Session A17: Quantum Machine Learning I

Challenges and opportunities for hybrid quantum-classical machine learning and optimization

We present an overview of our progress on quantum-inspired and quantum-assisted algorithms for optimization and machine learning at Quantum AI Lab at Google. We develop an end-to end quantum-inspired discrete optimization platform that uses an interplay of local and non-local thermal updates to sample from inaccessible low-energy states of spin-glass systems that encode high-quality solutions of certain hard combinatorial optimization. We introduce several new techniques for quantum circuit learning on Noisy Intermediate-Scale Quantum (NISQ) processors. We show how we can learn to learn on parameterized quantum circuits via classical recurrent neural networks. We apply this metalearning approach for efficient initialization of Quantum Approximate Optimization Algorithm for Sherrington-Kirkpatrick model and variational Quantum Eigensolver for the Hubbard model. Moreover, we introduce two different layerwise learning for quantum neural networks. In the first method we train layer-wise POVMs to perform variational quantum unsampling of unknown noisy quantum operations. In the second method, we are training varying subsets of the quantum circuit's parameters iteratively while increasing the circuit depth to have sufficient representation of classical or quantum data. In our approach the problem of vanishing gradients or barren plateaus of training landscape can be avoided to a large extent. We provide several applications of such quantum models for characterization of NISQ devices and classification of quantum data.   

Variational Quantum Unsampling on an Photonic Quantum Processor

Quantum algorithms for Noisy Intermediate-Scale Quantum (NISQ) processors have emerged as promising routes towards demonstrating practical advantage over classical machines. In these systems samples are typically drawn from probability distributions which — under plausible complexity-theoretic conjectures — cannot be efficiently generated classically. Rather than first define a physical system and then determine computational features of the output state, we ask the converse question: given direct access to the quantum state, what features of the generating system can we efficiently learn? Here, we introduce the Variational Quantum Unsampling (VQU) protocol, a nonlinear quantum neural network approach for verification and inference of near-term quantum circuits outputs. We experimentally demonstrate this protocol on a quantum photonic processor. Alongside quantum verification, our protocol has broad applications; including optimal quantum measurement and tomography, quantum sensing and imaging, and ansatz validation.   

Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm

We introduce a new class of generative quantum-neural-network-based models called Quantum Hamiltonian-Based Models (QHBMs). In doing so, we establish a paradigmatic approach for quantum-probabilistic hybrid variational learning, where we efficiently decompose the tasks of learning classical and quantum correlations in a way which maximizes the utility of both classical and quantum processors. In addition, we introduce the Variational Quantum Thermalizer (VQT) for generating the thermal state of a given Hamiltonian and target temperature, a task for which QHBMs are naturally well-suited. The VQT can be seen as a generalization of the Variational Quantum Eigensolver (VQE) to thermal states: we show that the VQT converges to the VQE in the zero temperature limit. We provide numerical results demonstrating the efficacy of these techniques in illustrative examples. We use QHBMs and the VQT on Heisenberg spin systems, we apply QHBMs to learn entanglement Hamiltonians and compression codes in simulated free Bosonic systems, and finally we use the VQT to prepare thermal Fermionic Gaussian states for quantum simulation.   

Variational Fast Forwarding for Quantum Simulation Beyond the Coherence Time

Trotterization-based, iterative approaches to quantum simulation are restricted to simulation times less than the coherence time of the quantum computer, which limits their utility in the near term. Here, we present a hybrid quantum-classical algorithm, called Variational Fast Forwarding (VFF), for decreasing the quantum circuit depth of quantum simulations. VFF seeks an approximate diagonalization of a short-time simulation to enable longer-time simulations using a constant number of gates. Our error analysis provides two results: (1) the simulation error of VFF scales at worst linearly in the fast-forwarded simulation time, and (2) our cost function's operational meaning as an upper bound on average-case simulation error provides a natural termination condition for VFF. We implement VFF for the Hubbard, Ising, and Heisenberg models on a simulator. Finally, we implement VFF on Rigetti's quantum computer to show simulation beyond the coherence time. See full paper at: https://arxiv.org/abs/1910.04292   

Stochastic Gradient Descent for Hybrid Quantum-Classical Optimization

Gradient-based methods for hybrid quantum-classical optimization typically rely on expectation values with respect to the outcome of parameterized quantum circuits. In this work, we investigate the fact that the estimation of these quantities on quantum hardware leads to a form of stochastic gradient descent. In many relevant cases estimating expectation values with k measurements results in optimization algorithms whose convergence properties can be rigorously understood, for any value of k≥1. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values and we show that in these cases k-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain doubly stochastic gradient descent. For all algorithms we prove convergence guarantees. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for significantly fewer circuit executions.   

Robust and efficient algorithms for high-dimensional black-box quantum optimization

Hybrid quantum-classical optimization using near-term quantum technology is an emerging direction for exploring quantum advantage in high-dimensional systems. However, precise characterization of all experimental parameters is often impractical and challenging. A viable approach is to use algorithms that rely only on black-box inference rather than analytical gradients. Here, we combine randomized perturbation gradient estimation with adaptive momentum gradient updates to create the AdamSPSA and AdamRSGF algorithms. We prove the asymptotic convergence of our algorithms in a convex setting, and we benchmark them against other gradient-based optimization algorithms on non-convex optimal control tasks. Our results show that these new algorithms accelerate the convergence rate, decrease the variance of loss trajectories, and efficiently tune up high-fidelity (above 99.9%) Hann-window single-qubit gates from trivial initial conditions with twenty variables.

