reformulation of Hilbert's tenth problem through quantum mechanics. Proc. Roy. Soc., A 460:1535- 1545, 2004.
- T.D. Kieu. Hypercomputability of quantum adiabatic processes: Fact versus prejudices. ArXiv:quant-ph/0504101, 2005.
- T.D. Kieu. Mathematical computability questions for some classes of linear and non-linear differential equations originated from Hilbert's tenth problem. arXiv:math.GM/0507109, 2005.
- T.D. Kieu. A mathematical proof for a ground-state identification criterion. arXiv:quant-ph/0602146, 2006.
- Warren D. Smith. Three counterexamples refuting kieu's plan for "quantum adiabatic hypercomputation"; and some uncomputable quantum mechanical tasks. http://math.temple.edu/ wds/homepage/works.html, #85, 2005. To appear in the Journal of the Association for Computing Machinery.
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