Drawing independent samples from high-dimensional probability distributions represents the major computational bottleneck for modern algorithms, including powerful machine learning frameworks such as deep learning. The quest for discovering larger families of distributions for which sampling can be efficiently realized has inspired an exploration beyond established computing methods and turning to novel physical devices that leverage the principles of quantum computation. Quantum annealing embodies a promising computational paradigm that is intimately related to the complexity of energy landscapes in Gibbs distributions, which relate the probabilities of system states to the energies of these states. Here, we study the sampling properties of physical realizations of quantum annealers which are implemented through programmable lattices of superconducting flux qubits. Comprehensive statistical analysis of the data produced by these quantum machines shows that quantum annealers behave as samplers that generate independent configurations from low-temperature noisy Gibbs distributions. We show that the structure of the output distribution probes the intrinsic physical properties of the quantum device such as effective temperature of individual qubits and magnitude of local qubit noise, which result in a non-linear response function and spurious interactions that are absent in the hardware implementation. We anticipate that our methodology will find widespread use in characterization of future generations of quantum annealers and other emerging analog computing devices.
Two objects can be distinguished if they have different measurable properties. Thus, distinguishability depends on the Physics of the objects. In considering graphs, we revisit the Ising model as a framework to define physically meaningful spectral invariants. In this context, we introduce a family of refinements of the classical spectrum and consider the quantum partition function. We demonstrate that the energy spectrum of the quantum Ising Hamiltonian is a stronger invariant than the classical one without refinements. For the purpose of implementing the related physical systems, we perform …
Calibration of quantum computing technologies is essential to the effective utilization of their quantum resources. Specifically, the performance of quantum annealers is likely to be significantly impaired by noise in their programmable parameters, effectively misspecification of the computational problem to be solved, often resulting in spurious suboptimal solutions. We developed a strategy to determine and correct persistent, systematic biases between the actual values of the programmable parameters and their user-specified values. We applied the recalibration strategy to two D-Wave Two quan…
Boltzmann Sampling of Frustrated J1 - J2 Ising Models with Programmable Quantum Annealers
Information Systems & Modeling, Los Alamos National Laboratory（信息系统与建模，洛斯阿拉莫斯国家实验室）；Theoretical Division, Quantum & Condensed Matter Physics, Los Alamos National Laboratory（理论部，量子与凝聚态物理，洛斯阿拉莫斯国家实验室）；Center for Quantum Computing, Los Alamos National Laboratory（量子计算中心，洛斯阿拉莫斯国家实验室）
Quantum annealing is a promising approach to heuristically solving difficult combinatorial optimization problems. However, the connectivity limitations in current devices lead to an exponential degradation of performance on general problems. We propose an architecture for a quantum annealer that achieves full connectivity and full programmability while using a number of physical resources only linear in the number of spins. We do so by application of carefully engineered periodic modulations of oscillator-based qubits, resulting in a Floquet Hamiltonian in which all the interactions are tunabl…
Motivated by recent experiments in which specific thermal properties of complex many-body systems were successfully reproduced on a commercially available quantum annealer, we examine the extent to which quantum annealing hardware can reliably sample from the thermal state in a specific basis associated with a target quantum Hamiltonian. We address this question by studying the diagonal thermal properties of the canonical one-dimensional transverse-field Ising model on a D-Wave 2000Q quantum annealing processor. We find that the quantum processor fails to produce the correct expectation values…
Quantum annealers as programmable thermal machines
Wrocław University of Science and Technology（弗罗茨瓦夫理工大学）；Quantumz.io Sp. z o.o.（Quantumz.io有限责任公司）；Institute of Theoretical and Applied Informatics, Polish Academy of Sciences（波兰科学院理论与应用物理研究所）；Jülich Supercomputing Centre, Institute for Advanced Simulation, Forschungszentrum Jülich（于利希超算中心，先进模拟研究所，于利希研究中心）；University of Maryland, Baltimore County（马里兰大学巴尔的摩分校）；National Quantum Laboratory（国家量子实验室）；Quantum Science Institute（量子科学研究所）；Institut für Theoretische Physik, Universität Tübingen（蒂宾根大学理论物理研究所）