Leng et al. arXiv:1910.03591   

A Hybrid Quantum-Classical Algorithm for Training Quantum Boltzmann Machines

A Boltzmann Machine is a Machine Learning algorithm based on a measure from statistical mechanics, i.e. the Boltzmann distribution. The respective concepts can also be used with quantum computers which leads to Quantum Boltzmann Machines. These have the potential to outperform classical algorithms in a variety of learning problems and to enable classically intractable tasks, such as discriminative learning with quantum data and generative modeling of classically inaccessible structures.
Several implementations have been suggested, but they all face practical difficulties. Firstly, it is often challenging to generate the quantum Gibbs state representing the Boltzmann distribution. Secondly, the training performance is usually impaired as the problem formulation requires that the system must be trained with an upper bound instead of the actual loss function.
In this work, we use variational quantum Gibbs state preparation to enable gate-based Quantum Boltzmann Machines which can be trained with the true loss function. Moreover, the algorithm relies on a hybrid quantum-classical optimization algorithm and is, thus, compatible with near-term quantum computers. The applicability of this approach is demonstrated on illustrative examples.   

Quantum classifier with tailored quantum kernel

Kernel methods have broad applications in machine learning. Recently, a link between quantum computing and kernel theory has been formally established, opening up opportunities for quantum enhancements in various machine learning methods. We present a distance-based quantum binary classifier whose kernel is based on the quantum state fidelity between training and test data. The quantum kernel can be tailored systematically with a quantum circuit to assign an exponent to the kernel and assign weights to training data. Our classifier calculates the weighted power sum of fidelities of quantum data in parallel via a swap-test circuit and two single-qubit measurements, requiring only a constant number of repetitions regardless of the number of data. Furthermore, our classifier is equivalent to measuring the expectation value of a Helstrom operator, from which the optimal quantum state discrimination can be derived. We demonstrate the proof-of-principle via classical simulations with a realistic noise model and experiments using an IBM quantum computer.   

Machine learning with solid-state NMR using quantum kernel

We employ so-called quantum kernel estimation to exploit complex quantum dynamics of solid-state NMR for machine learning. Kernel method is a popular branch in machine learning algorithms where only the inner products among feature vectors each representing an input datum are required to construct a prediction model. We propose to map an input to a feature space by input-dependent Hamiltonian evolution, and the kernel is estimated by the interference of the evolution. Simple machine learning tasks, namely one-dimensional fitting tasks and two-dimensional classification tasks, are performed as demonstrations. The performance of the trained model tends to increase with the longer evolution time, or equivalently, with a larger number of spins involved in the dynamics. This work can be regarded as one of the baselines for this emerging field.   

Hybrid quantum-classical algorithms for generative models

Quantum machine learning is a field that combines machine learning techniques and quantum computation together. It has the potential of enjoying impressive data analysis power while improving the time efficiency greatly. We propose a new hybrid quantum-classical circuit design for one major problem from machine learning aspect: generative models. We will discuss different ways to construct generative models using quantum algorithms. We will also apply this new design in example datasets and compare the complexity and the results. This work might help to find hidden patterns behind data and offer applications for near-term quantum devices.   

Quantum-tailored machine-learning architectures

Future quantum technologies are expected to be fundamentally challenging to characterize and calibrate. For this reason, a heuristic will most likely be required for this task, and machine learning provides an attractive framework for developing such a heuristic. To this end, we introduce a machine-learning architecture for inferring the dynamics of a quantum device from time-series measurement data. Our architecture is recurrent in nature and leverages quantum-mechanical structure in its design to interpret measurement data from complex quantum devices more efficiently. We investigate how the architectural structure influences the way we learn from data generated by quantum experiments and address applications of our techniques to the calibration and characterization of superconducting quantum devices.   

Scalable Quantum State Tomography with Attention Network

The problem of many-body wavefunction reconstruction, which suffers from exponential scaling in system size as well as noisy state preparation and measurement, remains a major obstacle to the study of intermediate-scale quantum systems. Recent works found success by recasting the problem of reconstruction to learning the probability distribution of quantum state measurement vectors, a natural task for generative neural network models. Networks based on the attention mechanism, designed to learn long-range correlations in natural language sentences, appear especially well-suited to the task of learning highly entangled wavefunctions. In this work, we demonstrate that an attention mechanism-based generative network, based on the model proposed in ``Attention is all you need’’ by Vishwani et al (2017), can outperform previous neural network based approaches to quantum state tomography. Specifically, in addition to working with state-of-the-art system sizes, the attention mechanism is able to accommodate noise by directly reconstructing the density matrix of mixed states. This work represents an important step forward in the applicability of machine learning to the study of noisy intermediate-scale quantum systems.   

Implementing perceptron models with qubits

We propose a method for learning a quantum probabilistic model of a perceptron. By considering a cross entropy between two density matrices we can learn a model that takes noisy output labels into account while learning. Although some work has been done that aims to utilize the curious properties of quantum systems to build a quantum perceptron, these proposals rely on the ad hoc introduction of a classical cost functionfor the optimization procedure. We demonstrate the usage of a quantum probabilistic model by considering a quantum equivalent of the classical log-likelihood, which allows for both a quantum model and training procedure. We show that this allows us to better capture noisyness in data compared to a classical perceptron. By considering entangled qubits we can learn nonlinear separation boundaries, such as XOR.