Johannes Bausch, Andrew W Senior, Francisco J H Heras et al.
Quantum error-correction is a prerequisite for reliable quantum computation. Towards this goal, we present a recurrent, transformer-based neural network which learns to decode the surface code, the leading quantum error-correction code. Our decoder outperforms state-of-the-art algorithmic decoders on real-world data from Google's Sycamore quantum processor for distance 3 and 5 surface codes. On distances up to 11, the decoder maintains its advantage on simulated data with realistic noise including cross-talk, leakage, and analog readout signals, and sustains its accuracy far beyond the 25 cycles it was trained on. Our work illustrates the ability of machine learning to go beyond human-designed algorithms by learning from data directly, highlighting machine learning as a strong contender for decoding in quantum computers.
In living organisms, homeostasis is the natural regulation of internal states aimed at maintaining conditions compatible with life. Typical artificial systems are not equipped with comparable regulatory features. Here, we introduce an artificial neural network that incorporates homeostatic features. Its own computing substrate is placed in a needful and vulnerable relation to the very objects over which it computes. For example, artificial neurons performing classification of MNIST digits or Fashion-MNIST articles of clothing may receive excitatory or inhibitory effects, which alter their own learning rate as a direct result of perceiving and classifying the digits. In this scenario, accurate recognition is desirable to the agent itself because it guides decisions to regulate its vulnerable internal states and functionality. Counterintuitively, the addition of vulnerability to a learner does not necessarily impair its performance. On the contrary, self-regulation in response to vulnerability confers benefits under certain conditions. We show that homeostatic design confers increased adaptability under concept shift, in which the relationships between labels and data change over time, and that the greatest advantages are obtained under the highest rates of shift. This necessitates the rapid un-learning of past associations and the re-learning of new ones. We also demonstrate the superior abilities of homeostatic learners in environments with dynamically changing rates of concept shift. Our homeostatic design exposes the artificial neural network's thinking machinery to the consequences of its own "thoughts", illustrating the advantage of putting one's own "skin in the game" to improve fluid intelligence.
Michael Broughton, Guillaume Verdon, Trevor McCourt et al.
We introduce TensorFlow Quantum (TFQ), an open source library for the rapid prototyping of hybrid quantum-classical models for classical or quantum data. This framework offers high-level abstractions for the design and training of both discriminative and generative quantum models under TensorFlow and supports high-performance quantum circuit simulators. We provide an overview of the software architecture and building blocks through several examples and review the theory of hybrid quantum-classical neural networks. We illustrate TFQ functionalities via several basic applications including supervised learning for quantum classification, quantum control, simulating noisy quantum circuits, and quantum approximate optimization. Moreover, we demonstrate how one can apply TFQ to tackle advanced quantum learning tasks including meta-learning, layerwise learning, Hamiltonian learning, sampling thermal states, variational quantum eigensolvers, classification of quantum phase transitions, generative adversarial networks, and reinforcement learning. We hope this framework provides the necessary tools for the quantum computing and machine learning research communities to explore models of both natural and artificial quantum systems, and ultimately discover new quantum algorithms which could potentially yield a quantum advantage.
Masoud Mohseni, Daniel Eppens, Johan Strumpfer et al.
Optimizing highly complex cost/energy functions over discrete variables is at the heart of many open problems across different scientific disciplines and industries. A major obstacle is the emergence of many-body effects among certain subsets of variables in hard instances leading to critical slowing down or collective freezing for known stochastic local search strategies. An exponential computational effort is generally required to unfreeze such variables and explore other unseen regions of the configuration space. Here, we introduce a quantum-inspired family of nonlocal Nonequilibrium Monte Carlo (NMC) algorithms by developing an adaptive gradient-free strategy that can efficiently learn key instance-wise geometrical features of the cost function. That information is employed on-the-fly to construct spatially inhomogeneous thermal fluctuations for collectively unfreezing variables at various length scales, circumventing costly exploration versus exploitation trade-offs. We apply our algorithm to two of the most challenging combinatorial optimization problems: random k-satisfiability (k-SAT) near the computational phase transitions and Quadratic Assignment Problems (QAP). We observe significant speedup and robustness over both specialized deterministic solvers and generic stochastic solvers. In particular, for 90% of random 4-SAT instances we find solutions that are inaccessible for the best specialized deterministic algorithm known as Survey Propagation (SP) with an order of magnitude improvement in the quality of solutions for the hardest 10% instances. We also demonstrate two orders of magnitude improvement in time-to-solution over the state-of-the-art generic stochastic solver known as Adaptive Parallel Tempering (APT).
