| Authors: |
G.E. Santoro
and T. Caneva
and R. Fazio
|
| Editors: |
Luciano Pietronero
and Vittorio Loreto
and Stefano Zapperi
|
| URL: |
http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1009 |
| Tags: |
adiabatic
annealing
computation
disordered
quantum
spin
statphys23
systems
topic-8
|
| 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. |
@incollection{statphys23_1009,
title = {Quantum annealing of a random Ising chain},
address = {Genova, Italy},
author = {G.E. Santoro and T. Caneva and R. Fazio},
booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
editor = {Luciano Pietronero and Vittorio Loreto and Stefano Zapperi},
month = {9-13 July},
url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1009},
year = {2007},
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.},
keywords = {adiabatic annealing computation disordered quantum spin statphys23 systems topic-8 }
}
::