Adiabatic computation: A toy model

Abstract

We discuss a toy model for adiabatic quantum computation which displays some phenomenological properties expected in more realistic implementations. This model has two free parameters: the adiabatic evolution parameter s and the α parameter which emulates many-variables constrains in the classical computational problem. The proposed model presents, in the s − α plane, a line of first order quantum phase transition that ends at a second order point. The relation between computation complexity and the occurrence of quantum phase transitions is discussed. We analyze the behavior of the ground and first excited states near the quantum phase transition, the gap and the entanglement content of the ground state.

The quantum adiabatic algorithm applied to random optimization problems: The quantum spin glass perspective

“…The choice of H i is more crucial, and for general optimization problems, it is mainly an open problem to understand whether the modification of the annealing path can change the scaling of the time needed for the adiabatic condition to hold. Such a possible change was argued for in [110,111] to avoid gaps at a first order phase transition for fully connected models by introducing a twoparameter annealing path (see also [112] for another example of a two-parameter annealing path). Alternatively, a randomization of H i was proposed in [41] to avoid gaps of the second type in the classification above, that appear within the classical spin glass phase, for a particular problem; but its efficiency for more general optimization problems is still an open question.…”

On quantum mean-field models and their quantum annealing

“…To avoid the phase transitions that appear when Ĥi is a transverse field, it was, for instance, proposed in [79] to randomize the direction of the transverse fields on each spin. Very recently another proposal was to include antiferromagnetic couplings in the interpolating Hamiltonian [36]; in this way it is possible to avoid the first-order phase transition by making a detour in the (s, λ) plane (see also [32] for a similar phenomenon). The annealing towards the ferromagnet for p ≥ 3 studied in the paper was particularly inefficient because the groundstate of the initial Hamiltonian (the transverse field − mx ) remained metastable all the way to s = 1.…”

“…However, despite their simplicity that allows for an analytical resolution, they exhibit some of the features expected also in more realistic optimization problems, and therefore constitute useful toy-models to study. The statics [30][31][32][33][34][35][36] and the dynamics, both for quantum annealings [37][38][39][40] and for quantum quenches [41][42][43], of this kind of models have been largely studied. From a technical point of view these models are relatively simple because their mean-field character allows for a semi-classical treatment, the small parameter in this limit being the inverse of the size of the system (instead of in usual semi-classical computations).…”