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Traditional simulated annealing uses thermal fluctuations for convergence in optimization problems. Quantum tunneling provides a different mechanism for moving between states, with the potential for reduced time scales. Thermal and quantum annealing are compared in a model disordered magnet, where the effects of quantum mechanics can be tuned by varying an applied magnetic field. The results indicate that quantum annealing hastens convergence to the optimum state.

Glassy behavior (including hysteresis, irreversibility, a peak in the zero-field-cooled magnetization, and nonexponential relaxation) is observed in a quenched ferrofluid system consisting of 50-A magnetite particles. An

The classical, thermally driven transition in the dipolar-coupled Ising ferromagnet LiHoF 4 ͑T c 1.53 K͒ can be converted into a quantum transition driven by a transverse magnetic field H t at T 0. The transverse field, applied perpendicular to the Ising axis, introduces channels for quantum relaxation, thereby depressing T c . We have determined the phase diagram in the H t -T plane via magnetic susceptibility measurements. The critical exponent, g 1, has a mean-field value in both the classical and quantum limits. A solution of the full mean-field Hamiltonian using the known LiHoF 4 crystal-field wave functions, including nuclear hyperfine terms, accurately matches experiment. [S0031-9007(96) Quantum phase transitions can differ fundamentally from their classical counterparts because of the unparalleled influence of the dynamics on the T 0 static critical behavior [1]. In addition, unusual electronic and magnetic behavior can arise at nonzero temperature. Thus includes the peculiar mix of the spin and charge degrees of freedom in transition-metal oxides [2], the apparent "non-Fermi-liquid" behavior of highly correlated f-electron compounds [3,4], and the unusual normal-state properties of the high-T c superconducting cuprates [5][6][7][8]. The remarkable properties of these systems have been ascribed in each case to the proximity of a T 0 quantum critical point.There remain considerable experimental and theoretical barriers to describing quantum phase transitions with fidelity and precision. In the high-temperature superconductors, for example, the superconductivity masks the direct study of the quantum order-disorder transition. In heavy-fermion materials, characterization of the T 0 magnetic instability is complicated by the presence of charge carriers and by substitutional disorder. In spin glasses [9], the combination of frustration and disorder impedes consensus on a correct description of even the thermally driven transition. Moreover, despite their power and elegance, pressure-tuning studies of quantum critical points [2,3,10] cannot approach the exactitude which has become the hallmark of experiments on classical critical phenomena.High-precision measurements of quantum critical behavior in clean, insulating magnets simply do not exist, even with the great current interest in quantum phase transitions. Therefore, we have carried out such measurements for a model magnet LiHoF 4 . The key conclusion is that the quantum critical behavior is mean-fieldlike, as predicted by long-standing and elegant theory identifying ͑T 0͒ quantum phase transitions in d dimensions with thermal phase transitions in d 1 1 dimensions [11]. Furthermore, a mean-field theory using known magnetic parameters quantitatively describes the observed magnetic susceptibility in both the quantum and the thermal regimes.LiHoF 4 in an external field H t is the experimental realization of the simplest quantum spin model, namely, the Ising magnet in a transverse magnetic field. The corresponding Hamiltonian iswhere the s's are Pauli spin mat...