By contrasting the performance of two quantum annealers operating at different temperatures, we address recent questions related to the role of temperature in these devices and their function as `Boltzmann samplers'. Using a method to reliably calculate the degeneracies of the energy levels of large-scale spin-glass instances, we are able to estimate the instance-dependent effective temperature from the output of annealing runs. Our results show that the output distributions of the annealers do not in general correspond to classical Boltzmann distributions. For the small fraction of the instan…
We present and apply a general-purpose, multi-start algorithm for improving the performance of low-energy samplers used for solving optimization problems. The algorithm iteratively fixes the value of a large portion of the variables to values that have a high probability of being optimal. The resulting problems are smaller and less connected, and samplers tend to give better low-energy samples for these problems. The algorithm is trivially parallelizable, since each start in the multi-start algorithm is independent, and could be applied to any heuristic solver that can be run multiple times to…
We present \emph{multi-qubit correction} (MQC) as a novel postprocessing method for quantum annealers that views the evolution in an open-system as a Gibbs sampler and reduces a set of excited states to a new synthetic state with lower energy value. After sampling from the ground state of a given (Ising) Hamiltonian, MQC compares pairs of excited states to recognize virtual tunnels--i.e., a group of qubits that changing their states simultaneously can result in a new state with lower energy value--and successively converges to the ground state. Experimental results using D-Wave 2000Q quantum a…
The debate around the potential superiority of quantum annealers over their classical counterparts has been ongoing since the inception of the field by Kadowaki and Nishimori close to two decades ago. Recent technological breakthroughs in the field, which have led to the manufacture of experimental prototypes of quantum annealing optimizers with sizes approaching the practical regime, have reignited this discussion. However, the demonstration of quantum annealing speedups remains to this day an elusive albeit coveted goal. Here, we examine the power of quantum annealers to provide a different …
We propose a novel method for reducing the number of variables in quadratic unconstrained binary optimization problems, using a quantum annealer (or any sampler) to fix the value of a large portion of the variables to values that have a high probability of being optimal. The resulting problems are usually much easier for the quantum annealer to solve, due to their being smaller and consisting of disconnected components. This approach significantly increases the success rate and number of observations of the best known energy value in samples obtained from the quantum annealer, when compared wi…
Optimisation problems in science and engineering typically involve finding the ground state (i.e. the minimum energy configuration) of a cost function with respect to many variables. If the variables are corrupted by noise then this approach maximises the likelihood that the solution found is correct. An alternative approach is to make use of prior statistical information about the noise in conjunction with Bayes's theorem. The maximum entropy solution to the problem then takes the form of a Boltzmann distribution over the ground and excited states of the cost function. Here we use a programma…
For randomly selected couplers and fields, the D-Wave device typically yields a highly Boltzmann like distribution [ indicating equilibration. These equilibrated data however do not contain much useful information about the dynamics which lead to equilibration. To illuminate the dynamics, special Hamiltonians can be chosen which contain large energy barriers. In this paper we generalize this approach by considering a class of Hamiltonians which map clusters of spin-like qubits into 'superspins', thereby creating an energy landscape where local minima are separated by large energy barriers. The…
Graduate School of Information Sciences, Tohoku University（东北大学信息科学研究科）；Institute for Innovative Research, Tokyo Institute of Technology（东京工业大学创新研究所）；Sigma-i Co., Ltd.（Sigma-i 有限公司）
Quantum annealing was originally proposed as an approach for solving combinatorial optimisation problems using quantum effects. D-Wave Systems has released a production model of quantum annealing hardware. However, the inherent noise and various environmental factors in the hardware hamper the determination of optimal solutions. In addition, the freezing effect in regions with weak quantum fluctuations generates outputs approximately following a Gibbs--Boltzmann distribution at an extremely low temperature. Thus, a quantum annealer may also serve as a fast sampler for the Ising spin-glass prob…
