Modern deep learning models are extraordinarily expensive to run and train, raising a natural question: can quantum computers offer an exponential advantage for inference and learning for specific classes of deep neural networks? Indeed, no-go theorems prevent the direct implementation of nonlinear transformations on quantum amplitudes, so naively quantizing a feedforward network cannot yield an exponential speedup. In this talk, I’ll present a route around this obstacle. Building on the Neural ODE perspective, we view a deep residual network as a continuous-time dynamical system, turning the forward pass into an ordinary differential equation simulation problem.

With this formulation in hand, I’ll describe quantum algorithms that (i) run the forward pass of an exponentially wide continuous-time network at a cost only polylogarithmic in its width, using quantum differential equation solvers as subroutines, and (ii) sample classification labels directly by preparing a softmax-weighted quantum state. We show the underlying simulation problem is BQP-complete, giving strong evidence that sampling from these networks is classically hard. Beyond inference, I’ll show how to learn such networks from data without a variational measure-and-update loop.

By formulating training as regression problems with closed-form solutions, we solve them end-to-end with quantum linear algebra techniques and extract parameters via shadow tomography. Together, these results connect deep learning, nonlinear dynamics, and quantum algorithms, and suggest a setting where exponential quantum advantage in machine learning may be provably attainable. Based on joint work with Nathan Wiebe.