Session T37: Quantum Machine Learning II

Theory of overparametrization in quantum neural networks

The prospect of achieving quantum advantage with Quantum Neural Networks (QNNs) is exciting. Understanding how QNN properties affect the loss landscape is crucial to the design of scalable architectures. Here, we rigorously analyze the overparametrization phenomenon in QNNs with periodic structure. We define overparametrization as the regime where the QNN has more than a critical number of parameters Mc that allows it to explore all relevant directions in state space. Our main results show that the dimension of the Lie algebra obtained from the generators of the QNN is an upper bound for Mc, and for the maximal rank that the quantum Fisher information and Hessian matrices can reach. Underparametrized QNNs have spurious local minima that start disappearing when M> Mc. Thus, the overparametrization onset corresponds to a computational phase transition where the trainability is greatly improved by a more favorable landscape. We connect the notion of overparametrization to that of capacity, so that when a QNN is overparametrized its capacity achieves its maximum possible value. We run numerical simulations for eigensolver, compilation, and autoencoding applications to showcase the phase transition. Our results also apply to variational quantum algorithms and quantum optimal control.   

Experimentally Realizable Continuous-variable Quantum Neural Networks

We build a continuous variable (CV) hybrid quantum-classical neural network protocol and show how CV supervised learning with hybrid networks can be used for fraud detection. Previous work on CV neural networks protocols for fraud detection required implementation of non-Gaussian operators which are hard to realize experimentally. Our protocol uses Gaussian gates only with the addition of ancillary qumodes. We achieve non-linearity, an essential feature of neural networks, through measurements on ancillary qumodes. Our gates can be implemented with squeezers and beam splitters; hence our protocol can be realized experimentally with current photonic quantum hardware.   

Classifying Many-Body Wavefunctions by Means of a Quantum Convolutional Neural Network

We aim to use Quantum Machine Learning by attempting to classify quantum mechanical data. The goal is to train a Quantum Convolutional Neural Network (QCNN) to detect the zero temperature quantum phases, in particular the quantum critical point, of the Transverse Field Ising model (TFIM), an archetypal quantum many-body system. This approach allows us to investigate whether a QCNN can be used as a classifier for quantum phase transitions. The TFIM is solved by the Variational Quantum Eigensolver (VQE). The VQE is a quantum algorithm that uses the variational method to find close approximations to the ground state of the corresponding system. The outcome of the VQE is a set of datapoints, represented as quantum circuits, which can be used as input data for the QCNN with the intent of correctly labeling the phase of the system. The promising results of this study allowed us to display the effectiveness of the QCNN as a quantum classifier. We find that the accuracy of our model proves to be consistently high, demonstrating the future potential that Quantum Machine Learning offers as a quantum classifier. Larger, more complex systems, such as the Periodic Anderson model and the Hubbard model, will become the next targets of interest.   

Diagnosing barren plateaus with tools from quantum optimal control

Variational Quantum Algorithms (VQAs) have received considerable attention due to their potential for achieving near-term quantum advantage. However, more work is needed to understand their scalability. One known scaling result for VQAs is barren plateaus, where certain circumstances lead to exponentially vanishing gradients. It is common folklore that problem-inspired ansatzes avoid barren plateaus, but in fact, very little is known about their gradient scaling. In this work we employ tools from quantum optimal control to develop a framework that can diagnose the presence or absence of barren plateaus for problem-inspired ansatzes. Such ansatzes include the Quantum Alternating Operator Ansatz (QAOA), the Hamiltonian Variational Ansatz (HVA), and others. With our framework, we prove that avoiding barren plateaus for these ansatzes is not always guaranteed. Specifically, we show that the gradient scaling of the VQA depends on the controllability of the system, and hence can be diagnosed trough the dynamical Lie algebra g obtained from the generators of the ansatz. We analyze the existence of barren plateaus in QAOA and HVA ansatzes, and we highlight the role of the input state, as different initial states can lead to the presence or absence of barren plateaus. Taken together, our results provide a framework for trainability-aware ansatz design strategies that do not come at the cost of extra quantum resources. Moreover, we prove no-go results for obtaining ground states with variational ansatzes for controllable system such as spin glasses. We finally provide evidence that barren plateaus can be linked to dimension of g.   

