results for quantum machine learning

- Mar 17 2017 quant-ph arXiv:1703.05402v1Efficiently characterising quantum systems, verifying operations of quantum devices and validating underpinning physical models, are central challenges for the development of quantum technologies and for our continued understanding of foundational physics. Machine-learning enhanced by quantum simulators has been proposed as a route to improve the computational cost of performing these studies. Here we interface two different quantum systems through a classical channel - a silicon-photonics quantum simulator and an electron spin in a diamond nitrogen-vacancy centre - and use the former to learn the latter's Hamiltonian via Bayesian inference. We learn the salient Hamiltonian parameter with an uncertainty of approximately $10^{-5}$. Furthermore, an observed saturation in the learning algorithm suggests deficiencies in the underlying Hamiltonian model, which we exploit to further improve the model itself. We go on to implement an interactive version of the protocol and experimentally show its ability to characterise the operation of the quantum photonic device. This work demonstrates powerful new quantum-enhanced techniques for investigating foundational physical models and characterising quantum technologies.
- The experimental realization of increasingly complex synthetic quantum systems calls for the development of general theoretical methods, to validate and fully exploit quantum resources. Quantum-state tomography (QST) aims at reconstructing the full quantum state from simple measurements, and therefore provides a key tool to obtain reliable analytics. Brute-force approaches to QST, however, demand resources growing exponentially with the number of constituents, making it unfeasible except for small systems. Here we show that machine learning techniques can be efficiently used for QST of highly-entangled states in arbitrary dimension. Remarkably, the resulting approach allows one to reconstruct traditionally challenging many-body quantities - such as the entanglement entropy - from simple, experimentally accessible measurements. This approach can benefit existing and future generations of devices ranging from quantum computers to ultra-cold atom quantum simulators.
- Molecular machine learning has been maturing rapidly over the last few years. Improved methods and the presence of larger datasets have enabled machine learning algorithms to make increasingly accurate predictions about molecular properties. However, algorithmic progress has been limited due to the lack of a standard benchmark to compare the efficacy of proposed methods; most new algorithms are benchmarked on different datasets making it challenging to gauge the quality of proposed methods. This work introduces MoleculeNet, a large scale benchmark for molecular machine learning. MoleculeNet curates multiple public datasets, establishes metrics for evaluation, and offers high quality open-source implementations of multiple previously proposed molecular featurization and learning algorithms (released as part of the DeepChem open source library). MoleculeNet benchmarks demonstrate that learnable representations, and in particular graph convolutional networks, are powerful tools for molecular machine learning and broadly offer the best performance. However, for quantum mechanical and biophysical datasets, the use of physics-aware featurizations can be significantly more important than choice of particular learning algorithm.
- Feb 21 2017 cond-mat.mtrl-sci physics.chem-ph arXiv:1702.05771v1High-throughput computational screening has emerged as a critical component of materials discovery. Direct density functional theory (DFT) simulation of inorganic materials and molecular transition metal complexes is often used to describe subtle trends in inorganic bonding and spin-state ordering, but these calculations are computationally costly and properties are sensitive to the exchange-correlation functional employed. To begin to overcome these challenges, we trained artificial neural networks (ANNs) to predict quantum-mechanically-derived properties, including spin-state ordering, sensitivity to Hartree-Fock exchange, and spin- state specific bond lengths in transition metal complexes. Our ANN is trained on a small set of inorganic-chemistry-appropriate empirical inputs that are both maximally transferable and do not require precise three-dimensional structural information for prediction. Using these descriptors, our ANN predicts spin-state splittings of single-site transition metal complexes (i.e., Cr-Ni) at arbitrary amounts of Hartree-Fock exchange to within 3 kcal/mol accuracy of DFT calculations. Our exchange-sensitivity ANN enables improved predictions on a diverse test set of experimentally-characterized transition metal complexes by extrapolation from semi-local DFT to hybrid DFT. The ANN also outperforms other machine learning models (i.e., support vector regression and kernel ridge regression), demonstrating particularly improved performance in transferability, as measured by prediction errors on the diverse test set. We establish the value of new uncertainty quantification tools to estimate ANN prediction uncertainty in computational chemistry, and we provide additional heuristics for identification of when a compound of interest is likely to be poorly predicted by the ANN.
- Feb 21 2017 physics.chem-ph arXiv:1702.05532v1We investigate the impact of choosing regressors and molecular representations for the construction of fast machine learning (ML) models of thirteen electronic ground-state properties of organic molecules. The performance of each regressor/representation/property combination is assessed with learning curves which report approximation errors as a function of training set size. Molecular structures and properties at hybrid density functional theory (DFT) level of theory used for training and testing come from the QM9 database [Ramakrishnan et al, Scientific Data 1 140022 (2014)] and include dipole moment, polarizability, HOMO/LUMO energies and gap, electronic spatial extent, zero point vibrational energy, enthalpies and free energies of atomization, heat capacity and the highest fundamental vibrational frequency. Various representations from the literature have been studied (Coulomb matrix, bag of bonds, BAML and ECFP4, molecular graphs (MG)), as well as newly developed distribution based variants including histograms of distances (HD), and angles (HDA/MARAD), and dihedrals (HDAD). Regressors include linear models (Bayesian ridge regression (BR) and linear regression with elastic net regularization (EN)), random forest (RF), kernel ridge regression (KRR) and two types of neural networks, graph convolutions (GC) and gated graph networks (GG). We present numerical evidence that ML model predictions for all properties can reach an approximation error to DFT which is on par with chemical accuracy. These findings indicate that ML models could be more accurate than DFT if explicitly electron correlated quantum (or experimental) data was provided.
- This paper surveys quantum learning theory: the theoretical aspects of machine learning using quantum computers. We describe the main results known for three models of learning: exact learning from membership queries, and Probably Approximately Correct (PAC) and agnostic learning from classical or quantum examples.
- Jan 19 2017 cond-mat.dis-nn quant-ph arXiv:1701.05039v1The challenge of quantum many-body problems comes from the difficulty to represent large-scale quantum states, which in general requires an exponentially large number of parameters. Recently, a connection has been made between quantum many-body states and the neural network representation (\textitarXiv:1606.02318). An important open question is what characterizes the representational power of deep and shallow neural networks, which is of fundamental interest due to popularity of the deep learning methods. Here, we give a rigorous proof that a deep neural network can efficiently represent most physical states, including those generated by any polynomial size quantum circuits or ground states of many body Hamiltonians with polynomial-size gaps, while a shallow network through a restricted Boltzmann machine cannot efficiently represent those states unless the polynomial hierarchy in computational complexity theory collapses.
- Superconducting circuit technologies have recently achieved quantum protocols involving closed feedback loops. Quantum artificial intelligence and quantum machine learning are emerging fields inside quantum technologies which may enable quantum devices to acquire information from the outer world and improve themselves via a learning process. Here we propose the implementation of basic protocols in quantum reinforcement learning, with superconducting circuits employing feedback-loop control. We introduce diverse scenarios for proof-of-principle experiments with state-of-the-art superconducting circuit technologies and analyze their feasibility in presence of imperfections. The field of quantum artificial intelligence implemented with superconducting circuits paves the way for enhanced quantum control and quantum computation protocols.
