Classical neural networks with random initialization famously behave as Gaussian processes in the limit of many neurons, which allows one to completely characterize their training and generalization behavior. No such general understanding exists for quantum neural networks (QNNs), which -- outside of certain special cases -- are known to not behave as Gaussian processes when randomly initialized. We here prove that QNNs and their first two derivatives instead generally form what we call "Wishart processes," where certain algebraic properties of the network determine the hyperparameters of the process. This Wishart process description allows us to, for the first time: give necessary and sufficient conditions for a QNN architecture to have a Gaussian process limit; calculate the full gradient distribution, generalizing previously known barren plateau results; and calculate the local minima distribution of algebraically constrained QNNs. Our unified framework suggests a certain simple operational definition for the "trainability" of a given QNN model using a newly introduced, experimentally accessible quantity we call the "degrees of freedom" of the network architecture.
Joshua Viszlai, Teague Tomesh, Pranav Gokhale et al.
Recent work has proposed and explored using coreset techniques for quantum algorithms that operate on classical data sets to accelerate the applicability of these algorithms on near-term quantum devices. We apply these ideas to Quantum Boltzmann Machines (QBM) where gradient-based steps which require Gibbs state sampling are the main computational bottleneck during training. By using a coreset in place of the full data set, we try to minimize the number of steps needed and accelerate the overall training time. In a regime where computational time on quantum computers is a precious resource, we propose this might lead to substantial practical savings. We evaluate this approach on 6x6 binary images from an augmented bars and stripes data set using a QBM with 36 visible units and 8 hidden units. Using an Inception score inspired metric, we compare QBM training times with and without using coresets.
Hilbert space fragmentation is a phenomenon in which the Hilbert space of a quantum system is dynamically decoupled into exponentially many Krylov subspaces. We can define the Schur transform as a unitary operation mapping some set of preferred bases of these Krylov subspaces to computational basis states labeling them. We prove that this transformation can be efficiently learned via gradient descent from a set of training data using quantum neural networks, provided that the fragmentation is sufficiently strong such that the summed dimension of the unique Krylov subspaces is polynomial in the system size. To demonstrate this, we analyze the loss landscapes of random quantum neural networks constructed out of Hilbert space fragmented systems. We prove that in this setting, it is possible to eliminate barren plateaus and poor local minima, suggesting efficient trainability when using gradient descent. Furthermore, as the algebra defining the fragmentation is not known a priori and not guaranteed to have sparse algebra elements, to the best of our knowledge there are no existing efficient classical algorithms generally capable of simulating expectation values in these networks. Our setting thus provides a rare example of a physically motivated quantum learning task with no known dequantization.
M. Cerezo, Martin Larocca, Diego García-Martín et al.
A large amount of effort has recently been put into understanding the barren plateau phenomenon. In this perspective article, we face the increasingly loud elephant in the room and ask a question that has been hinted at by many but not explicitly addressed: Can the structure that allows one to avoid barren plateaus also be leveraged to efficiently simulate the loss classically? We collect evidence-on a case-by-case basis-that many commonly used models whose loss landscapes avoid barren plateaus can also admit classical simulation, provided that one can collect some classical data from quantum devices during an initial data acquisition phase. This follows from the observation that barren plateaus result from a curse of dimensionality, and that current approaches for solving them end up encoding the problem into some small, classically simulable, subspaces. Thus, while stressing that quantum computers can be essential for collecting data, our analysis sheds doubt on the information processing capabilities of many parametrized quantum circuits with provably barren plateau-free landscapes. We end by discussing the (many) caveats in our arguments including the limitations of average case arguments, the role of smart initializations, models that fall outside our assumptions, the potential for provably superpolynomial advantages and the possibility that, once larger devices become available, parametrized quantum circuits could heuristically outperform our analytic expectations.
Recent theoretical results in quantum machine learning have demonstrated a general trade-off between the expressive power of quantum neural networks (QNNs) and their trainability; as a corollary of these results, practical exponential separations in expressive power over classical machine learning models are believed to be infeasible as such QNNs take a time to train that is exponential in the model size. We here circumvent these negative results by constructing a hierarchy of efficiently trainable QNNs that exhibit unconditionally provable, polynomial memory separations of arbitrary constant degree over classical neural networks -- including state-of-the-art models, such as Transformers -- in performing a classical sequence modeling task. This construction is also computationally efficient, as each unit cell of the introduced class of QNNs only has constant gate complexity. We show that contextuality -- informally, a quantitative notion of semantic ambiguity -- is the source of the expressivity separation, suggesting that other learning tasks with this property may be a natural setting for the use of quantum learning algorithms.
Generative modeling using samples drawn from the probability distribution constitutes a powerful approach for unsupervised machine learning. Quantum mechanical systems can produce probability distributions that exhibit quantum correlations which are difficult to capture using classical models. We show theoretically that such quantum correlations provide a powerful resource for generative modeling. In particular, we provide an unconditional proof of separation in expressive power between a class of widely-used generative models, known as Bayesian networks, and its minimal quantum extension. We show that this expressivity advantage is associated with quantum nonlocality and quantum contextuality. Furthermore, we numerically test this separation on standard machine learning data sets and show that it holds for practical problems. The possibility of quantum advantage demonstrated in this work not only sheds light on the design of useful quantum machine learning protocols but also provides inspiration to draw on ideas from quantum foundations to improve purely classical algorithms.
Teague Tomesh, Pranav Gokhale, Eric R. Anschuetz et al.
Many quantum algorithms for machine learning require access to classical data in superposition. However, for many natural data sets and algorithms, the overhead required to load the data set in superposition can erase any potential quantum speedup over classical algorithms. Recent work by Harrow introduces a new paradigm in hybrid quantum-classical computing to address this issue, relying on coresets to minimize the data loading overhead of quantum algorithms. We investigate using this paradigm to perform $k$-means clustering on near-term quantum computers, by casting it as a QAOA optimization instance over a small coreset. We compare the performance of this approach to classical $k$-means clustering both numerically and experimentally on IBM Q hardware. We are able to find data sets where coresets work well relative to random sampling and where QAOA could potentially outperform standard $k$-means on a coreset. However, finding data sets where both coresets and QAOA work well--which is necessary for a quantum advantage over $k$-means on the entire data set--appears to be challenging.
We introduce a new hybrid quantum-classical adversarial machine learning architecture called a quantum-classical associative adversarial network (QAAN). This architecture consists of a classical generative adversarial network with a small auxiliary quantum Boltzmann machine that is simultaneously trained on an intermediate layer of the discriminator of the generative network. We numerically study the performance of QAANs compared to their classical counterparts on the MNIST and CIFAR-10 data sets, and show that QAANs attain a higher quality of learning when evaluated using the Inception score and the Fréchet Inception distance. As the QAAN architecture only relies on sampling simple local observables of a small quantum Boltzmann machine, this model is particularly amenable for implementation on the current and next generations of quantum devices.