Avoiding The Barren Plateau in the Variational Quantum Circuits With Bayesian Learning

 In the era of Noisy Intermediate scale quantum (NISQ) hardware, due to a limited number of qubits and hardware noise, hybrid classical-quantum algorithms are promising strategies for practical applications. Hybrid algorithms are based on variational quantum circuits (VQC), which their flexibility to tune the gates' parameters provide the compelling properties to be robust and adaptive to hardware limitations while giving access to solve a class of problems. However, training the VQCs over the parameters landscape requires classical optimization, which can be a classically hard task. The Effect of a barren plateau and localization of local minima of Hamiltonian, far from the global minima make the regular optimization, are two of the challenges that make regular gradient methods ineffective as the number of qubits increases.  In this work, we introduce the Fast-Slow algorithm based on the Bayesian Learning method to mitigate the mentioned issues.   

A novel approach to time series modeling using quantum generative netoworks

Modeling time series in a quantum setting has been proposed using various learning techniques including Quantum Generative Adversarial Network and Quantum Boltzmann Machines. In this work, we introduce a new paradigm of quantum generative algorithm that is inherently quantum native. We model a stochastic time series by a quantum process with the goal of learning the underlying Hamiltonian H, which governs this process. We explore how this technique generalizes to generate new series of the learned processes as well as learn/generate correlations between multivariate time series.   

Barren plateaus preclude learning scramblers

Scrambling, the process by which quantum information is rapidly spread through many-body quantum systems, has proven central not only to understanding quantum chaos but also to the study of the dynamics of quantum information, the black hole information paradox, random circuits and entropic uncertainty relations. However, the complexity of strongly-interacting many-body quantum systems makes scrambling rather challenging to study analytically. Recently, quantum machine learning (QML) has emerged as an exciting new paradigm for the study of complex physical processes. It is therefore natural to ask whether QML could be used to study scrambling. 

In this talk we will present a no-go theorem for the use of QML to study quantum scrambling. Namely, we show that any QML approach used to learn the unitary dynamics implemented by a typical scrambler will exhibit a barren plateau and thus be untrainable in the absence of further prior knowledge. Crucially, in contrast to previously established barren plateau phenomena, which are a consequence of the ansatz structure and parameter initialization strategy, our barren plateaus holds for any choice of ansatz and any initialization of parameters. Thus, previously proposed strategies for avoiding barren plateaus do not work here. 

More generally, given the close connection between scrambling and randomness, our no-go theorem also applies to learning random and pseudo-random unitaries. Consequently, our result implies that to efficiently learn an unknown unitary process using QML, prior information about that process is required. Thus, our result provides a fundamental limit on the domain of applicability of QML.   

Storage capacity and learning capability of quantum neural networks

We study the storage capacity of quantum neural networks (QNNs), described by completely positive trace preserving (CPTP) maps acting on a $N$-dimensional Hilbert space. We demonstrate that attractor QNNs can store in a non-trivial manner up to $N$ linearly independent pure states. For $n$ qubits, QNNs can reach an exponential storage capacity, $\mathcal O(2^{n})$, clearly outperforming standard classical neural networks whose storage capacity scales linearly with the number of neurons $n$. We estimate, employing the Gardner program, the relative volume of CPTP maps with $M\leq N$ stationary states and show that this volume decreases exponentially with $M$ and shrinks to zero for $M\geq N+1$. We generalize our results to QNNs storing mixed states as well as input-output relations for feed-forward QNNs. Our approach opens the path to relate storage properties of QNNs to the quantum features of the input-output states.  This work is dedicated to the memory of Peter Wittek.   

Realization of a quantum neural network by repeat-until-success circuits in a superconducting quantum processor

We present the experimental realization of a quantum neural network capable of non-linear classification. By implementing control-flow feedback in a superconducting quantum processor, we synthesize non-linear activation functions using repeat-until-success circuits, demonstrating the elementary building blocks for quantum machine learning. In particular, we realize a quantum circuit that reproduces a variety of classical feed-forward neural network constructions and can learn from superpositions of training data.   