- Restricted Boltzmann machine (RBM) is one of the fundamental building blocks of deep learning. RBM finds wide applications in dimensional reduction, feature extraction, and recommender systems via modeling the probability distributions of a variety of input data including natural images, speech signals, and customer ratings, etc. We build a bridge between RBM and tensor network states (TNS) widely used in quantum many-body physics research. We devise efficient algorithms to translate an RBM into the commonly used TNS. Conversely, we give sufficient and necessary conditions to determine whether a TNS can be transformed into an RBM of given architectures. Revealing these general and constructive connections can cross-fertilize both deep learning and quantum-many body physics. Notably, by exploiting the entanglement entropy bound of TNS, we can rigorously quantify the expressive power of RBM on complex datasets. Insights into TNS and its entanglement capacity can guide the design of more powerful deep learning architectures. On the other hand, RBM can represent quantum many-body states with fewer parameters compared to TNS, which may allow more efficient classical simulations.
- Machine learning, one of today's most rapidly growing interdisciplinary fields, promises an unprecedented perspective for solving intricate quantum many-body problems. Understanding the physical aspects of the representative artificial neural-network states is recently becoming highly desirable in the applications of machine learning techniques to quantum many-body physics. Here, we study the quantum entanglement properties of neural-network states, with a focus on the restricted-Boltzmann-machine (RBM) architecture. We prove that the entanglement of all short-range RBM states satisfies an area law for arbitrary dimensions and bipartition geometry. For long-range RBM states we show by using an exact construction that such states could exhibit volume-law entanglement, implying a notable capability of RBM in representing efficiently quantum states with massive entanglement. We further examine generic RBM states with random weight parameters. We find that their averaged entanglement entropy obeys volume-law scaling and meantime strongly deviates from the Page-entropy of the completely random pure states. We show that their entanglement spectrum has no universal part associated with random matrix theory and bears a Poisson-type level statistics. Using reinforcement learning, we demonstrate that RBM is capable of finding the ground state (with power-law entanglement) of a model Hamiltonian with long-range interaction. In addition, we show, through a concrete example of the one-dimensional symmetry-protected topological cluster states, that the RBM representation may also be used as a tool to analytically compute the entanglement spectrum. Our results uncover the unparalleled power of artificial neural networks in representing quantum many-body states, which paves a novel way to bridge computer science based machine learning techniques to outstanding quantum condensed matter physics problems.
- Jan 18 2017 stat.ML arXiv:1701.04503v1The rise and fall of artificial neural networks is well documented in the scientific literature of both computer science and computational chemistry. Yet almost two decades later, we are now seeing a resurgence of interest in deep learning, a machine learning algorithm based on multilayer neural networks. Within the last few years, we have seen the transformative impact of deep learning in many domains, particularly in speech recognition and computer vision, to the extent that the majority of expert practitioners in those field are now regularly eschewing prior established models in favor of deep learning models. In this review, we provide an introductory overview into the theory of deep neural networks and their unique properties that distinguish them from traditional machine learning algorithms used in cheminformatics. By providing an overview of the variety of emerging applications of deep neural networks, we highlight its ubiquity and broad applicability to a wide range of challenges in the field, including QSAR, virtual screening, protein structure prediction, quantum chemistry, materials design and property prediction. In reviewing the performance of deep neural networks, we observed a consistent outperformance against non-neural networks state-of-the-art models across disparate research topics, and deep neural network based models often exceeded the "glass ceiling" expectations of their respective tasks. Coupled with the maturity of GPU-accelerated computing for training deep neural networks and the exponential growth of chemical data on which to train these networks on, we anticipate that deep learning algorithms will be a valuable tool for computational chemistry.
- We investigate whether quantum annealers with select chip layouts can outperform classical computers in reinforcement learning tasks. We associate a transverse field Ising spin Hamiltonian with a layout of qubits similar to that of a deep Boltzmann machine (DBM) and use simulated quantum annealing (SQA) to numerically simulate quantum sampling from this system. We design a reinforcement learning algorithm in which the set of visible nodes representing the states and actions of an optimal policy are the first and last layers of the deep network. In absence of a transverse field, our simulations show that DBMs train more effectively than restricted Boltzmann machines (RBM) with the same number of weights. Since sampling from Boltzmann distributions of a DBM is not classically feasible, this is evidence of advantage of a non-Turing sampling oracle. We then develop a framework for training the network as a quantum Boltzmann machine (QBM) in the presence of a significant transverse field for reinforcement learning. This further improves the reinforcement learning method using DBMs.
- We propose a quantum machine learning algorithm for efficiently solving a class of problems encoded in quantum controlled unitary operations. The central physical mechanism of the protocol is the iteration of a quantum time-delayed equation that introduces feedback in the dynamics and eliminates the necessity of intermediate measurements. The performance of the quantum algorithm is analyzed by comparing the results obtained in numerical simulations with the outcome of classical machine learning methods for the same problem. The use of time-delayed equations enhances the toolbox of the field of quantum machine learning, which may enable unprecedented applications in quantum technologies.
- Dec 16 2016 cond-mat.str-el stat.ML arXiv:1612.04895v1We present a machine learning approach to the inversion of Fredholm integrals of the first kind. The approach provides a natural regularization in cases where the inverse of the Fredholm kernel is ill-conditioned. It also provides an efficient and stable treatment of constraints. The key observation is that the stability of the forward problem permits the construction of a large database of outputs for physically meaningful inputs. We apply machine learning to this database to generate a regression function of controlled complexity, which returns approximate solutions for previously unseen inputs; the approximate solutions are then projected onto the subspace of functions satisfying relevant constraints. We also derive and present uncertainty estimates. We illustrate the approach by applying it to the analytical continuation problem of quantum many-body physics, which involves reconstructing the frequency dependence of physical excitation spectra from data obtained at specific points in the complex frequency plane. Under standard error metrics the method performs as well or better than the Maximum Entropy method for low input noise and is substantially more robust to increased input noise. We expect the methodology to be similarly effective for any problem involving a formally ill-conditioned inversion, provided that the forward problem can be efficiently solved.
- Dec 16 2016 quant-ph arXiv:1612.05204v1The promise of quantum neural nets, which utilize quantum effects to model complex data sets, has made their development an aspirational goal for quantum machine learning and quantum computing in general. Here we provide new methods of training quantum Boltzmann machines, which are a class of recurrent quantum neural network. Our work generalizes existing methods and provides new approaches for training quantum neural networks that compare favorably to existing methods. We further demonstrate that quantum Boltzmann machines enable a form of quantum state tomography that not only estimates a state but provides a perscription for generating copies of the reconstructed state. Classical Boltzmann machines are incapable of this. Finally we compare small non-stoquastic quantum Boltzmann machines to traditional Boltzmann machines for generative tasks and observe evidence that quantum models outperform their classical counterparts.
- Dec 13 2016 quant-ph arXiv:1612.03713v2The support vector machine (SVM) is a popular machine learning classification method which produces a nonlinear decision boundary in a feature space by constructing linear boundaries in a transformed Hilbert space. It is well known that these algorithms when executed on a classical computer do not scale well with the size of the feature space both in terms of data points and dimensionality. One of the most significant limitations of classical algorithms using non-linear kernels is that the kernel function has to be evaluated for all pairs of input feature vectors which themselves may be of substantially high dimension. This can lead to computationally excessive times during training and during the prediction process for a new data point. Here, we propose using both canonical and generalized coherent states to rapidly calculate specific nonlinear kernel functions. The key link will be the reproducing kernel Hilbert space (RKHS) property for SVMs that naturally arise from canonical and generalized coherent states. Specifically, we discuss the fast evaluation of radial kernels through a positive operator valued measure (POVM) on a quantum optical system based on canonical coherent states. A similar procedure may also lead to fast calculations of kernels not usually used in classical algorithms such as those arising from generalized coherent states.
