In a recent preprint (arXiv:1305.4904) entitled "Classical signature of quantum annealing" Smolin and Smith point out that a bimodal distribution presented in (arXiv:1304.4595) for the success probability in the D-Wave device does not in itself provide sufficient evidence for quantum annealing, by presenting a classical model that also exhibits bimodality. Here we analyze their model and in addition present a similar model derived from the semi-classical limit of quantum spin dynamics, which also exhibits a bimodal distribution. We find that in both cases the correlations between the success probabilities of these classical models and the D-Wave device are weak compared to the correlations between a simulated quantum annealer and the D-Wave device. Indeed, the evidence for quantum annealing presented in arXiv:1304.4595 is not limited to the bimodality, but relies in addition on the success probability correlations between the D-Wave device and the simulated quantum annealer. The Smolin-Smith model and our semi-classical spin model both fail this correlation test.
In a recent preprint (arXiv:1305.4904) entitled "Classical signature of quantum annealing" Smolin and Smith point out that a bimodal distribution presented in (arXiv:1304.4595) for the success probability in the D-Wave device does not in itself provide sufficient evidence for quantum annealing, by presenting a classical model that also exhibits bimodality. Here we analyze their model and in addition present a similar model derived from the semi-classical limit of quantum spin dynamics, which also exhibits a bimodal distribution. We find that in both cases the correlations between the success probabilities of these classical models and the D-Wave device are weak compared to the correlations between a simulated quantum annealer and the D-Wave device. Indeed, the evidence for quantum annealing presented in arXiv:1304.4595 is not limited to the bimodality, but relies in addition on the success probability correlations between the D-Wave device and the simulated quantum annealer. The Smolin-Smith model and our semi-classical spin model both fail this correlation test.
Reverse annealing is a variant of quantum annealing, in which the system is prepared in a classical state, reverse-annealed to an inversion point, and then forward-annealed. We report on reverse annealing experiments using the D-Wave 2000Q device, with a focus on the p=2p-spin problem, which undergoes a second order quantum phase transition with a gap that closes polynomially in the number of spins. We concentrate on the total and partial success probabilities, the latter being the probabilities of finding each of two degenerate ground states of all spins up or all spins down, the former being their sum. The empirical partial success probabilities exhibit a strong asymmetry between the two degenerate ground states, depending on the initial state of the reverse anneal. To explain these results, we perform open-system simulations using master equations in the limits of weak and strong coupling to the bath. The former, known as the adiabatic master equation (AME), with decoherence in the instantaneous energy eigenbasis, predicts perfect symmetry between the two degenerate ground states, thus failing to agree with the experiment. In contrast, the latter, known as the polaron transformed Redfield equation (PTRE), is in close agreement with experiment. The same is true for a purely classical model known as spin-vector Monte Carlo with transverse-field-dependent updates (SVMC-TF). Thus our results present a strong challenge to the sufficiency of the weak system-bath coupling limit in describing the dynamics of current experimental quantum annealers, at least for annealing on timescales of a μsec or longer.
A recent experiment [T. Lanting et al., Phys. Rev. X 4, 021041 (2014)] claimed to provide evidence of up to eight-qubit entanglement in a D-Wave quantum annealing device. However, entanglement was measured using qubit tunneling spectroscopy, a technique that provides indirect access to the state of the system at intermediate times during the anneal by performing measurements at the end of the anneal with a probe qubit. In addition, an underlying assumption was that the quantum transverse-field Ising Hamiltonian, whose ground states are already highly entangled, is an appropriate model of the device and not some other (possibly classical) model. This begs the question of whether alternative classical or semiclassical models would be equally effective at predicting the observed spectrum and thermal state populations. To check this, we consider a recently proposed classical rotor model with classical Monte Carlo updates, which has been successfully employed in describing features of earlier experiments involving the device. We also consider simulated quantum annealing with quantum Monte Carlo updates, an algorithm that samples from the instantaneous Gibbs state of the device Hamiltonian. Finally, we use the quantum adiabatic master equation, which cannot be efficiently simulated classically and which has previously been used to successfully capture the open-system quantum dynamics of the device. We find that only the master equation is able to reproduce the features of the tunneling spectroscopy experiment, while both the classical rotor model and simulated quantum annealing fail to reproduce the experimental results. We argue that this bolsters the evidence for the reported entanglement.
