  Recently, a programmable quantum annealing machine has been built that
minimizes the cost function of hard optimization problems by adiabatically
quenching quantum fluctuations. Tests performed by different research teams
have shown that, indeed, the machine seems to exploit quantum effects. However
experiments on a class of random-bond instances have not yet demonstrated an
advantage over classical optimization algorithms on traditional computer
hardware. Here we present evidence as to why this might be the case. These
engineered quantum annealing machines effectively operate coupled to a
decohering thermal bath. Therefore, we study the finite-temperature critical
behavior of the standard benchmark problem used to assess the computational
capabilities of these complex machines. We simulate both random-bond Ising
models and spin glasses with bimodal and Gaussian disorder on the D-Wave
Chimera topology. Our results show that while the worst-case complexity of
finding a ground state of an Ising spin glass on the Chimera graph is not
polynomial, the finite-temperature phase space is likely rather simple: Spin
glasses on Chimera have only a zero-temperature transition. This means that
benchmarking optimization methods using spin glasses on the Chimera graph might
not be the best benchmark problems to test quantum speedup. We propose
alternative benchmarks by embedding potentially harder problems on the Chimera
topology. Finally, we also study the (reentrant) disorder-temperature phase
diagram of the random-bond Ising model on the Chimera graph and show that a
finite-temperature ferromagnetic phase is stable up to 19.85(15)%
antiferromagnetic bonds. Beyond this threshold the system only displays a
zero-temperature spin-glass phase. Our results therefore show that a careful
design of the hardware architecture and benchmark problems is key when building
quantum annealing machines.