- The task of reconstructing a low rank matrix from incomplete linear measurements arises in areas such as machine learning, quantum state tomography and in the phase retrieval problem. In this note, we study the particular setup that the measurements are taken with respect to rank one matrices constructed from the elements of a random tight frame. We consider a convex optimization approach and show both robustness of the reconstruction with respect to noise on the measurements as well as stability with respect to passing to approximately low rank matrices. This is achieved by establishing a version of the null space property of the corresponding measurement map.
- Dec 07 2016 quant-ph arXiv:1612.01789v1Solving optimization problems in disciplines such as machine learning is commonly done by iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to homogeneous polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum principal component analysis scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector, an exponential speed-up over classical algorithms, which scale polynomially. The quantum algorithm can therefore be beneficial for high dimensional problems where a relatively small number of iterations is sufficient.
- Martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning. Here we apply techniques borrowed from this field to quantum hypothesis testing, which is the problem of discriminating quantum states belonging to two different sequences $\{\rho_n\}_{n}$ and $\{\sigma_n\}_n$. We obtain upper bounds on the finite blocklength type II Stein- and Hoeffding errors, which, for i.i.d. states, are in general tighter than the corresponding bounds obtained by Audenaert, Mosonyi and Verstraete [Journal of Mathematical Physics, 53(12), 2012]. We also derive finite blocklength bounds and moderate deviation results for pairs of sequences of correlated states satisfying a (non-homogeneous) factorization property. Examples of such sequences include Gibbs states of spin chains with translation-invariant finite range interaction, as well as finitely correlated quantum states. We apply our results to find bounds on the capacity of a certain class of classical-quantum channels with memory, which satisfy a so-called channel factorization property- both in the finite blocklength and moderate deviation regimes.
- Recent progress implies that a crossover between machine learning and quantum information processing benefits both fields. Traditional machine learning has dramatically improved the benchmarking and control of experimental quantum computing systems, including adaptive quantum phase estimation and designing quantum computing gates. On the other hand, quantum mechanics offers tantalizing prospects to enhance machine learning, ranging from reduced computational complexity to improved generalization performance. The most notable examples include quantum enhanced algorithms for principal component analysis, quantum support vector machines, and quantum Boltzmann machines. Progress has been rapid, fostered by demonstrations of midsized quantum optimizers which are predicted to soon outperform their classical counterparts. Further, we are witnessing the emergence of a physical theory pinpointing the fundamental and natural limitations of learning. Here we survey the cutting edge of this merger and list several open problems.
- Markov logic networks (MLNs) reconcile two opposing schools in machine learning and artificial intelligence: causal networks, which account for uncertainty extremely well, and first-order logic, which allows for formal deduction. An MLN is essentially a first-order logic template to generate Markov networks. Inference in MLNs is probabilistic and it is often performed by approximate methods such as Markov chain Monte Carlo (MCMC) Gibbs sampling. An MLN has many regular, symmetric structures that can be exploited at both first-order level and in the generated Markov network. We analyze the graph structures that are produced by various lifting methods and investigate the extent to which quantum protocols can be used to speed up Gibbs sampling with state preparation and measurement schemes. We review different such approaches, discuss their advantages, theoretical limitations, and their appeal to implementations. We find that a straightforward application of a recent result yields exponential speedup compared to classical heuristics in approximate probabilistic inference, thereby demonstrating another example where advanced quantum resources can potentially prove useful in machine learning.
- Nov 23 2016 physics.comp-ph physics.chem-ph arXiv:1611.07435v2The training of molecular models of quantum mechanical properties based on statistical machine learning requires large datasets which exemplify the map from chemical structure to molecular property. Intelligent a priori selection of training examples is often difficult or impossible to achieve as prior knowledge may be sparse or unavailable. Ordinarily representative selection of training molecules from such datasets is achieved through random sampling. We use genetic algorithms for the optimization of training set composition consisting of tens of thousands of small organic molecules. The resulting machine learning models are considerably more accurate with respect to small randomly selected training sets: mean absolute errors for out-of-sample predictions are reduced to ~25% for enthalpies, free energies, and zero-point vibrational energy, to ~50% for heat-capacity, electron-spread, and polarizability, and by more than ~20% for electronic properties such as frontier orbital eigenvalues or dipole-moments. We discuss and present optimized training sets consisting of 10 molecular classes for all molecular properties studied. We show that these classes can be used to design improved training sets for the generation of machine learning models of the same properties in similar but unrelated molecular sets.
- Nov 16 2016 physics.chem-ph arXiv:1611.04678v3Using conservation of energy -- a fundamental property of closed classical and quantum mechanical systems -- we develop an efficient gradient-domain machine learning (GDML) approach to construct accurate molecular force fields using a restricted number of samples from ab initio molecular dynamics (AIMD) trajectories. The GDML implementation is able to reproduce global potential-energy surfaces of intermediate-size molecules with an accuracy of 0.3 kcal/mol for energies and 1 kcal/mol/Å for atomic forces using only 1000 conformational geometries for training. We demonstrate this accuracy for AIMD trajectories of molecules, including benzene, toluene, naphthalene, ethanol, uracil, and aspirin. The challenge of constructing conservative force fields is accomplished in our work by learning in a Hilbert space of vector-valued functions that obey the law of energy conservation. The GDML approach enables quantitative molecular dynamics simulations for molecules at a fraction of cost of explicit AIMD calculations, thereby allowing the construction of efficient force fields with the accuracy and transferability of high-level ab initio methods.
- Nov 15 2016 physics.chem-ph cond-mat.mtrl-sci arXiv:1611.03877v1We present a novel scheme to accurately predict atomic forces as vector quantities, rather than sets of scalar components, by Gaussian Process (GP) Regression. This is based on matrix-valued kernel functions, to which we impose that the predicted force rotates with the target configuration and is independent of any rotations applied to the configuration database entries. We show that such "covariant" GP kernels can be obtained by integration over the elements of the rotation group SO(d) for the relevant dimensionality d. Remarkably, in specific cases the integration can be carried out analytically and yields a conservative force field that can be recast into a pair interaction form. Finally, we show that restricting the integration to a summation over the elements of a finite point group relevant to the target system is sufficient to recover an accurate GP. The accuracy of our kernels in predicting quantum-mechanical forces in real materials is investigated by tests on pure and defective Ni and Fe crystalline systems.
- Despite rapidly growing interest in harnessing machine learning in the study of quantum many-body systems, training neural networks to identify quantum phases is a nontrivial challenge. The key challenge is in efficiently extracting essential information from the many-body Hamiltonian or wave function and turning the information into an image that can be fed into a neural network. When targeting topological phases, this task becomes particularly challenging as topological phases are defined in terms of non-local properties. Here we introduce quantum loop topography (QLT): a procedure of constructing a multi-dimensional image from the "sample" Hamiltonian or wave function by evaluating two-point operators that form loops at independent Monte Carlo steps. The loop configuration is guided by characteristic response for defining the phase, which is Hall conductivity for the cases at hand. Feeding QLT to a fully-connected neural network with a single hidden layer, we demonstrate that the architecture can be effectively trained to distinguish Chern insulator and fractional Chern insulator from trivial insulators with high fidelity. In addition to establishing the first case of obtaining a phase diagram with topological quantum phase transition with machine learning, the perspective of bridging traditional condensed matter theory with machine learning will be broadly valuable.
