The D-wave processor is a partially controllable open quantum system which exchanges energy with its surrounding environment (in the form of heat) and with the external time dependent control fields (in the form of work). Despite being rarely thought as such, it is a thermodynamic machine. Here we investigate the properties of the D-Wave quantum annealers from a thermodynamical perspective. We performed a number of reverse-annealing experiments on the D-Wave 2000Q via the open access cloud server Leap, with the aim of understanding what type of thermal operation the machine performs, and quantifying the degree of dissipation that accompanies it, as well as the amount of heat and work that it exchanges. The latter is a challenging task in view of the fact that one can experimentally access only the overall energy change occurring in the processor, (which is the sum of heat and work it receives). However, recent results of non-equilibrium thermodynamics(namely, the fluctuation theorem and the thermodynamic uncertainty relations), allow to calculate lower bounds on the average entropy production (which quantifies the degree of dissipation) as well as the average heat and work exchanges. The analysis of the collected experimental data shows that 1) in a reverse annealing process the D-Wave processor works as a thermal accelerator and 2) its evolution involves an increasing amount of dissipation with increasing transverse field.
Quantum devices are affected by intrinsic and environmental noises. An in-depth characterization of noise effects is essential for exploiting noisy quantum computing. To this end, we studied the energy dissipative behavior of a quantum annealer via experiments and numerical simulations. Our investigation adopts a recently proposed technique that interpolates between pure quantum dynamics and pure thermodynamics. Experiments were conducted on a quantum annealer with an anneal pause function, which inserts a thermal relaxation period into the annealing schedule by pausing the transverse field, w…
Thermodynamic significance of QUBO encoding on quantum annealers
University of Massachusetts, Boston（马萨诸塞大学波士顿分校）；University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校）；Quantum Science Institute, University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校量子科学研究所）；Universität Tübingen（蒂宾根大学）；Forschungszentrum Jülich（于利希研究中心）；Polish Academy of Sciences（波兰科学院）
Conventional quantum annealing does not sample all ground states fairly. We demonstrate that fair sampling can be achieved by performing simulated annealing on a quantum annealer. We discuss the problems that occur when implementing this method and propose an alternative way to overcome them. We numerically verify the fair sampling ability of our method in a small-scale toy model.
Near term quantum hardware promises unprecedented computational advantage. Crucial in its development is the characterization and minimization of computational errors. We propose the use of the quantum fluctuation theorem to benchmark the performance of quantum annealers. This versatile tool provides simple means to determine whether the quantum dynamics are unital, unitary, and adiabatic, or whether the system is prone to thermal noise. Our proposal is experimentally tested on two generations of the D-Wave machine, which illustrates the sensitivity of the fluctuation theorem to the smallest a…
We propose a new method for solving binary optimization problems under inequality constraints using a quantum annealer. To deal with inequality constraints, we often use slack variables, as in previous approaches. When we use slack variables, we usually conduct a binary expansion, which requires numerous physical qubits. Therefore, the problem of the current quantum annealer is limited to a small scale. In this study, we employ the alternating direction method of multipliers. This approach allows us to deal with various types using constraints in the current quantum annealer without slack vari…
Boltzmann sampling with quantum annealers via fast Stein correction
Graduate School of Frontier Sciences, The University of Tokyo（东京大学前沿科学研究生院）；Center for Basic Research on Materials, National Institute for Materials Science（物质材料研究机构基础材料研究中心）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
In solving optimization problems, objective functions generally need to be minimized or maximized. However, objective functions cannot always be formulated explicitly in a mathematical form for complicated problem settings. Although several regression techniques infer the approximate forms of objective functions, they are at times expensive to evaluate. Optimal points of "black-box" objective functions are computed in such scenarios, while effectively using a small number of clues. Recently, an efficient method by use of inference by sparse prior for a black-box objective function with binary …
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…
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）
The solution of physical problems discretized using the finite element methods using quantum computers remains relatively unexplored. Here, we present a unified formulation (FEqa) to solve such problems using quantum annealers. FEqa is a hybrid technique in which the finite element problem is formulated on a classical computer, and the residual is minimized using a quantum annealer. The advantages of FEqa include utilizing a single qubit per degree of freedom, enforcing Dirichlet boundary conditions a priori, reaching arbitrary solution precision, and eliminating the possibility of the anneale…