Given the success of deep learning in classical machine learning, quantum algorithms for traditional neural network architectures may provide one of the most promising settings for quantum machine learning. Considering a fully-connected feedforward neural network, we show that conditions amenable to classical trainability via gradient descent coincide with those necessary for efficiently solving quantum linear systems. We propose a quantum algorithm to approximately train a wide and deep neural network up to $O(1/n)$ error for a training set of size $n$ by performing sparse matrix inversion in $O(\log n)$ time. To achieve an end-to-end exponential speedup over gradient descent, the data distribution must permit efficient state preparation and readout. We numerically demonstrate that the MNIST image dataset satisfies such conditions; moreover, the quantum algorithm matches the accuracy of the fully-connected network. Beyond the proven architecture, we provide empirical evidence for $O(\log n)$ training of a convolutional neural network with pooling.
Given a quantum circuit, a quantum computer can sample the output distribution exponentially faster in the number of bits than classical computers. A similar exponential separation has yet to be established in generative models through quantum sample learning: given samples from an n-qubit computation, can we learn the underlying quantum distribution using models with training parameters that scale polynomial in n under a fixed training time? We study four kinds of generative models: Deep Boltzmann machine (DBM), Generative Adversarial Networks (GANs), Long Short-Term Memory (LSTM) and Autoregressive GAN, on learning quantum data set generated by deep random circuits. We demonstrate the leading performance of LSTM in learning quantum samples, and thus the autoregressive structure present in the underlying quantum distribution from random quantum circuits. Both numerical experiments and a theoretical proof in the case of the DBM show exponentially growing complexity of learning-agent parameters required for achieving a fixed accuracy as n increases. Finally, we establish a connection between learnability and the complexity of generative models by benchmarking learnability against different sets of samples drawn from probability distributions of variable degrees of complexities in their quantum and classical representations.
Murphy Yuezhen Niu, Vadim Smelyanskyi, Paul Klimov et al.
Two-level-system (TLS) defects in amorphous dielectrics are a major source of noise and decoherence in solid-state qubits. Gate-dependent non-Markovian errors caused by TLS-qubit coupling are detrimental to fault-tolerant quantum computation and have not been rigorously treated in the existing literature. In this work, we derive the non-Markovian dynamics between TLS and qubits during a SWAP-like two-qubit gate and the associated average gate fidelity for frequency-tunable Transmon qubits. This gate dependent error model facilitates using qubits as sensors to simultaneously learn practical imperfections in both the qubit's environment and control waveforms. We combine the-state-of-art machine learning algorithm with Moiré-enhanced swap spectroscopy to achieve robust learning using noisy experimental data. Deep neural networks are used to represent the functional map from experimental data to TLS parameters and are trained through an evolutionary algorithm. Our method achieves the highest learning efficiency and robustness against experimental imperfections to-date, representing an important step towards in-situ quantum control optimization over environmental and control defects.
Guillaume Verdon, Michael Broughton, Jarrod R. McClean et al.
Quantum Neural Networks (QNNs) are a promising variational learning paradigm with applications to near-term quantum processors, however they still face some significant challenges. One such challenge is finding good parameter initialization heuristics that ensure rapid and consistent convergence to local minima of the parameterized quantum circuit landscape. In this work, we train classical neural networks to assist in the quantum learning process, also know as meta-learning, to rapidly find approximate optima in the parameter landscape for several classes of quantum variational algorithms. Specifically, we train classical recurrent neural networks to find approximately optimal parameters within a small number of queries of the cost function for the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, QAOA for Sherrington-Kirkpatrick Ising model, and for a Variational Quantum Eigensolver for the Hubbard model. By initializing other optimizers at parameter values suggested by the classical neural network, we demonstrate a significant improvement in the total number of optimization iterations required to reach a given accuracy. We further demonstrate that the optimization strategies learned by the neural network generalize well across a range of problem instance sizes. This opens up the possibility of training on small, classically simulatable problem instances, in order to initialize larger, classically intractably simulatable problem instances on quantum devices, thereby significantly reducing the number of required quantum-classical optimization iterations.
James King, Masoud Mohseni, William Bernoudy et al.
Genetic algorithms, which mimic evolutionary processes to solve optimization problems, can be enhanced by using powerful semi-local search algorithms as mutation operators. Here, we introduce reverse quantum annealing, a class of quantum evolutions that can be used for performing families of quasi-local or quasi-nonlocal search starting from a classical state, as novel sources of mutations. Reverse annealing enables the development of genetic algorithms that use quantum fluctuation for mutations and classical mechanisms for the crossovers -- we refer to these as Quantum-Assisted Genetic Algorithms (QAGAs). We describe a QAGA and present experimental results using a D-Wave 2000Q quantum annealing processor. On a set of spin-glass inputs, standard (forward) quantum annealing finds good solutions very quickly but struggles to find global optima. In contrast, our QAGA proves effective at finding global optima for these inputs. This successful interplay of non-local classical and quantum fluctuations could provide a promising step toward practical applications of Noisy Intermediate-Scale Quantum (NISQ) devices for heuristic discrete optimization.
