Alhussein Fawzi

Fabian Finger, Frederic Rapp, Pranav Kalidindi et al.

Designing quantum algorithms is a complex and counterintuitive task, making it an ideal candidate for AI-driven algorithm discovery. To this end, we employ the Hive, an AI platform for program synthesis, which utilises large language models to drive a highly distributed evolutionary process for discovering new algorithms. We focus on the ground state problem in quantum chemistry, and discover efficient quantum heuristic algorithms that solve it for molecules LiH, H2O, and F2 while exhibiting significant reductions in quantum resources relative to state-of-the-art near-term quantum algorithms. Further, we perform an interpretability study on the discovered algorithms and identify the key functions responsible for the efficiency gains. Finally, we benchmark the Hive-discovered circuits on the Quantinuum System Model H2 quantum computer and identify minimum system requirements for chemical precision. We envision that this novel approach to quantum algorithm discovery applies to other domains beyond chemistry, as well as to designing quantum algorithms for fault-tolerant quantum computers.

Francisco J. R. Ruiz, Tuomas Laakkonen, Johannes Bausch et al.

A key challenge in realizing fault-tolerant quantum computers is circuit optimization. Focusing on the most expensive gates in fault-tolerant quantum computation (namely, the T gates), we address the problem of T-count optimization, i.e., minimizing the number of T gates that are needed to implement a given circuit. To achieve this, we develop AlphaTensor-Quantum, a method based on deep reinforcement learning that exploits the relationship between optimizing T-count and tensor decomposition. Unlike existing methods for T-count optimization, AlphaTensor-Quantum can incorporate domain-specific knowledge about quantum computation and leverage gadgets, which significantly reduces the T-count of the optimized circuits. AlphaTensor-Quantum outperforms the existing methods for T-count optimization on a set of arithmetic benchmarks (even when compared without making use of gadgets). Remarkably, it discovers an efficient algorithm akin to Karatsuba's method for multiplication in finite fields. AlphaTensor-Quantum also finds the best human-designed solutions for relevant arithmetic computations used in Shor's algorithm and for quantum chemistry simulation, thus demonstrating it can save hundreds of hours of research by optimizing relevant quantum circuits in a fully automated way.

Alhussein Fawzi, Mateusz Malinowski, Hamza Fawzi et al.

Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of semi-algebraic proof systems that manipulate polynomial inequalities via elementary inference rules that infer new inequalities from the premises. These proof systems are known to be very powerful, but searching for proofs remains a major difficulty. In this work, we introduce a machine learning based method to search for a dynamic proof within these proof systems. We propose a deep reinforcement learning framework that learns an embedding of the polynomials and guides the choice of inference rules, taking the inherent symmetries of the problem as an inductive bias. We compare our approach with powerful and widely-studied linear programming hierarchies based on static proof systems, and show that our method reduces the size of the linear program by several orders of magnitude while also improving performance. These results hence pave the way towards augmenting powerful and well-studied semi-algebraic proof systems with machine learning guiding strategies for enhancing the expressivity of such proof systems.

Alhussein Fawzi, Omar Fawzi, Pascal Frossard

The goal of this paper is to analyze an intriguing phenomenon recently discovered in deep networks, namely their instability to adversarial perturbations (Szegedy et. al., 2014). We provide a theoretical framework for analyzing the robustness of classifiers to adversarial perturbations, and show fundamental upper bounds on the robustness of classifiers. Specifically, we establish a general upper bound on the robustness of classifiers to adversarial perturbations, and then illustrate the obtained upper bound on the families of linear and quadratic classifiers. In both cases, our upper bound depends on a distinguishability measure that captures the notion of difficulty of the classification task. Our results for both classes imply that in tasks involving small distinguishability, no classifier in the considered set will be robust to adversarial perturbations, even if a good accuracy is achieved. Our theoretical framework moreover suggests that the phenomenon of adversarial instability is due to the low flexibility of classifiers, compared to the difficulty of the classification task (captured by the distinguishability). Moreover, we show the existence of a clear distinction between the robustness of a classifier to random noise and its robustness to adversarial perturbations. Specifically, the former is shown to be larger than the latter by a factor that is proportional to \sqrt{d} (with d being the signal dimension) for linear classifiers. This result gives a theoretical explanation for the discrepancy between the two robustness properties in high dimensional problems, which was empirically observed in the context of neural networks. To the best of our knowledge, our results provide the first theoretical work that addresses the phenomenon of adversarial instability recently observed for deep networks. Our analysis is complemented by experimental results on controlled and real-world data.