We are at the verge of a new era, which will be dominated by Noisy Intermediate-Scale Quantum Devices. Prototypical examples for these new technologies are present-day quantum annealers. In the present work, we investigate to what extent static disorder generated by an external source of noise does not have to be detrimental, but can actually assist quantum annealers in achieving better performance. In particular, we analyze the graph coloring problem that can be solved on a sparse topology (i.e. chimera graph) via suitable embedding. We show that specifically tailored disorder can enhance the…
Quantum particle statistics fundamentally controls the way particles interact, and plays an essential role in determining the properties of the system at low temperature. Here we study how the quantum statistics affects the computational power of quantum annealing. We propose an annealing Hamiltonian describing quantum particles moving on a square lattice and compare the computational performance of the atomic quantum annealers between two statistically-different components: spinless fermions and hard-core bosons. In addition, we take an Ising quantum annealer driven by traditional transverse-…
Quantum annealers are special-purpose quantum computers that primarily target solving Ising optimization problems. Theoretical work has predicted that the probability of a quantum annealer ending in a ground state can be dramatically improved if the spin driving terms, which play a crucial role in the functioning of a quantum annealer, have different strengths for different spins; that is, they are inhomogeneous. In this paper we describe a time-shift-based protocol for inhomogeneous driving and demonstrate, using an experimental quantum annealer, the performance of our protocol on a range of …
Quantum computers are ideal for solving chemistry problems due to their polynomial scaling with system size in contrast to classical computers which scale exponentially. Until now molecular energy calculations using quantum computing hardware have been limited to quantum simulators. In this paper, a new methodology is presented to calculate the vibrational spectrum of a molecule on a quantum annealer. The key idea of the method is a mapping of the ground state variational problem onto an Ising or quadratic unconstrained binary optimization (QUBO) problem by expressing the expansion coefficient…
Quantum annealing devices have been subject to various analyses in order to classify their usefulness for practical applications. While it has been successfully proven that such systems can in general be used for solving combinatorial optimization problems, they have not been used to solve chemistry applications. In this paper we apply a mapping, put forward by Xia et al. (The Journal of Physical Chemistry B 122.13 (2017): 3384-3395.), from a quantum chemistry Hamiltonian to an Ising spin glass formulation and find the ground state energy with a quantum annealer. Additionally we investigate th…
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）
We report on a case study in programming an early quantum annealer to attack optimization problems related to operational planning. While a number of studies have looked at the performance of quantum annealers on problems native to their architecture, and others have examined performance of select problems stemming from an application area, ours is one of the first studies of a quantum annealer's performance on parametrized families of hard problems from a practical domain. We explore two different general mappings of planning problems to quadratic unconstrained binary optimization (QUBO) prob…
Recent years have witnessed an unprecedented increase in experiments and hybrid simulations involving quantum computers. In particular, quantum annealers. Although quantum supremacy has not been established thus far, there exist a plethora of algorithms promising to outperform classical computers in the near-term future. Here, we propose a parallel in time approach to simulate dynamical systems designed to be executed already on present-day quantum annealers. In essence, purely classical methods for solving dynamics systems are serial. Therefore, their parallelization is substantially limited.…
We develop and apply several strategies for setting physical parameters on quantum annealers for application problems that do not fit natively on the hardware graph. The strategies are tested with a culled random set of mixed satisfiability problems, yielding results that generalize to guidelines regarding which parameter setting strategies to use for different classes of problems, and how to choose other necessary hardware quantities as well. Alternate methods of changing the hardware implementation of an application problem are also considered and their utility discussed.
Beyond Optimization: Harnessing Quantum Annealer Dynamics for Machine Learning
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Toyota Central R&D Labs., Inc.（丰田中央研究所）；DENSO CORPORATION（电装株式会社）；Global Research and Development Center for Business by Quantum-AI technology (G-QuAT), National Institute of Advanced Industrial Science and Technology (AIST)（产业技术综合研究所量子AI技术商业化全球研发中心）
We demonstrate that the output statistics of a quantum annealing protocol run on D-Wave 2000Q can be explained by static disorder garnishing an otherwise ideal device hardware. A Boltzmann-like distribution over distinct output states emerges with increasing problem size, and significantly reduces the chances for a correct identification of the sought-after optimal solutions.