- Laplacian eigenmap algorithm is a typical nonlinear model for dimensionality reduction in classical machine learning. We propose an efficient quantum Laplacian eigenmap algorithm to exponentially speed up the original counterparts. In our work, we demonstrate that the Hermitian chain product proposed in quantum linear discriminant analysis (arXiv:1510.00113,2015) can be applied to implement quantum Laplacian eigenmap algorithm. While classical Laplacian eigenmap algorithm requires polynomial time to solve the eigenvector problem, our algorithm is able to exponentially speed up nonlinear dimensionality reduction.
- The emerging field of quantum machine learning has the potential to substantially aid in the problems and scope of artificial intelligence. This is only enhanced by recent successes in the field of classical machine learning. In this work we propose an approach for the systematic treatment of machine learning, from the perspective of quantum information. Our approach is general and covers all three main branches of machine learning: supervised, unsupervised and reinforcement learning. While quantum improvements in supervised and unsupervised learning have been reported, reinforcement learning has received much less attention. Within our approach, we tackle the problem of quantum enhancements in reinforcement learning as well, and propose a systematic scheme for providing improvements. As an example, we show that quadratic improvements in learning efficiency, and exponential improvements in performance over limited time periods, can be obtained for a broad class of learning problems.
- Oct 18 2016 cond-mat.mtrl-sci arXiv:1610.04684v1Surface phenomena are increasingly becoming important in exploring nanoscale materials growth and characterization. Consequently, the need for atomistic based simulations is increasing. Nevertheless, relying entirely on quantum mechanical methods limits the length and time scales one can consider, resulting in an ever increasing dependence on alternative machine learning based force fields. Recently, we proposed a machine learning approach, known as AGNI, that allows fast and accurate atomic force predictions given the atom's neighborhood environment. Here, we make use of such force fields to study and characterize the nanoscale diffusion and growth processes occurring on an Al (111) surface. In particular we focus on the adatom ripening phenomena, confirming past experimental findings, wherein a low and high temperature growth regime were observed, using entirely molecular dynamics simulations.
- Classifying phases of matter is a central problem in physics. For quantum mechanical systems, this task can be daunting owing to the exponentially large Hilbert space. Thanks to the available computing power and access to ever larger data sets, classification problems are now routinely solved using machine learning techniques. Here, we propose to use a neural network based approach to find phase transitions depending on the performance of the neural network after training it with deliberately incorrectly labelled data. We demonstrate the success of this method on the topological phase transition in the Kitaev chain, the thermal phase transition in the classical Ising model, and the many-body-localization transition in a disordered quantum spin chain. Our method does not depend on order parameters, knowledge of the topological content of the phases, or any other specifics of the transition at hand. It therefore paves the way to a generic tool to identify unexplored phase transitions.
- Oct 10 2016 cond-mat.mtrl-sci arXiv:1610.02098v2Force fields developed with machine learning methods in tandem with quantum mechanics are beginning to find merit, given their (i) low cost, (ii) accuracy, and (iii) versatility. Recently, we proposed one such approach, wherein, the vectorial force on an atom is computed directly from its environment. Here, we discuss the multi-step workflow required for their construction, which begins with generating diverse reference atomic environments and force data, choosing a numerical representation for the atomic environments, down selecting a representative training set, and lastly the learning method itself, for the case of Al. The constructed force field is then validated by simulating complex materials phenomena such as surface melting and stress-strain behavior - that truly go beyond the realm of $ab\ initio$ methods both in length and time scales. To make such force fields truly versatile an attempt to estimate the uncertainty in force predictions is put forth, allowing one to identify areas of poor performance and paving the way for their continual improvement.
- Decompositions of tensors into factor matrices, which interact through a core tensor, have found numerous applications in signal processing and machine learning. A more general tensor model which represents data as an ordered network of sub-tensors of order-2 or order-3 has, so far, not been widely considered in these fields, although this so-called tensor network decomposition has been long studied in quantum physics and scientific computing. In this study, we present novel algorithms and applications of tensor network decompositions, with a particular focus on the tensor train decomposition and its variants. The novel algorithms developed for the tensor train decomposition update, in an alternating way, one or several core tensors at each iteration, and exhibit enhanced mathematical tractability and scalability to exceedingly large-scale data tensors. The proposed algorithms are tested in classic paradigms of blind source separation from a single mixture, denoising, and feature extraction, and achieve superior performance over the widely used truncated algorithms for tensor train decomposition.
- Artificial neural networks play a prominent role in the rapidly growing field of machine learning and are recently introduced to quantum many-body systems to tackle complex problems. Here, we find that even topological states with long-range quantum entanglement can be represented with classical artificial neural networks. This is demonstrated by using two concrete spin systems, the one-dimensional (1D) symmetry-protected topological cluster state and the 2D toric code state with an intrinsic topological order. For both cases we show rigorously that the topological ground states can be represented by short-range neural networks in an \it exact fashion. This neural network representation, in addition to being exact, is surprisingly \it efficient as the required number of hidden neurons is as small as the number of physical spins. Our exact construction of topological-order neuron-representation demonstrates explicitly the exceptional power of neural networks in describing exotic quantum states, and at the same time provides valuable topological data to supervise machine learning topological quantum orders in generic lattice models.
- Sep 28 2016 physics.chem-ph arXiv:1609.08259v4Learning from data has led to paradigm shifts in a multitude of disciplines, including web, text, and image search, speech recognition, as well as bioinformatics. Can machine learning enable similar breakthroughs in understanding quantum many-body systems? Here we develop an efficient deep learning approach that enables spatially and chemically resolved insights into quantum-mechanical observables of molecular systems. We unify concepts from many-body Hamiltonians with purpose-designed deep tensor neural networks (DTNN), which leads to size-extensive and uniformly accurate (1 kcal/mol) predictions in compositional and configurational chemical space for molecules of intermediate size. As an example of chemical relevance, the DTNN model reveals a classification of aromatic rings with respect to their stability -- a useful property that is not contained as such in the training dataset. Further applications of DTNN for predicting atomic energies and local chemical potentials in molecules, reliable isomer energies, and molecules with peculiar electronic structure demonstrate the high potential of machine learning for revealing novel insights into complex quantum-chemical systems.
- There is an enormous amount of information that can be extracted from the data of a quantum gas microscope that has yet to be fully explored. The quantum gas microscope has been used to directly measure magnetic order, dynamic correlations, Pauli blocking, and many other physical phenomena in several recent groundbreaking experiments. However, the analysis of the data from a quantum gas microscope can be pushed much further, and when used in conjunction with theoretical constructs it is possible to measure virtually any observable of interest in a wide range of systems. We focus on how to measure quantum entanglement in large interacting quantum systems. In particular, we show that quantum gas microscopes can be used to measure the entanglement of interacting boson systems exactly, where previously it had been thought this was only possible for non-interacting systems. We consider algorithms that can work for large experimental data sets which are similar to theoretical variational Monte Carlo techniques, and more data limited sets using properties of correlation functions.
