Quantum annealing of a random Ising chain
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 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 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.