Realizing Quantum Convolutional Neural Networks on a Superconducting Quantum Processor to Recognize Quantum Phases

Quantum computing crucially relies on the ability to efficiently characterize the quantum states output by quantum hardware. Conventional methods which probe these states through direct measurements and classically computed correlations become computationally expensive when increasing the system size. Quantum neural networks tailored to recognize specific features of quantum states by combining unitary operations, measurements and feedforward promise to require fewer measurements and to tolerate errors. Here, we realize a quantum convolutional neural network (QCNN) on a 7-qubit superconducting quantum processor to identify symmetry-protected topological (SPT) phases of a spin model characterized by a non-zero string order parameter. We benchmark the performance of the QCNN based on approximate ground states of a family of cluster-Ising Hamiltonians which we prepare using a hardware-efficient, low-depth state preparation circuit. We find that, despite being composed of finite-fidelity gates itself, the QCNN recognizes the topological phase with higher fidelity than direct measurements of the string order parameter for the prepared states.   

Shortening Quantum Convolutional Neural Networks to Constant Depth

The quantum convolutional neural network (QCNN) is a quantum circuit that detects symmetry protected topological (SPT) phases, with designs drawing from renormalisation theory. In this talk I will discuss a special class of these circuits that are equivalent to constant depth circuits, local measurements, and classical post-processing. Although the quantum circuit is constant depth on N-qubits, we still observe a provable exponential (in N) time speed up compared to local Pauli measurement and post-processing of the input state. Despite the constant depth, an exponential speed up arises due to inputing non-trivial quantum states. The speed-up therefore bounded by the natural classical representation complexity of the input quantum state.

Our circuit also improves on earlier approaches for a Z2 x Z2 SPT phase detection circuit, both in time complexity and signal fidelity. This improvement arises from a quantum error correction analysis of the circuit, and works by providing protection against error channels common to NISQ devices.

This talk compliments the experimental talk by J. Herrmann (XXXX) and provides the underlying theory.   

Evaluating Generalization in Classical and Quantum Generative Machine Learning Models: Part I

In many applications, the real advantage behind machine learning is unlocking the model's ability to generalize. In supervised ML, this power is given by the model's capacity to correctly classify unseen data. In contrast, generalization within generative unsupervised ML determines the model's capability to generate valuable data beyond the training set. In the latter, this property remains highly unexplored, both in the classical and quantum ML communities, given the challenge to propose robust metrics for evaluating the quality of newly generated samples. In this work, we present a novel approach for comparing and contrasting the generalization capabilities of generative models.

In part I of this two-part presentation, we introduce new benchmarks and metrics that allow us to conduct a 3D evaluation of a model's ability to generalize efficiently. Additionally, by using data sets where we can measure the quality of the generated samples, we provide a metric for quantitatively assessing the practical value of generalization.

In part II, we use the problem agnostic metrics to study a specific data set from a financial application. Here, we evaluate and compare the generalization power of three types of generative models (classical, quantum-inspired, and hybrid quantum-classical).   

Evaluating Generalization in Classical and Quantum Generative Machine Learning Models: Part II

In many applications, the real advantage behind machine learning is unlocking the model's ability to generalize. In supervised ML, this power is given by the model's capacity to correctly classify unseen data. In contrast, generalization within generative unsupervised ML determines the model's capability to generate valuable data beyond the training set. In the latter, this property remains highly unexplored, both in the classical and quantum ML communities, given the challenge to propose robust metrics for evaluating the quality of newly generated samples. In this work, we present a novel approach for comparing and contrasting the generalization capabilities of generative models.

In part I of this two-part presentation, we introduce new benchmarks and metrics that allow us to conduct a 3D evaluation of a model's ability to generalize efficiently. Additionally, by using data sets where we can measure the quality of the generated samples, we provide a metric for quantitatively assessing the practical value of generalization.

In part II, we use the problem agnostic metrics to study a specific data set from a financial application. Here, we evaluate and compare the generalization power of three types of generative models (classical, quantum-inspired, and hybrid quantum-classical).