%0 Book Section %1 statphys23_1009 %A Santoro, G.E. %A Caneva, T. %A Fazio, R. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K adiabatic annealing computation disordered quantum spin statphys23 systems topic-8 %T Quantum annealing of a random Ising chain %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1009 %X We present results concerning the application of a Quantum Annealing (QA) strategy (alias Adiabatic Quantum Computation) to the determination of the trivial classical ground state of the one-dimensional random Ising ferromagnet $-\sum_i J_i \sigma^z_i \sigma^z_i+1$. The QA approach consists in adding to the classical Hamiltonian a source of time-dependent quantum fluctuations, for instance a transverse field term $-\Gamma(t)\sum_i \sigma^x_i$, transforming the classical ground state search into a time-dependent Schroedinger dynamics where the quantum fluctuations are switched off. The one-dimensional case is particularly useful because, due to the quadratic nature of the problem in terms of Wigner-Jordan fermions, one can follow the time-dependent Scroedinger dynamics in an essentially exact way, even for large chain sizes. We show that the presence, in the quantum Hamiltonian, of an infinite randomness critical point --- separating the large-$\Gamma$ paramagnetic phase from the small-$\Gamma$ ferromagnetic one, and analyzed in detail by D.S. Fisher in PRB 51, 6411 (1995) --- makes the Schroedinger dynamics intrinsically slow in attaining the correct classical ferromagnetic state: indeed, the residual energy $E_res$ after annealing decreases as an inverse power of the logarithm of the annealing time $\tau$ \ E_res(\tau) 1łog^\zeta(\tau) \ in a way that is qualitatively not different (although quantitatively better, because of a larger $\zeta$) from what classical simulated annealing would do (see D.A. Huse and D.S. Fisher, PRL 57, 2203 (1986)). We believe that this represents a paradigmatic illustration of how a computationally simple problem can become highly non-trivial for a quantum dynamical approach whenever disorder plays a role.

@incollection{statphys23_1009, abstract = {We present results concerning the application of a Quantum Annealing (QA) strategy (alias Adiabatic Quantum Computation) to the determination of the trivial classical ground state of the one-dimensional random Ising ferromagnet $-\sum_i J_i \sigma^z_i \sigma^z_{i+1}$. The QA approach consists in adding to the classical Hamiltonian a source of time-dependent quantum fluctuations, for instance a transverse field term $-\Gamma(t)\sum_i \sigma^x_i$, transforming the classical ground state search into a time-dependent Schroedinger dynamics where the quantum fluctuations are switched off. The one-dimensional case is particularly useful because, due to the quadratic nature of the problem in terms of Wigner-Jordan fermions, one can follow the time-dependent Scroedinger dynamics in an essentially exact way, even for large chain sizes. We show that the presence, in the quantum Hamiltonian, of an infinite randomness critical point --- separating the large-$\Gamma$ paramagnetic phase from the small-$\Gamma$ ferromagnetic one, and analyzed in detail by D.S. Fisher in PRB {\bf 51}, 6411 (1995) --- makes the Schroedinger dynamics intrinsically slow in attaining the correct classical ferromagnetic state: indeed, the residual energy $E_{res}$ after annealing decreases as an inverse power of the {\em logarithm} of the annealing time $\tau$ \[ E_{res}(\tau) \propto \frac{1}{\log^{\zeta}{(\gamma \tau)}} \] in a way that is qualitatively not different (although quantitatively better, because of a larger $\zeta$) from what classical simulated annealing would do (see D.A. Huse and D.S. Fisher, PRL {\bf 57}, 2203 (1986)). We believe that this represents a paradigmatic illustration of how a computationally simple problem can become highly non-trivial for a quantum dynamical approach whenever disorder plays a role.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Santoro, G.E. and Caneva, T. and Fazio, R.}, biburl = {https://www.bibsonomy.org/bibtex/22799f575fbfde3908942f29d37871cf9/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {5e787b58ae980b39513bf33b474d4b3a}, intrahash = {2799f575fbfde3908942f29d37871cf9}, keywords = {adiabatic annealing computation disordered quantum spin statphys23 systems topic-8}, month = {9-13 July}, timestamp = {2007-06-20T10:16:36.000+0200}, title = {Quantum annealing of a random Ising chain}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1009}, year = 2007 }