We study the performance of quantum annealing for systems with ground-state degeneracy by directly solving the Schrödinger equation for small systems and quantum Monte Carlo simulations for larger systems. The results indicate that naive quantum annealing using a transverse field may not be well suited to identify all degenerate ground-state configurations, although the value of the ground-state energy is often efficiently estimated. An introduction of quantum transitions to all states with equal weights is shown to greatly improve the situation but with a sacrifice in the annealing time. We also clarify the relation between the spin configurations in the degenerate ground states and the probabilities that those states are obtained by quantum annealing. The strengths and weaknesses of quantum annealing for problems with degenerate ground states are discussed in comparison with classical simulated annealing.

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We demonstrate feedback cooling of the center-of-mass motion of single charged nanoparticles to millikelvin temperatures in three dimensions via applying oscillating electric fields synchronized to their optically observed motion. The observed motional temperatures at weak feedback agree with a simple model and allow us to estimate the charge number of trapped nanoparticles. The agreement between our model and experiments is confirmed by independent measurements of the charge numbers based on a shift in the oscillation frequency induced by a constant electric field. The demonstrated temperature of below 10 mK at 4 × 10 −3 Pa is lower than that with the conventional optical cooling approach at this pressure by one to two orders of magnitude. Our results form the basis of manipulating cold charged nanoparticles and paves the way to quantum mechanical studies with trapped nanoparticles near their ground state.Manipulating the motion of objects near their quantum ground state has been a crucial subject in diverse fields from quantum simulations [1-3] and quantum information processing [4] to precision measurements [5,6]. Cooling atomic ions and ensembles of neutral atoms to their motional ground state has been successful [2,7]. Specific vibrational modes of nanoand micromechanical oscillators have been brought to their quantum ground state [8,9]. However, cooling the motion of particles including more than a few atoms to their motional ground state has been an elusive goal. The main difficulty lies in the absence of an efficient mechanism for cooling.Cold nanoparticles are expected to possess various applications such as testing quantum mechanics for macroscopic objects [10,11], ultrasensitive force and mass sensing [12][13][14][15][16][17][18], and the laboratory test of the collisional dynamics of interstellar materials [19]. Up to now, cooling the motion of nanoparticles to millikelvin temperatures has been demonstrated via all-optical approaches, where trapping, observing, and cooling them are all based on light scattering [20][21][22][23][24][25]. The lowest temperature achieved with all-optical approaches is finally limited by random photon recoils [26]. To overcome the limitation from photon recoils, an all-electrical approach for highly charged particles has been proposed [27].Here, we show that the motional temperature of single charged nanoparticles in an optical trap is efficiently lowered via the optical measurement of the nanoparticle's position and the application of oscillating electric fields synchronized to their motion. The observed motional temperatures with the electric feedback T eff agree with a simple model only when the mass of the nanoparticle is properly estimated through the time scale of the rethermalization of the motion after it is cooled. The agreement between our model and experimental results is confirmed by independent measurements of the charge number based on the electric-field-induced shift in the oscillation frequency.Compared to the conventional all-optical cooling method, param...