Nested quantum annealing correction (NQAC) is an error correcting scheme for quantum annealing that allows for the encoding of a logical qubit into an arbitrarily large number of physical qubits. The encoding replaces each logical qubit by a complete graph of degree $C$. The nesting level $C$ represents the distance of the error-correcting code and controls the amount of protection against thermal and control errors. Theoretical mean-field analyses and empirical data obtained with a D-Wave Two quantum annealer (supporting up to $512$ qubits) showed that NQAC has the potential to achieve a scal…
We demonstrate experimentally the ability of a quantum annealer to distinguish between sets of non-isomorphic graphs that share the same classical Ising spectrum. Utilizing the pause-and-quench features recently introduced into D-Wave quantum annealing processors, which allow the user to probe the quantum Hamiltonian realized in the middle of an anneal, we show that obtaining thermal averages of diagonal observables of classically indistinguishable non-isomorphic graphs encoded into transverse-field Ising Hamiltonians enable their discrimination. We discuss the significance of our results in t…
Quantum computing is developing fast. Real world applications are within reach in the coming years. One of the most promising areas is combinatorial optimisation, where the Quadratic Unconstrained Binary Optimisation (QUBO) problem formulation is used to get good approximate solutions. Both the universal quantum computer as the quantum annealer can handle this kind of problems well. In this paper, we present an application on multimodal container planning. We show how to map this problem to a QUBO problem formulation and how the practical implementation can be done on the quantum annealer prod…
Adiabatic quantum computing is an analog quantum computing scheme with various applications in solving optimization problems. In the parity picture of quantum optimization, the problem is encoded in local fields that act on qubits which are connected via local 4-body terms. We present an implementation of a parity annealer with Transmon qubits with a specifically tailored Ising interaction from Josephson ring modulators.
The application in cryptography of quantum algorithms for prime factorization fostered the interest in quantum computing. However, quantum computers, and particularly quantum annealers, can also be helpful to construct secure cryptographic keys. Indeed, finding robust Boolean functions for cryptography is an important problem in sequence ciphers, block ciphers, and hash functions, among others. Due to the super-exponential size $\mathcal{O}(2^{2^n})$ of the associated space, finding $n$-variable Boolean functions with global cryptographic constraints is computationally hard. This problem has a…
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…
We propose a novel method using a quantum annealer -- an analog quantum computer based on the principles of quantum adiabatic evolution -- to solve the Graph Isomorphism problem, in which one has to determine whether two graphs are isomorphic (i.e., can be transformed into each other simply by a relabeling of the vertices). We demonstrate the capabilities of the method by analyzing several types of graph families, focusing on graphs with particularly high symmetry called strongly regular graphs (SRG's). We also show that our method is applicable, within certain limitations, to currently availa…
Quantum chemistry is regarded to be one of the first disciplines that will be revolutionized by quantum computing. Although universal quantum computers of practical scale may be years away, various approaches are currently being pursued to solve quantum chemistry problems on near-term gate-based quantum computers and quantum annealers by developing the appropriate algorithm and software base. This work implements the general Quantum Annealer Eigensolver (QAE) algorithm to solve the molecular electronic Hamiltonian eigenvalue-eigenvector problem on a D-Wave 2000Q quantum annealer. The approach …
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 benchmark the quantum processing units of the largest quantum annealers to date, the 5000+ qubit quantum annealer Advantage and its 2000+ qubit predecessor D-Wave 2000Q, using tail assignment and exact cover problems from aircraft scheduling scenarios. The benchmark set contains small, intermediate, and large problems with both sparsely connected and almost fully connected instances. We find that Advantage outperforms D-Wave 2000Q for almost all problems, with a notable increase in success rate and problem size. In particular, Advantage is also able to solve the largest problems with 120 lo…
The recent availability of quantum annealers has fueled a new area of information technology where such devices are applied to address practically motivated and computationally difficult problems with hardware that exploits quantum mechanical phenomena. D-Wave annealers are promising platforms to solve these problems in the form of quadratic unconstrained binary optimization. Here we provide a formulation of the Chinese postman problem that can be used as a tool for probing the local connectivity of graphs and networks. We treat the problem classically with a tabu algorithm and using a D-Wave …
Quantum(-inspired) annealers show promise in solving combinatorial optimisation problems in practice. There has been extensive researches demonstrating the utility of D-Wave quantum annealer and quantum-inspired annealer, i.e., Fujitsu Digital Annealer on various applications, but few works are comparing these platforms. In this paper, we benchmark quantum(-inspired) annealers with three combinatorial optimisation problems ranging from generic scientific problems to complex problems in practical use. In the case where the problem size goes beyond the capacity of a quantum(-inspired) computer, …
Quantum chemistry calculations for small molecules on quantum hardware have been demonstrated to date only on universal-gate quantum computers, not quantum annealers. The latter devices are limited to finding the lowest eigenstate of the Ising Hamiltonian whereas the electronic Hamiltonian could not be mapped to the Ising form without exponential growth of the Ising Hamiltonian with the size of the system [J. Phys. Chem. B 122, 3384 (2018)]. Here we propose a novel mixed discrete-continuous optimization algorithm, which finds the lowest eigenstate of the qubit coupled cluster (QCC) method usin…