- The current work addresses quantum machine learning in the context of Quantum Artificial Neural Networks such that the networks' processing is divided in two stages: the learning stage, where the network converges to a specific quantum circuit, and the backpropagation stage where the network effectively works as a self-programing quantum computing system that selects the quantum circuits to solve computing problems. The results are extended to general architectures including recurrent networks that interact with an environment, coupling with it in the neural links' activation order, and self-organizing in a dynamical regime that intermixes patterns of dynamical stochasticity and persistent quasiperiodic dynamics, making emerge a form of noise resilient dynamical record.
- We use density-matrix renormalization group, applied to a one-dimensional model of continuum Hamiltonians, to accurately solve chains of hydrogen atoms of various separations and numbers of atoms. We train and test a machine-learned approximation to $F[n]$, the universal part of the electronic density functional, to within quantum chemical accuracy. Our calculation (a) bypasses the standard Kohn-Sham approach, avoiding the need to find orbitals, (b) includes the strong correlation of highly-stretched bonds without any specific difficulty (unlike all standard DFT approximations) and (c) is so accurate that it can be used to find the energy in the thermodynamic limit to quantum chemical accuracy.
- Last year, at least 30,000 scientific papers used the Kohn-Sham scheme of density functional theory to solve electronic structure problems in a wide variety of scientific fields, ranging from materials science to biochemistry to astrophysics. Machine learning holds the promise of learning the kinetic energy functional via examples, by-passing the need to solve the Kohn-Sham equations. This should yield substantial savings in computer time, allowing either larger systems or longer time-scales to be tackled, but attempts to machine-learn this functional have been limited by the need to find its derivative. The present work overcomes this difficulty by directly learning the density-potential and energy-density maps for test systems and various molecules. Both improved accuracy and lower computational cost with this method are demonstrated by reproducing DFT energies for a range of molecular geometries generated during molecular dynamics simulations. Moreover, the methodology could be applied directly to quantum chemical calculations, allowing construction of density functionals of quantum-chemical accuracy.
- There is increasing interest in the potential advantages of using quantum computing technologies as sampling engines to speedup machine learning and probabilistic programming tasks. However, some pressing challenges in state-of-the-art quantum annealers have to be overcome before we can assess their actual performance. Most notably, the effective temperature at which samples are generated is instance-dependent and unknown, the interaction graph is sparse, the parameters are noisy, and the dynamic range of the parameters is finite. Of all these limitations, the sparse connectivity resulting from the local interaction between quantum bits in physical hardware implementations, is considered the most severe limitation to the quality of constructing powerful machine learning models. Here we show how to surpass this "curse of limited connectivity" bottleneck and illustrate our findings by training probabilistic generative models with arbitrary pairwise connectivity. Our model can be trained in quantum hardware without full knowledge of the effective parameters specifying the corresponding Boltzmann-like distribution. Therefore, inference of the effective temperature is avoided and the effect of noise in the parameters is mitigated. We illustrate our findings by successfully training hardware-embedded models with all-to-all connectivity on a real dataset of handwritten digits and two synthetic datasets. In each of these datasets we show the generative capabilities of the models learned with the assistance of the quantum annealer in experiments with up to 940 quantum bits. Additionally, we show a visual Turing test with handwritten digit data, where the machine generating the digits is a quantum processor. Such digits, with a remarkable similarity to those generated by humans, are extracted from the experiments with 940 quantum bits.
- Sep 09 2016 cond-mat.str-el arXiv:1609.02552v2Machine learning offers an unprecedented perspective for the problem of classifying phases in condensed matter physics. We employ neural network machine learning techniques to distinguish finite-temperature phases of the strongly-correlated fermions on cubic lattices. We show that a three-dimensional convolutional network trained on auxiliary field configurations produced by quantum Monte Carlo simulations of the Hubbard model can correctly predict the magnetic phase diagram of the model at the average density of one (half filling). We then use the network, trained at half filling, to explore the trend in the transition temperature as the system is doped away from half filling. This transfer learning approach predicts that the instability to the magnetic phase extends to at least 5% doping in this region. Our results pave the way for other machine learning applications in correlated quantum many-body systems.
- State-of-the-art machine learning techniques promise to become a powerful tool in statistical mechanics via their capacity to distinguish different phases of matter in an automated way. Here we demonstrate that convolutional neural networks (CNN) can be optimized for quantum many-fermion systems such that they correctly identify and locate quantum phase transitions in such systems. Using auxiliary-field quantum Monte Carlo (QMC) simulations to sample the many-fermion system, we show that the Green's function (but not the auxiliary field) holds sufficient information to allow for the distinction of different fermionic phases via a CNN. We demonstrate that this QMC + machine learning approach works even for systems exhibiting a severe fermion sign problem where conventional approaches to extract information from the Green's function, e.g.~in the form of equal-time correlation functions, fail. We expect that this capacity of hierarchical machine learning techniques to circumvent the fermion sign problem will drive novel insights into some of the most fundamental problems in statistical physics.
- Aug 29 2016 cond-mat.mtrl-sci arXiv:1608.07374v1Recently, machine learning has emerged as an alternative, powerful approach for predicting quantum-mechanical properties of molecules and solids. Here, using kernel ridge regression and atomic fingerprints representing local environments of atoms, we trained a machine-learning model on a crystalline silicon system in order to directly predict the atomic forces at a wide range of temperatures. Our idea is to construct a machine-learning model using a quantum-mechanical data set taken from canonical-ensemble simulations at a higher temperature, or an upper bound of the temperature range. With our model, the force prediction errors were about 2% or smaller with respect to the corresponding force ranges, in the temperature region between 300 and 1650 K. We also verified the applicability to a larger system, ensuring the transferability with respect to system size.
- Aug 23 2016 physics.chem-ph arXiv:1608.06194v3The predictive accuracy of Machine Learning (ML) models of molecular properties depends on the choice of the molecular representation. Based on the postulates of quantum mechanics, we introduce a hierarchy of representations which meet uniqueness and target similarity criteria. To systematically control target similarity, we rely on interatomic many body expansions, as implemented in universal force-fields, including Bonding, Angular, and higher order terms (BA). Addition of higher order contributions systematically increases similarity to the true potential energy and predictive accuracy of the resulting ML models. We report numerical evidence for the performance of BAML models trained on molecular properties pre-calculated at electron-correlated and density functional theory level of theory for thousands of small organic molecules. Properties studied include enthalpies and free energies of atomization, heatcapacity, zero-point vibrational energies, dipole-moment, polarizability, HOMO/LUMO energies and gap, ionization potential, electron affinity, and electronic excitations. After training, BAML predicts energies or electronic properties of out-of-sample molecules with unprecedented accuracy and speed.
- Aug 02 2016 physics.comp-ph physics.chem-ph arXiv:1608.00316v1Density-functional theory is a formally exact description of a many-body quantum system in terms of its density; in practice, however, approximations to the universal density functional are required. In this work, a model based on deep learning is developed to approximate this functional. Deep learning allows computational models that are capable of naturally discovering intricate structure in large and/or high-dimensional data sets, with multiple levels of abstraction. As no assumptions are made as to the form of this structure, this approach is much more powerful and flexible than traditional approaches. As an example application, the model is shown to perform well on approximating the kinetic-energy density functional for noninteracting electrons. The model is analyzed in detail, and its advantages over conventional machine learning are discussed.
