Many interesting but practically intractable problems can be reduced to that of finding the ground state of a system of interacting spins; however, finding such a ground state remains computationally difficult. It is believed that the ground state of some naturally occurring spin systems can be effectively attained through a process called quantum annealing. If it could be harnessed, quantum annealing might improve on known methods for solving certain types of problem. However, physical investigation of quantum annealing has been largely confined to microscopic spins in condensed-matter systems. Here we use quantum annealing to find the ground state of an artificial Ising spin system comprising an array of eight superconducting flux quantum bits with programmable spin-spin couplings. We observe a clear signature of quantum annealing, distinguishable from classical thermal annealing through the temperature dependence of the time at which the system dynamics freezes. Our implementation can be configured in situ to realize a wide variety of different spin networks, each of which can be monitored as it moves towards a low-energy configuration. This programmable artificial spin network bridges the gap between the theoretical study of ideal isolated spin networks and the experimental investigation of bulk magnetic samples. Moreover, with an increased number of spins, such a system may provide a practical physical means to implement a quantum algorithm, possibly allowing more-effective approaches to solving certain classes of hard combinatorial optimization problems.

Efforts to develop useful quantum computers have been blocked primarily by environmental noise. Quantum annealing is a scheme of quantum computation that is predicted to be more robust against noise, because despite the thermal environment mixing the system's state in the energy basis, the system partially retains coherence in the computational basis, and hence is able to establish well-defined eigenstates. Here we examine the environment's effect on quantum annealing using 16 qubits of a superconducting quantum processor. For a problem instance with an isolated small-gap anticrossing between the lowest two energy levels, we experimentally demonstrate that, even with annealing times eight orders of magnitude longer than the predicted single-qubit decoherence time, the probabilities of performing a successful computation are similar to those expected for a fully coherent system. Moreover, for the problem studied, we show that quantum annealing can take advantage of a thermal environment to achieve a speedup factor of up to 1,000 over a closed system.

We have developed a method for extracting the high-frequency noise spectral density of an rf-SQUID flux qubit from macroscopic resonant tunneling (MRT) rate measurements. The extracted noise spectral density is consistent with that of an ohmic environment up to frequencies ∼ 4 GHz. We have also derived an expression for the MRT lineshape expected for a noise spectral density consisting of such a broadband ohmic component and an additional strongly peaked low-frequency component. This hybrid model provides an excellent fit to experimental data across a range of tunneling amplitudes and temperatures.

We report measurements of macroscopic resonant tunneling between the two lowest energy states of a pair of magnetically coupled rf-SQUID flux qubits. This technique provides a direct means of observing two-qubit dynamics and a probe of the environment coupled to the pair of qubits. Measurements of the tunneling rate as a function of qubit flux bias show a Gaussian line shape that is well matched to theoretical predictions. Moreover, the peak widths indicate that each qubit is coupled to a local environment whose fluctuations are uncorrelated with that of the other qubit.Superconducting circuits have played an essential role in realizing quantum mechanical phenomena in macroscopic systems. One such example is the observation of macroscopic resonant tunneling (MRT) of magnetic flux between the lowest energy states of single rf-SQUID flux qubits, as demonstrated by several groups [1][2][3][4]. These measurements provide both a clear signature of quantum mechanics in a macroscopic circuit at a finite temperature and in the presence of noise and a direct means of determining the tunneling energy between states. Theoretical descriptions of the MRT rate have been presented [5,6] and indicate a direct connection between the profile of the MRT rate peaks and properties of the environment. Analogous measurements of the tunneling of magnetization in nanomagnets [7,8] suggest that MRT is responsible for dynamics in these materials as well.In this work, we extend measurements of MRT to inductively coupled pairs of flux qubits. We present experimental observations of tunneling between the two lowest energy states of the coupled system for several coupling strengths. These data yield two-qubit energy gaps that match those predicted by the independently calibrated Hamiltonian of the coupled system. Moreover, measurements of the two-qubit energy gap are used to infer single qubit energy gaps at ∼ h × 10 9 Hz without the use of microwave lines. Finally, the profile of the MRT rate versus qubit flux bias has a Gaussian lineshape with a width that is a factor of √ 2 larger than that of a single qubit. We argue that this observation indicates that the environments coupled to each qubit are uncorrelated.For a single flux qubit, an MRT experiment consists of measuring the rate of tunneling of flux between two wells of the double-well potential of the rf SQUID when the lowest energy levels of each well are closely aligned. Restricting the dynamics of the single rf SQUID to its two lowest energy states allows one to map the physics of this device onto the canonical qubit Hamiltonian:where σ x,z are Pauli matrices, ǫ ≡ 2I p (Φ (1) is valid when |ǫ|, ∆ ≪hω p , wherehω p is the energy spacing to the next excited state of the rf SQUID. For a non-Markovian environment [9], the initial tunneling rate from |0 to |1 (eigenstates of σ z ) versus ǫ has a Gaussian profile, as given by Eq. (2) in Ref. [2].A natural extension to the single qubit MRT experiment is to add a second qubit that is inductively coupled to a first qubit via a mutual induct...