We demonstrate that a quantum annealer can be used to solve the NP-complete problem of graph partitioning into subgraphs containing Hamiltonian cycles of constrained length. We present a method to find a partition of a given directed graph into Hamiltonian subgraphs with three or more vertices, called vertex 3-cycle cover. We formulate the problem as a quadratic unconstrained binary optimisation and run it on a D-Wave Advantage quantum annealer. We test our method on synthetic graphs constructed by adding a number of random edges to a set of disjoint cycles. We show that the probability of sol…
A successful application of quantum annealing to machine learning is training restricted Boltzmann machines (RBM). However, many neural networks for vision applications are feedforward structures, such as multilayer perceptrons (MLP). Backpropagation is currently the most effective technique to train MLPs for supervised learning. This paper aims to be forward-looking by exploring the training of MLPs using quantum annealers. We exploit an equivalence between MLPs and energy-based models (EBM), which are a variation of RBMs with a maximum conditional likelihood objective. This leads to a strate…
Quantum annealers are an alternative approach to quantum computing which make use of the adiabatic theorem to efficiently find the ground state of a physically realizable Hamiltonian. Such devices are currently commercially available and have been successfully applied to several combinatorial and discrete optimization problems. However, the application of quantum annealers to problems in chemistry remains a relatively sparse area of research due to the difficulty in mapping molecular systems to the Ising model Hamiltonian. In this paper we review two different methods for finding the ground st…
We argue that a quantum annealer at very long annealing times is likely to experience a quasistatic evolution, returning a final population that is close to a Boltzmann distribution of the Hamiltonian at a single (freeze-out) point during the annealing. Such a system is expected to correlate well with a proper quantum Monte Carlo simulation. It may also correlate with simulated annealing if at the freeze-out point the quantum energy eigenvalues are close to the classical ones. These correlations are just signatures of equilibration and do not mean that the evolution of the quantum annealer is …
School of Computing Science（计算科学学院）；University of Glasgow（格拉斯哥大学）；OxBrdgRbtx Ltd（OxBrdgRbtx有限公司）
We demonstrate the feasibility of framing a classically learned deep neural network as an energy based model that can be processed on a one-step quantum annealer in order to exploit fast sampling times. We propose approaches to overcome two hurdles for high resolution image classification on a quantum processing unit (QPU): the required number and binary nature of the model states. With this novel method we successfully transfer a convolutional neural network to the QPU and show the potential for classification speedup of at least one order of magnitude.
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 …
In an effort to overcome the limitations of random spin-glass benchmarks for quantum annealers, focus has shifted to carefully-crafted gadget-based problems whose logical structure has typically a planar topology. Recent experiments on these gadget problems using a commercially-available quantum annealer have demonstrated an impressive performance over a selection of commonly-used classical optimization heuristics. Here we show that efficient classical optimization techniques, such as minimum-weight perfect matching, can solve these gadget problems exactly and in polynomial time. We present ap…
We outline an algorithm for the Quantum Counting problem using Adiabatic Quantum Computation (AQC). We show that using local adiabatic evolution, a process in which the adiabatic procedure is performed at a variable rate, the problem is solved with the same complexity as the analogous circuit-based algorithm, i.e., quadratically faster than the corresponding classical algorithm. The above algorithm provides further evidence for the potentially powerful capabilities of AQC as a paradigm for more efficient problem solving on a quantum computer, and may be used as the basis for solving more sophi…
In this paper an application of D-Wave Systems quantum annealer for optimizing initial line-up of a soccer team is presented. Players and their playing position are selected to maximize the sum of players ratings for both a 4-3-3 attack formation and a 4-2-3-1 medium defensive formation. The problem is stated as a binary quadratic model (BQM) and it is solved in a D-Wave Leap Hybrid Solver. Results are presented and compared with already published results obtained with a classical solver.
We perform an in-depth comparison of quantum annealing with several classical optimisation techniques, namely thermal annealing, Nelder-Mead, and gradient descent. We begin with a direct study of the 2D Ising model on a quantum annealer, and compare its properties directly with those of the thermal 2D Ising model. These properties include an Ising-like phase transition that can be induced by either a change in 'quantum-ness' of the theory, or by a scaling the Ising couplings up or down. This behaviour is in accord with what is expected from the physical understanding of the quantum system. We …
Systems of linear equations are employed almost universally across a wide range of disciplines, from physics and engineering to biology, chemistry and statistics. Traditional solution methods such as Gaussian elimination become very time consuming for large matrices, and more efficient computational methods are desired. In the twilight of Moore's Law, quantum computing is perhaps the most direct path out of the darkness. There are two complementary paradigms for quantum computing, namely, gated systems and quantum annealers. In this paper, we express floating point operations such as division …