- Determinantal Point Processes (DPPs) are probabilistic models of repulsion that originate in quantum physics and random matrix theory and have been of recent interest in computer science. DPPs define distributions over subsets of a given ground set, and exhibit interesting properties such as negative correlation. When applied to kernel methods in machine learning, DPPs give rise to an efficient algorithm to select a small, diverse sample of the given data. Kulesza and Taskar [KT12] posed a natural open question: Can we sample from DPPs when there are additional constraints on the allowable subsets? In this paper, we study the complexity of sampling from DPPs over combinatorially-constrained families of subsets and present several connections and applications. We start by showing that it is at least as hard to sample from combinatorial DPPs as it is to compute the mixed discriminant of a tuple of positive semidefinite matrices. Subsequently, we give a polynomial time algorithm for sampling from combinatorial DPPs with a constant number of linear constraints; thus, we make significant progress towards answering the question of [KT12]. These results can lead to several non-trivial applications; e.g., we show how to derandomize a result of Nikolov and Singh [NS16] for maximizing subdeterminants under (a constant number of) partition constraints and -- motivated by making the [MSS13] proof for the Kadison-Singer problem algorithmic -- give an algorithm to compute the higher-order coefficients of mixed characteristic polynomials.
- Belief propagation is a powerful tool in statistical physics, machine learning, and modern coding theory. As a decoding method, it is ubiquitous in classical error correction and has also been applied to stabilizer-based quantum error correction. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding, in some cases even up to the Shannon capacity of the channel. Here we construct a belief propagation algorithm which passes quantum messages on the factor graph and is capable of decoding the classical-quantum channel with pure state outputs. This gives explicit decoding circuits whose number of gates is quadratic in the blocklength of the code. We also show that this decoder can be modified to work with polar codes for the pure state channel and as part of a polar decoder for transmitting quantum information over the amplitude damping channel. These represent the first explicit capacity-achieving decoders for non-Pauli channels.
- Quantum control is valuable for various quantum technologies such as high-fidelity gates for universal quantum computing, adaptive quantum-enhanced metrology, and ultra-cold atom manipulation. Although supervised machine learning and reinforcement learning are widely used for optimizing control parameters in classical systems, quantum control for parameter optimization is mainly pursued via gradient-based greedy algorithms. Although the quantum fitness landscape is often compatible with greedy algorithms, sometimes greedy algorithms yield poor results, especially for large-dimensional quantum systems. We employ differential evolution algorithms to circumvent the stagnation problem of non-convex optimization. We improve quantum control fidelity for noisy system by averaging over the objective function. To reduce computational cost, we introduce heuristics for early termination of runs and for adaptive selection of search subspaces. Our implementation is massively parallel and vectorized to reduce run time even further. We demonstrate our methods with two examples, namely quantum phase estimation and quantum gate design, for which we achieve superior fidelity and scalability than obtained using greedy algorithms.
- Jun 21 2016 cs.ET arXiv:1606.06123v1The D-Wave is an adiabatic quantum computer. It is an understatement to say that it is not a traditional computer. It can be viewed as a computational accelerator or more precisely a computational oracle, where one asks it a relevant question and it returns a useful answer. The question is how do you ask a relevant question and how do you use the answer it returns. This paper addresses these issues in a way that is pertinent to machine learning. A Boltzmann machine is implemented with the D-Wave since the D-Wave is merely a hardware instantiation of a partially connected Boltzmann machine. This paper presents a prototype implementation of a 3-layered neural network where the D-Wave is used as the middle (hidden) layer of the neural network. This paper also explains how the D-Wave can be utilized in a multi-layer neural network (more than 3 layers) and one in which each layer may be multiple times the size of the D-Wave being used.
- Jun 09 2016 cond-mat.dis-nn quant-ph arXiv:1606.02318v1The challenge posed by the many-body problem in quantum physics originates from the difficulty of describing the non-trivial correlations encoded in the exponential complexity of the many-body wave function. Here we demonstrate that systematic machine learning of the wave function can reduce this complexity to a tractable computational form, for some notable cases of physical interest. We introduce a variational representation of quantum states based on artificial neural networks with variable number of hidden neurons. A reinforcement-learning scheme is then demonstrated, capable of either finding the ground-state or describing the unitary time evolution of complex interacting quantum systems. We show that this approach achieves very high accuracy in the description of equilibrium and dynamical properties of prototypical interacting spins models in both one and two dimensions, thus offering a new powerful tool to solve the quantum many-body problem.
- Jun 06 2016 cs.DS arXiv:1606.00942v2Computation of the trace of a matrix function plays an important role in many scientific computing applications, including applications in machine learning, computational physics (e.g., lattice quantum chromodynamics), network analysis and computational biology (e.g., protein folding), just to name a few application areas. We propose a linear-time randomized algorithm for approximating the trace of matrix functions of large symmetric matrices. Our algorithm is based on coupling function approximation using Chebyshev interpolation with stochastic trace estimators (Hutchinson's method), and as such requires only implicit access to the matrix, in the form of a function that maps a vector to the product of the matrix and the vector. We provide rigorous approximation error in terms of the extremal eigenvalue of the input matrix, and the Bernstein ellipse that corresponds to the function at hand. Based on our general scheme, we provide algorithms with provable guarantees for important matrix computations, including log-determinant, trace of matrix inverse, Estrada index, Schatten p-norm, and testing positive definiteness. We experimentally evaluate our algorithm and demonstrate its effectiveness on matrices with tens of millions dimensions.
- May 19 2016 quant-ph arXiv:1605.05370v1We study a variant of the quantum approximate optimization algorithm [ E. Farhi, J. Goldstone, and S. Gutmann, arXiv:1411.4028] with slightly different parametrization and different objective: rather than looking for a state which approximately solves an optimization problem, our goal is to find a quantum algorithm that, given an instance of MAX-2-SAT, will produce a state with high overlap with the optimal state. Using a machine learning approach, we chose a "training set" of instances and optimized the parameters to produce large overlap for the training set. We then tested these optimized parameters on a larger instance set. As a training set, we used a subset of the hard instances studied by E. Crosson, E. Farhi, C. Yen-Yu Lin, H.-H. Lin, and P. Shor (CFLLS) [arXiv:1401.7320]. When tested on the full set, the parameters that we find produce significantly larger overlap than the optimized annealing times of CFLLS. Testing on other random instances from $20$ to $28$ bits continues to show improvement over annealing, with the improvement being most notable on the hardest instances. Further tests on instances of MAX-3-SAT also showed improvement on the hardest instances. This algorithm may be a possible application for near-term quantum computers with limited coherence times.
- Apr 20 2016 cond-mat.stat-mech quant-ph arXiv:1604.05354v2We construct a planar vertex model that encodes the result of a universal reversible classical computation in its ground state. The approach involves Boolean variables (spins) placed on links of a two-dimensional lattice, with vertices representing logic gates. Large short-ranged interactions between at most two spins implement the operation of each gate. The lattice is anisotropic with one direction corresponding to "computational" time, and with transverse boundaries storing the computation's input and output. While we show that the model displays no finite temperature phase transitions, independent of circuit, the computational complexity is encoded in the scaling of the relaxation rate into the ground state with the system size. The paper uses thermal annealing and a novel and more efficient heuristic "annealing with learning" to study various computational problems. To explore faster relaxation routes, we construct an explicit mapping of the vertex model into the Chimera architecture of the D-Wave machine, initiating a novel approach to reversible classical computation based on state-of-the-art implementations of quantum annealing.