Recently the question of whether the D-Wave processors exhibit large-scale quantum behavior or can be described by a classical model has attracted significant interest. In this work we address this question by studying a 503-qubit D-Wave Two device in the ``black box'' model i.e., by studying its input-output behavior. Our work generalizes an approach introduced in Boixo et al. [Nat. Commun. 4, 2067 (2013)] and uses groups of up to 20 qubits to realize a transverse Ising model evolution with a ground-state degeneracy whose distribution acts as a sensitive probe that distinguishes classical and quantum models for the D-Wave device. Our findings rule out all classical models proposed to date for the device and provide evidence that an open-system quantum dynamical description of the device that starts from a quantized energy level structure is well justified, even in the presence of relevant thermal excitations and a small value of the ratio of the single-qubit decoherence time to the annealing time.
Quantum computation performs calculations by using quantum devices instead of electronic devices following classical physics and used by classical computers. Although general purpose quantum computers of practical scale may be many years away, special purpose quantum computers are being built with capabilities exceeding classical computers. One prominent case is the so-called D-Wave quantum computer, which is a computing hardware device built to implement quantum annealing for solving combinatorial optimization problems. Whether D-Wave computing hardware devices display a quantum behavior or can be described by a classical model has attracted tremendous attention, and it remains controversial to determine whether quantum or classical effects play a crucial role in exhibiting the computational input–output behaviors of the D-Wave devices. This paper consists of two parts where the first part provides a review of quantum annealing and its implementations, and the second part proposes statistical methodologies to analyze data generated from annealing experiments. Specifically, we introduce quantum annealing to solve optimization problems and describe D-Wave computing devices to implement quantum annealing. We illustrate implementations of quantum annealing using Markov chain Monte Carlo (MCMC) simulations carried out by classical computers. Computing experiments have been conducted to generate data and compare quantum annealing with classical annealing. We propose statistical methodologies to analyze computing experimental data from a D-Wave device and simulated data from the MCMC based annealing methods, and establish asymptotic theory and check finite sample performances for the proposed statistical methodologies. Our findings confirm bimodal histogram patterns displayed in input–output data from the D-Wave device and both U-shape and unimodal histogram patterns exhibited in input–output data from the MCMC based annealing methods. Further statistical explorations reveal possible sources for the U-shape patterns. On the other hand, our statistical analysis produces statistical evidence to indicate that input–output data from the D-Wave device are not consistent with the stochastic behaviors of any MCMC based annealing models under the study. We present a list of statistical research topics for the future study on quantum annealing and MCMC simulations.
Recent tests performed on the D-Wave Two quantum annealer have revealed no clear evidence of speedup over conventional silicon-based technologies. Here we present results from classical parallel-tempering Monte Carlo simulations combined with isoenergetic cluster moves of the archetypal benchmark problem---an Ising spin glass---on the native chip topology. Using realistic uncorrelated noise models for the D-Wave Two quantum annealer, we study the best-case resilience, i.e., the probability that the ground-state configuration is not affected by random fields and random-bond fluctuations found on the chip. We thus compute classical upper-bound success probabilities for different types of disorder used in the benchmarks and predict that an increase in the number of qubits will require either error correction schemes or a drastic reduction of the intrinsic noise found in these devices. We restrict this study to the exact ground state, however, the approach can be trivially extended to the inclusion of excited states if the success metric is relaxed. We outline strategies to develop robust, as well as hard benchmarks for quantum annealing devices, as well as any other (black box) computing paradigm affected by noise.
Reverse annealing is a variant of quantum annealing, in which the system is prepared in a classical state, reverse annealed to an inversion point, and then forward annealed. We report on reverse annealing experiments using the D-Wave 2000Q device, with a focus on the p=2p -spin problem, which undergoes a second-order quantum phase transition with a gap that closes polynomially in the number of spins. We concentrate on the total and partial success probabilities, with the latter being the probabilities of finding each of two degenerate ground states of all spins up or all spins down, and the former being their sum. The empirical partial success probabilities exhibit a strong asymmetry between the two degenerate ground states, depending on the initial state of the reverse anneal. To explain these results, we perform open-system simulations using master equations in the limits of weak and strong coupling to the bath. The former, known as the adiabatic master equation, with decoherence in the instantaneous energy eigenbasis, predicts perfect symmetry between the two degenerate ground states, thus failing to agree with the experiment. In contrast, the latter, known as the polaron transformed Redfield equation, is in close agreement with experiment. Thus, our results present a challenge to the sufficiency of the weak system-bath coupling limit in describing the dynamics of current experimental quantum annealers, at least for reverse annealing on timescales of a microsecond or longer.