Quantum energy gaps and first-order mean-field transitions

We study first-order mean-field quantum phase transitions, in particular the exponentially fast closure of the energy gap with the system size. We consider exactly solvable ferromagnetic models and discuss their relation to the Grover problem to which they reduce in a particular limit. We compute the coefficient in the exponential closure of the gap using an instantonic approach. We also discuss the (dire) consequences this has for quantum annealing and combinatorial optimization. Many important practical problems involve the minimization of a function of discrete variables. Solving such combinatorial problems by temperature annealing is a classical strategy in computer science [1]: the idea is to use thermal fluctuations to avoid trapping the system in local minima, and thereby efficiently visit the whole configuration space. It has been proposed to extend this approach to quantum fluctuations [2]; it is thus of interest to ask whether annealing by tuning down the amplitude of a quantum mechanical kinetic operator such as a transverse magnetic field can outperform the classical approach. In particular, can problems that normally take exponential time be solved in only polynomial time? Some considerable effort has been devoted to this question in the context of difficult combinatorial problems (see for instance [3]) which have a counterpart in statistical physics where they corresponds to mean-field spin-glass models [4, 5]. However, most of the studies were purely numerical and thus restricted to very small sizes due to the difficulty of simulating quantum mechanics without a quantum computer. In a recent Letter [6] (see also [7]), we argued that with the usual implementation of the quantum annealing it is likely that the most difficult systems undergo a quantum transition of the first order as the transverse field is tuned; this is a generic feature of a category of quantum spin glasses [8]. More recently, a first order transition has indeed been indentified in the phase diagram of some of the most studied random optimization problems, the XORSAT problem [9]. All this implies the failure of the quantum annealing algorithm for the hardest optimization problems. The goal of the present paper is to illustrate these features via a complete analytical and detailed numerical analysis of a family of models, in order to show how a precise estimate of the energy gap at the transition can be obtained. In a nutshell, the reason why quantum annealing is not an efficient strategy for finding the ground state across a first-order transition can be understood from a simple, qualitative argument. Quantum annealing could in principle be more efficient than thermal annealing for certain classes of problems: From the WKB approximation it is well known that a quantum particle tunnels rapidly through very high (in energy) but thin (in distance) energy barriers. Thermal annealing is much better at low, but deep barrier crossing. However, in a first-order transition the two states whose free energies cross are generally far from each other in the phase space; quantum tunneling must be inefficient. To make this argument more precise, one can consider the energy gap between the two lowest energy states using the standard implementation [2] for quantum spin annealing; the time needed to actually reach the ground state is bounded by the inverse of the gap as ! −2: a small gap implies a large running time. Mean-field first-order transitions have generically an exponentially small gap, typically Ne− N where N is the system size, and can be computed analytically in mean-field models using an instantonic approach. In turns, this implies ! eN , that is to say: quantum annealing is an exponentially slow algorithm for a mean-field system at a first-order transition1. In this Letter, we consider a family of simple ferromagnetic models that allow a detailed numerical and analytical analy1In finite dimensions one expects that nucleation will help.