- The wide-ranging adoption of quantum technologies requires practical, high-performance advances in our ability to maintain quantum coherence while facing the challenge of state collapse under measurement. Here we use techniques from control theory and machine learning to predict the future evolution of a qubit's state; we deploy this information to suppress stochastic, semiclassical decoherence, even when access to measurements is limited. First, we implement a time-division-multiplexed approach, interleaving measurement periods with periods of unsupervised but stabilised operation during which qubits are available, for e.g. quantum information experiments. Second, we employ predictive feedback during sequential but time delayed measurements to reduce the Dick effect as encountered in passive frequency standards. Both experiments demonstrate significant improvements in qubit phase stability over "traditional" measurement-based feedback approaches by exploiting time domain correlations in the noise processes. This technique requires no additional hardware and is applicable to all two-level quantum systems where projective measurements are possible.
- We utilize machine learning models which are based on recurrent neural networks to optimize dynamical decoupling (DD) sequences. DD is a relatively simple technique for suppressing the errors in quantum memory for certain noise models. In numerical simulations, we show that with minimum use of prior knowledge and starting from random sequences, the models are able to improve over time and eventually output DD-sequences with performance better than that of the well known DD-families. Furthermore, our algorithm is easy to implement in experiments to find solutions tailored to the specific hardware, as it treats the figure of merit as a black box.
- MicroRNAs (miRNAs) are short sequences of ribonucleic acids that control the expression of target messenger RNAs (mRNAs) by binding them. Robust prediction of miRNA-mRNA pairs is of utmost importance in deciphering gene regulations but has been challenging because of high false positive rates, despite a deluge of computational tools that normally require laborious manual feature extraction. This paper presents an end-to-end machine learning framework for miRNA target prediction. Leveraged by deep recurrent neural networks-based auto-encoding and sequence-sequence interaction learning, our approach not only delivers an unprecedented level of accuracy but also eliminates the need for manual feature extraction. The performance gap between the proposed method and existing alternatives is substantial (over 25% increase in F-measure), and deepTarget delivers a quantum leap in the long-standing challenge of robust miRNA target prediction.
- A recommendation system uses the past purchases or ratings of $n$ products by a group of $m$ users, in order to provide personalized recommendations to individual users. The information is modeled as an $m \times n$ preference matrix which is assumed to have a good rank-$k$ approximation, for a small constant $k$. In this work, we present a quantum algorithm for recommendation systems that has running time $O(\text{poly}(k)\text{polylog}(mn))$. All known classical algorithms for recommendation systems that work through reconstructing an approximation of the preference matrix run in time polynomial in the matrix dimension. Our algorithm provides good recommendations by sampling efficiently from an approximation of the preference matrix, without reconstructing the entire matrix. For this, we design an efficient quantum procedure to project a given vector onto the row space of a given matrix. This is the first algorithm for recommendation systems that runs in time polylogarithmic in the dimensions of the matrix and provides an example of a quantum machine learning algorithm for a real world application.
- Machine learning applications are limited by computational power. In this paper, we gain novel insights into the application of quantum annealing (QA) to machine learning (ML) through experiments in natural language processing (NLP), seizure prediction, and linear separability testing. These experiments are performed on QA simulators and early-stage commercial QA hardware and compared to an unprecedented number of traditional ML techniques. We extend QBoost, an early implementation of a binary classifier that utilizes a quantum annealer, via resampling and ensembling of predicted probabilities to produce a more robust class estimator. To determine the strengths and weaknesses of this approach, resampled QBoost (RQBoost) is tested across several datasets and compared to QBoost and traditional ML. We show and explain how QBoost in combination with a commercial QA device are unable to perfectly separate binary class data which is linearly separable via logistic regression with shrinkage. We further explore the performance of RQBoost in the space of NLP and seizure prediction and find QA-enabled ML using QBoost and RQBoost is outperformed by traditional techniques. Additionally, we provide a detailed discussion of algorithmic constraints and trade-offs imposed by the use of this QA hardware. Through these experiments, we provide unique insights into the state of quantum ML via boosting and the use of quantum annealing hardware that are valuable to institutions interested in applying QA to problems in ML and beyond.
- Mar 22 2016 quant-ph arXiv:1603.06222v2Machine learning is a fascinating and exciting field within computer science. Recently, this excitement has been transferred to the quantum information realm. Currently, all proposals for the quantum version of machine learning utilize the finite-dimensional substrate of discrete variables. Here we generalize quantum machine learning to the more complex, but still remarkably practical, infinite-dimensional systems. We present the critical subroutines of quantum machine learning algorithms for an all-photonic continuous-variable quantum computer that achieve an exponential speedup compared to their equivalent classical counterparts. Finally, we also map out an experimental implementation which can be used as a blueprint for future photonic demonstrations.
- Quantum computer has an amazing potential of fast information processing. However, realisation of a digital quantum computer is still a challenging problem requiring highly accurate controls and key application strategies. Here we propose a novel platform, quantum reservoir computing, to solve these issues successfully by exploiting natural quantum dynamics, which is ubiquitous in laboratories nowadays, for machine learning. In this framework, nonlinear dynamics including classical chaos can be universally emulated in quantum systems. A number of numerical experiments show that quantum systems consisting of at most seven qubits possess computational capabilities comparable to conventional recurrent neural networks of 500 nodes. This discovery opens up a new paradigm for information processing with artificial intelligence powered by quantum physics.
- Jan 29 2016 quant-ph arXiv:1601.07823v2We give an algorithm for prediction on a quantum computer which is based on a linear regression model with least squares optimisation. Opposed to related previous contributions suffering from the problem of reading out the optimal parameters of the fit, our scheme focuses on the machine learning task of guessing the output corresponding to a new input given examples of data points. Furthermore, we adapt the algorithm to process non-sparse data matrices that can be represented by low-rank approximations, and significantly improve the dependency on its condition number. The prediction result can be accessed through a single qubit measurement or used for further quantum information processing routines. The algorithm's runtime is logarithmic in the dimension of the input space provided the data is given as quantum information as an input to the routine.
- We consider a general class of models, where a reinforcement learning (RL) agent learns from cyclic interactions with an external environment via classical signals. Perceptual inputs are encoded as quantum states, which are subsequently transformed by a quantum channel representing the agent's memory, while the outcomes of measurements performed at the channel's output determine the agent's actions. The learning takes place via stepwise modifications of the channel properties. They are described by an update rule that is inspired by the projective simulation (PS) model and equipped with a glow mechanism that allows for a backpropagation of policy changes, analogous to the eligibility traces in RL and edge glow in PS. In this way, the model combines features of PS with the ability for generalization, offered by its physical embodiment as a quantum system. We apply the agent to various setups of an invasion game and a grid world, which serve as elementary model tasks allowing a direct comparison with a basic classical PS agent.