Hongxiang Chen, Leonard Wossnig, Simone Severini et al.
Quantum mechanics fundamentally forbids deterministic discrimination of quantum states and processes. However, the ability to optimally distinguish various classes of quantum data is an important primitive in quantum information science. In this work, we train near-term quantum circuits to classify data represented by non-orthogonal quantum probability distributions using the Adam stochastic optimization algorithm. This is achieved by iterative interactions of a classical device with a quantum processor to discover the parameters of an unknown non-unitary quantum circuit. This circuit learns to simulates the unknown structure of a generalized quantum measurement, or Positive-Operator-Value-Measure (POVM), that is required to optimally distinguish possible distributions of quantum inputs. Notably we use universal circuit topologies, with a theoretically motivated circuit design, which guarantees that our circuits can in principle learn to perform arbitrary input-output mappings. Our numerical simulations show that shallow quantum circuits could be trained to discriminate among various pure and mixed quantum states exhibiting a trade-off between minimizing erroneous and inconclusive outcomes with comparable performance to theoretically optimal POVMs. We train the circuit on different classes of quantum data and evaluate the generalization error on unseen mixed quantum states. This generalization power hence distinguishes our work from standard circuit optimization and provides an example of quantum machine learning for a task that has inherently no classical analogue.
Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy et al.
Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2-design characteristic of random circuits, and that solutions to this problem must be studied.
Vasil S. Denchev, Nan Ding, Shin Matsushima et al.
We propose a totally corrective boosting algorithm with explicit cardinality regularization. The resulting combinatorial optimization problems are not known to be efficiently solvable with existing classical methods, but emerging quantum optimization technology gives hope for achieving sparser models in practice. In order to demonstrate the utility of our algorithm, we use a distributed classical heuristic optimizer as a stand-in for quantum hardware. Even though this evaluation methodology incurs large time and resource costs on classical computing machinery, it allows us to gauge the potential gains in generalization performance and sparsity of the resulting boosted ensembles. Our experimental results on public data sets commonly used for benchmarking of boosting algorithms decidedly demonstrate the existence of such advantages. If actual quantum optimization were to be used with this algorithm in the future, we would expect equivalent or superior results at much smaller time and energy costs during training. Moreover, studying cardinality-penalized boosting also sheds light on why unregularized boosting algorithms with early stopping often yield better results than their counterparts with explicit convex regularization: Early stopping performs suboptimal cardinality regularization. The results that we present here indicate it is beneficial to explicitly solve the combinatorial problem still left open at early termination.
We consider classification problems in which the label space has structure. A common example is hierarchical label spaces, corresponding to the case where one label subsumes another (e.g., animal subsumes dog). But labels can also be mutually exclusive (e.g., dog vs cat) or unrelated (e.g., furry, carnivore). To jointly model hierarchy and exclusion relations, the notion of a HEX (hierarchy and exclusion) graph was introduced in [7]. This combined a conditional random field (CRF) with a deep neural network (DNN), resulting in state of the art results when applied to visual object classification problems where the training labels were drawn from different levels of the ImageNet hierarchy (e.g., an image might be labeled with the basic level category "dog", rather than the more specific label "husky"). In this paper, we extend the HEX model to allow for soft or probabilistic relations between labels, which is useful when there is uncertainty about the relationship between two labels (e.g., an antelope is "sort of" furry, but not to the same degree as a grizzly bear). We call our new model pHEX, for probabilistic HEX. We show that the pHEX graph can be converted to an Ising model, which allows us to use existing off-the-shelf inference methods (in contrast to the HEX method, which needed specialized inference algorithms). Experimental results show significant improvements in a number of large-scale visual object classification tasks, outperforming the previous HEX model.
Quantum annealing is a heuristic quantum algorithm which exploits quantum resources to minimize an objective function embedded as the energy levels of a programmable physical system. To take advantage of a potential quantum advantage, one needs to be able to map the problem of interest to the native hardware with reasonably low overhead. Because experimental considerations constrain our objective function to take the form of a low degree PUBO (polynomial unconstrained binary optimization), we employ non-convex loss functions which are polynomial functions of the margin. We show that these loss functions are robust to label noise and provide a clear advantage over convex methods. These loss functions may also be useful for classical approaches as they compile to regularized risk expressions which can be evaluated in constant time with respect to the number of training examples.