Advances in physics theories and applications

Adiabatic Quantum Computation In Satisfiability Problems, Quantum Chaos In The Quantum Computer, Quantum Chaos And Random Matrix Theory, Quantum Chaos And Imperfection Effects, Decoherence, Einselection And The Quantum Origins, Separability Of Very Noisy

K.N.P. Kumar

Published Jan 1, 2013• 0 citations

Abstract

Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Michael Sipser give a quantum algorithm for solving instances of (e) the satisfiability problem, based on (e) adiabatic evolution. The evolution of the quantum state is governed by (e&eb)a time-dependent Hamiltonian that interpolates between (e&eb) an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes (e) the satisfying assignment. To ensure that the system evolves to *(e&eb) the desired final ground state, the evolution time must be (eb) big enough. Time required depends on (=) the minimum energy differences between (e&eb) the two lowest.... The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-427314-431866.pdf

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Quantum Computation by Adiabatic Evolution

Edward Farhi, Jeffrey Goldstone, Sam Gutmann +1 more

We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time.

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Case studies in quantum adiabatic optimization

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Quantum adiabatic optimization is a quantum algorithm for solving classical optimization problems (E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. Quantum computation by adiabatic evolution, 2000. arXiv:quant-ph/0001106). The solution to an optimization prob-lem is encoded in the ground state of a "problem Hamiltonian " Hp which acts on the Hilbert space of n spin 1/2 particles and is diagonal in the Pauli z basis. To produce this ground state, one first initializes the quantum system in the ground state of a different Hamiltonian and then adiabatically changes the Hamiltonian into Hp. Farhi et al suggest the interpolating Hamiltonian i where the parameter s is slowly changed as a function of time between 0 and 1. The running time of this algorithm is related to the minimum spectral gap of H(s) for s E (0, 11. We study such transverse field spin Hamiltonians using both analytic and numerical techniques. Our approach is example-based, that is, we study some specific choices for the problem Hamiltonian Hp which illustrate the breadth of phenomena which can occur. We

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