- Jan 12 2016 quant-ph arXiv:1601.02036v1Inspired by the success of Boltzmann Machines based on classical Boltzmann distribution, we propose a new machine learning approach based on quantum Boltzmann distribution of a transverse-field Ising Hamiltonian. Due to the non-commutative nature of quantum mechanics, the training process of the Quantum Boltzmann Machine (QBM) can become nontrivial. We circumvent the problem by introducing bounds on the quantum probabilities. This allows us to train the QBM efficiently by sampling. We show examples of QBM training with and without the bound, using exact diagonalization, and compare the results with classical Boltzmann training. We also discuss the possibility of using quantum annealing processors like D-Wave for QBM training and application.
- Total energies of crystal structures can be calculated to high precision using quantum-based density functional theory (DFT) methods, but the calculations can be time consuming and scale badly with system size. Cluster expansions of total energy as a linear superposition of pair, triplet and higher interactions can efficiently approximate the total energies but are best suited to simple lattice structures. To model the total energy of boron carbide, with a complex crystal structure, we explore the utility of machine learning methods ($L_1$-penalized regression, neural network, Gaussian process and support vector regression) that capture certain non-linear effects associated with many-body interactions despite requiring only pair frequencies as input. Our interaction models are combined with Monte Carlo simulations to evaluate the thermodynamics of chemical ordering.
- Dec 21 2015 quant-ph arXiv:1512.06069v1The main promise of quantum computing is to efficiently solve certain problems that are prohibitively expensive for a classical computer. Most problems with a proven quantum advantage involve the repeated use of a black box, or oracle, whose structure encodes the solution. One measure of the algorithmic performance is the query complexity, i.e., the scaling of the number of oracle calls needed to find the solution with a given probability. Few-qubit demonstrations of quantum algorithms, such as Deutsch-Jozsa and Grover, have been implemented across diverse physical systems such as nuclear magnetic resonance, trapped ions, optical systems, and superconducting circuits. However, at the small scale, these problems can already be solved classically with a few oracle queries, and the attainable quantum advantage is modest. Here we solve an oracle-based problem, known as learning parity with noise, using a five-qubit superconducting processor. Running classical and quantum algorithms on the same oracle, we observe a large gap in query count in favor of quantum processing. We find that this gap grows by orders of magnitude as a function of the error rates and the problem size. This result demonstrates that, while complex fault-tolerant architectures will be required for universal quantum computing, a quantum advantage already emerges in existing noisy systems
- Gaussian processes (GP) are a widely used model for regression problems in supervised machine learning. Implementation of GP regression typically requires $O(n^3)$ logic gates. We show that the quantum linear systems algorithm [Harrow et al., Phys. Rev. Lett. 103, 150502 (2009)] can be applied to Gaussian process regression (GPR), leading to an exponential reduction in computation time in some instances. We show that even in some cases not ideally suited to the quantum linear systems algorithm, a polynomial increase in efficiency still occurs.
- For Machine Learning (ML) classification problem, where a vector of $\mathbf{x}$--observations (values of attributes) is mapped to a single $y$ value (class label), a generalized Radon--Nikodym type of solution is proposed. Quantum--mechanics --like probability states $\psi^2(\mathbf{x})$ are considered and "Cluster Centers", corresponding to the extremums of $<y\psi^2(\mathbf{x})>/<\psi^2(\mathbf{x})>$, are found from generalized eigenvalues problem. The eigenvalues give possible $y^{[i]}$ outcomes and corresponding to them eigenvectors $\psi^{[i]}(\mathbf{x})$ define "Cluster Centers". The projection of a $\psi$ state, localized at given $\mathbf{x}$ to classify, on these eigenvectors define the probability of $y^{[i]}$ outcome, thus avoiding using a norm ($L^2$ or other types), required for "quality criteria" in a typical Machine Learning technique. A coverage of each `Cluster Center" is calculated, what potentially allows to separate system properties (described by $y^{[i]}$ outcomes) and system testing conditions (described by $C^{[i]}$ coverage). As an example of such application $y$ distribution estimator is proposed in a form of pairs $(y^{[i]},C^{[i]})$, that can be considered as Gauss quadratures generalization. This estimator allows to perform $y$ probability distribution estimation in a strongly non--Gaussian case.
- Dec 10 2015 quant-ph arXiv:1512.02900v1Here we discuss advances in the field of quantum machine learning. The following document offers a hybrid discussion; both reviewing the field as it is currently, and suggesting directions for further research. We include both algorithms and experimental implementations in the discussion. The field's outlook is generally positive, showing significant promise. However, we believe there are appreciable hurdles to overcome before one can claim that it is a primary application of quantum computation.
- Three-qubit quantum gates are key ingredients for quantum error correction and quantum information processing. We generate quantum-control procedures to design three types of three-qubit gates, namely Toffoli, Controlled-Not-Not and Fredkin gates. The design procedures are applicable to a system comprising three nearest-neighbor-coupled superconducting artificial atoms. For each three-qubit gate, the numerical simulation of the proposed scheme achieves 99.9% fidelity, which is an accepted threshold fidelity for fault-tolerant quantum computing. We test our procedure in the presence of decoherence-induced noise as well as show its robustness against random external noise generated by the control electronics. The three-qubit gates are designed via the machine learning algorithm called Subspace-Selective Self-Adaptive Differential Evolution (SuSSADE).
- Nov 26 2015 physics.chem-ph physics.comp-ph arXiv:1511.07883v1Obtaining the exciton dynamics of large photosynthetic complexes by using mixed quantum mechanics/molecular mechanics (QM/MM) is computationally demanding. We propose a machine learning technique, multi-layer perceptrons, as a tool to reduce the time required to compute excited state energies. With this approach we predict time-dependent density functional theory (TDDFT) excited state energies of bacteriochlorophylls in the Fenna-Matthews-Olson (FMO) complex. Additionally we compute spectral densities and exciton populations from the predictions. Different methods to determine multi-layer perceptron training sets are introduced, leading to several initial data selections. In addition, we compute spectral densities and exciton populations. Once multi-layer perceptrons are trained, predicting excited state energies was found to be significantly faster than the corresponding QM/MM calculations. We showed that multi-layer perceptrons can successfully reproduce the energies of QM/MM calculations to a high degree of accuracy with prediction errors contained within 0.01 eV (0.5%). Spectral densities and exciton dynamics are also in agreement with the TDDFT results. The acceleration and accurate prediction of dynamics strongly encourage the combination of machine learning techniques with ab-initio methods.
- Oct 27 2015 quant-ph arXiv:1510.07611v4An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact on deep learning and other machine-learning applications. Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively. One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggest it will do so with an \it instance-dependent effective temperature, different from its physical temperature. Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling. In this work, we propose a strategy to overcome this challenge with a simple effective-temperature estimation algorithm. We provide a systematic study assessing the impact of the effective temperatures in the learning of a special class of a restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep-learning architectures. We also provide a comparison to $k$-step contrastive divergence (CD-$k$) with $k$ up to 100. Although assuming a suitable fixed effective temperature also allows us to outperform one step contrastive divergence (CD-1), only when using an instance-dependent effective temperature do we find a performance close to that of CD-100 for the case studied here.
- Oct 27 2015 physics.chem-ph arXiv:1510.07512v3We review recent studies dealing with the generation of machine learning models of molecular and solid properties. The models are trained and validated using standard quantum chemistry results obtained for organic molecules and materials selected from chemical space at random.