ar X iv : c on d - m at / 0 10 52 38 v 1 1 1 M ay 2 00 1 Quantum Annealing of a Disordered Spin System

Traditional simulated annealing utilizes thermal fluctuations for convergence in optimization problems. Quantum tunneling provides a different mechanism for moving between states, with the potential for reduced time scales. We compare thermal and quantum annealing in a model disordered Ising magnet, LiHo0.44Y0.56F4, where the effects of quantum mechanics can be tuned in the laboratory by varying a magnetic field applied transverse to the Ising axis. The results indicate that quantum annealing indeed hastens convergence to the optimum state. In their presentation of simulated annealing, Kirkpatrick, Gelatt and Vecchi [1] described a fundamental connection between statistical mechanics and combinatorial optimization. Complex systems subject to conflicting constraints, from the traveling salesman problem and circuit design on one hand to spin glasses and protein folding on the other, are difficult to solve because of the vast number of nearly degenerate solutions. The introduction of a variable “temperature” permits the simulation to naturally subdivide a problem by energy scale, and as the temperature approaches zero the system settles into a local minimum (Fig. 1) that should be comparable to the ground state of the system. If the settling is performed sufficiently slowly, then the minimum is guaranteed to be the ground state [2]. However, complex systems with many degrees of freedom may require impractically long annealing schedules to find the true lowest energy configuration. If barriers between adjacent energy minima are very high yet sufficiently narrow, it may be that quantum tunneling is a more effective means at energy minimization than pure thermal processes, with the potential to hasten convergence to the ground state. As an example, consider the two-state spin system (“up” and “down”) used to introduce tunneling in quantum mechanics. The application of a magnetic field perpendicular to the up-down axis induces off-diagonal terms in the Hamiltonian, and enables tunneling between the two measured states. The assembly of a macroscopic number of such quantum spins on a lattice represents Feynman’s original concept [3] of a quantum mechanical computer. Information at the inputs (the original spin state of the system) undergoes a series of quantum mechanical operations, with the final set of ones and zeroes read at the outputs (the optimized, low energy spin state). Our experiment investigated a nontrivial optimization problem in statistical mechanics, namely that of finding the ground state for a ferromagnet