A primary application of the paradigm combining generative AI with formal verification is the theory of quantum algorithms, with the Quantum Approximate Optimization Algorithm (QAOA) as our flagship case study. QAOA is among the central algorithms of near-term quantum computing - structured enough to admit deep analytical study, yet rich enough to expose central open questions: how approximation quality scales with circuit depth, which parameter regimes admit provable guarantees, and where efficient classical simulation gives way to genuinely quantum behavior. These questions sit at the intersection of optimization, probability, quantum many-body systems, and complexity theory.

As a first demonstration, we resolved a question open for over a decade - the Farhi-Goldstone-Gutmann conjecture that depth-p QAOA on the ring of disagrees attains an approximation ratio of exactly (2p+1)/(2p+2). The model discovered the decisive argument and Lean certified it end-to-end, uncovering a hidden dynamical symmetry that reduces the exponentially large evolution to a single effective qubit per momentum mode and recognizing it as a Quantum Signal Processing sequence. The longer-term ambition is a theorem-backed understanding of quantum algorithmic performance, including the regimes in which genuine quantum advantage may emerge.

In a recent work, carried out in collaboration with Wei Bu and Ziming Liu, we studied the training dynamics of artificial neural networks (ANNs). Focusing on an autoencoder architecture, we analyzed how the data distribution in the bottleneck layers evolves during training. Remarkably, we found that the training dynamics is accurately captured by the Callan–Symanzik equation, familiar from quantum field theory (QFT) as the equation governing renormalization group (RG) flows.

In close analogy with QFT, the network parameters are initialized from a random distribution, and their probability distribution evolves during training. This process can be viewed as the analogue of an RG flow from a free theory to an interacting quantum field theory.

We have recently reported the discovery of a novel type of critical phenomenon emerging in the dynamics of spinning black holes. The scattering dynamics is governed by the Teukolsky equation, whose solutions are organized according to the phase diagram depicted here. We identified a new critical phase, analogous to the quantum critical region in quantum many-body systems, in which the physics is governed exclusively by a set of critical exponents.

These results imply that spinning Kerr black holes fall into a universality class, leading to a strong and unique theoretical prediction that could potentially be tested using current gravitational-wave detectors (LIGO-Virgo-KAGRA), as well as future observatories such as LISA and the Einstein Telescope.

In an earlier work, carried out in collaboration with Alfredo Guevara, we showed that the dynamics of the self-dual black hole can be mapped onto those of the hydrogen atom. The broader paradigm emerging from our research program suggests that the self-dual black hole can be viewed as a fundamental configuration, while the real-world spinning Kerr black hole behaves as a composite many-body system, exhibiting properties reminiscent of quantum matter.

This paradigm fits naturally within the duality program that we developed in collaboration with Prof. Shing-Tung Yau, in which we draw on tools and techniques from quantum field theory to uncover the fundamental principles of quantum gravity.