NASA Ames Research Center（美国国家航空航天局艾姆斯研究中心）；Stinger Ghaffarian Technologies Inc.（斯汀格·加法里安技术公司）；D-Wave Systems Inc.（D-Wave系统公司）；University of Florence（佛罗伦萨大学）；Simon Fraser University（西蒙弗雷泽大学）
The development of quantum-classical hybrid (QCH) algorithms is critical to achieve state-of-the-art computational models. A QCH variational autoencoder (QVAE) was introduced in Ref. [1] by some of the authors of this paper. QVAE consists of a classical auto-encoding structure realized by traditional deep neural networks to perform inference to, and generation from, a discrete latent space. The latent generative process is formalized as thermal sampling from either a quantum or classical Boltzmann machine (QBM or BM). This setup allows quantum-assisted training of deep generative models by phy…
Recent advances in quantum technology have led to the development and manufacturing of experimental programmable quantum annealing optimizers that contain hundreds of quantum bits. These optimizers, named `D-Wave' chips, promise to solve practical optimization problems potentially faster than conventional `classical' computers. Attempts to quantify the quantum nature of these chips have been met with both excitement and skepticism but have also brought up numerous fundamental questions pertaining to the distinguishability of quantum annealers from their classical thermal counterparts. Here, we…
Quantum tunneling, a phenomenon in which a quantum state traverses energy barriers above the energy of the state itself, has been hypothesized as an advantageous physical resource for optimization. Here we show that multiqubit tunneling plays a computational role in a currently available, albeit noisy, programmable quantum annealer. We develop a non-perturbative theory of open quantum dynamics under realistic noise characteristics predicting the rate of many-body dissipative quantum tunneling. We devise a computational primitive with 16 qubits where quantum evolutions enable tunneling to the g…
Quantum simulation has emerged as a valuable arena for demonstrating and understanding the capabilities of near-term quantum computers. Quantum annealing has been used successfully in simulating a range of open quantum systems, both at equilibrium and out of equilibrium. However, in all previous experiments, annealing has been too slow to simulate a closed quantum system coherently, due to the onset of thermal effects from the environment. Here we demonstrate coherent evolution through a quantum phase transition in the paradigmatic setting of the 1D transverse-field Ising chain, using up to 20…
Experimental Measurement and Theoretical Analysis of Energy Relaxation Rates of an Interacting Two-Qubit System on a D-Wave Quantum Annealer
Tokyo University of Science（东京科学大学）；RIKEN（理化学研究所）；Department of Electrical, Electronic, and Communication Engineering, Faculty of Science and Engineering, Chuo University（中央大学理工学部电气电子通信工程系）
There is a significant ongoing effort in realizing quantum annealing with different physical platforms. The challenge is to achieve a fully programmable quantum device featuring coherent adiabatic quantum dynamics. Here we show that combining the well-developed quantum simulation toolbox for Rydberg atoms with the recently proposed Lechner-Hauke-Zoller~(LHZ) architecture allows one to build a prototype for a coherent adiabatic quantum computer with all-to-all Ising interactions and, therefore, a novel platform for quantum annealing. In LHZ a infinite-range spin-glass is mapped onto the low ene…
Temporal information processing on a 4,500-qubit quantum annealer
Institute for Cross-Disciplinary Physics and Complex Systems (IFISC) UIB-CSIC（跨学科物理与复杂系统研究所（IFISC） UIB-CSIC）；Theoretical Division, Los Alamos National Laboratory（理论部，洛斯阿拉莫斯国家实验室）；USRA Research Institute for Advanced Computer Science (RIACS)（美国大学空间研究协会先进计算机科学研究院（RIACS））；NEST, Scuola Normale Superiore（NEST，比萨高等师范学院）；D-Wave Quantum（D-Wave量子）；New Mexico Consortium（新墨西哥联盟）
Quantum annealing aims at solving optimization problems efficiently by preparing the ground state of an Ising spin-Hamiltonian quantum mechanically. A prerequisite of building a quantum annealer is the implementation of programmable long-range two-, three- or multi-spin Ising interactions. We discuss an architecture, where the required spin interactions are implemented via two-port, or in general multi-port quantum Ising wires connecting the spins of interest. This quantum annealing architecture of spins connected by Ising quantum wires can be realized by exploiting the three dimensional (3D) …
Quantum tunneling is a phenomenon in which a quantum state traverses energy barriers above the energy of the state itself. Tunneling has been hypothesized as an advantageous physical resource for optimization. Here we present the first experimental evidence of a computational role of multiqubit quantum tunneling in the evolution of a programmable quantum annealer. We develop a theoretical model based on a NIBA Quantum Master Equation to describe the multiqubit dissipative tunneling effects under the complex noise characteristics of such quantum devices. We start by considering a computational …
Resonant false vacuum decay in two dimensions on a 4000-qubit quantum annealer
Jožef Stefan Institute（约瑟夫·斯蒂芬研究所）；University of Ljubljana（卢布尔雅那大学）；Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；University of Padova（帕多瓦大学）；National Institute for Nuclear Physics (INFN)（意大利国家核物理研究所）；Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；University of Leeds（利兹大学）；Jülich Supercomputing Centre（于利希超级计算中心）；Forschungszentrum Jülich（于利希研究中心）；University of Cologne（科隆大学）；CENN Nanocenter（纳米技术卓越中心）
Recent advances in quantum technology have led to the development and manufacturing of experimental programmable quantum annealers that promise to solve certain combinatorial optimization problems of practical relevance faster than their classical analogues. The applicability of such devices for many theoretical and real-world optimization problems, which are often constrained, is severely limited by the sparse, rigid layout of the devices' quantum bits. Traditionally, constraints are addressed by the addition of penalty terms to the Hamiltonian of the problem, which in turn requires prohibiti…
With progress in quantum technology more sophisticated quantum annealing devices are becoming available. While they offer new possibilities for solving optimization problems, their true potential is still an open question. As the optimal design of adiabatic algorithms plays an important role in their assessment, we illustrate the aspects and challenges to consider when implementing optimization problems on quantum annealing hardware based on the example of the traveling salesman problem (TSP). We demonstrate that tunneling between local minima can be exponentially suppressed if the quantum dyn…
Filtering out mislabeled training instances using black-box optimization and quantum annealing
LiLz Inc.（LiLz公司）；Graduate School of Information Sciences, Tohoku University（东京大学信息科学研究生院）；Sigma-i Co., Ltd.（Sigma-i公司）；Department of Physics, Institute of Science Tokyo（科学东京系物理系）；Research and Education Institute for Semiconductors and Informatics, Kumamoto University（熊本大学半导体与信息学研究教育院）
We investigate a hybrid quantum-classical solution method to the mean-variance portfolio optimization problems. Starting from real financial data statistics and following the principles of the Modern Portfolio Theory, we generate parametrized samples of portfolio optimization problems that can be related to quadratic binary optimization forms programmable in the analog D-Wave Quantum Annealer 2000Q. The instances are also solvable by an industry-established Genetic Algorithm approach, which we use as a classical benchmark. We investigate several options to run the quantum computation optimally…
Recent advances in quantum technology have led to the development and the manufacturing of programmable quantum annealers that promise to solve certain combinatorial optimization problems faster than their classical counterparts. Semi-supervised learning is a machine learning technique that makes use of both labeled and unlabeled data for training, which enables a good classifier with only a small amount of labeled data. In this paper, we propose and theoretically analyze a graph-based semi-supervised learning method with the aid of the quantum annealing technique, which efficiently utilize th…
Quantum computing is no longer a nascent field. Programmable quantum annealing devices with more that 1000 qubits are commercially available. How does one know that a putative quantum annealing device is indeed quantum? How should one go about benchmarking its performance and compare it to classical algorithms? How can its performance be improved by error correction? In this contribution to the focus collection on "What would you do with 1000 qubits?", we review the work we and others have done in this area, since the first D-Wave quantum annealer with 108 qubits was made available to us. Some…
Quantum annealing promises to solve complex combinatorial optimization problems faster than current transistor-based computer technologies. Although to date only one commercially-available quantum annealer is procurable, one can already start to map out the application scope of these novel optimization machines. These mid-scale programmable analog special-purpose devices could, potentially, revolutionize optimization. However, their disruptive application domain remains to be found. While the commercial analog quantum optimization machine by D-Wave Systems Inc. already exceeds 1000 qubits, her…
