Algorithms 1

Abstract: We present an end-to-end quantum algorithm with provable performance guarantees for simulating a large class of classical nonlinear dynamical systems. The considered dynamical systems are described by stochastic dissipative differential equations with a quadratic nonlinearity satisfying certain sparsity and divergence-free conditions. Our algorithm approximates the expected value of any sparse low-degree polynomial evaluated at the solution of the classical system. The runtime scales poly-logarithmically with the system size and polynomially with the evolution time, inverse error tolerance, and parameters quantifying sparsity, dissipation, and nonlinearity strength. The considered simulation problem is shown to be BQP-complete, providing a strong evidence for a quantum advantage. We benchmark the quantum algorithm via numerical experiments by simulating a vortex flow in the 2D Navier Stokes equation.

Abstract: We provide several quantum algorithms for continuous optimization that do not require gradient estimation. Instead, we encode the optimization problem into the dynamics of a physical system and coherently simulate the time evolution. Our first two algorithms can find local optima of a differentiable function $f: \mathbb{R}^N \rightarrow \mathbb{R}$ by simulating either classical or quantum dynamics with friction via a time-dependent Hamiltonian. We show that for the benchmark problem of optimizing a locally quadratic objective function, these methods require a total of $O(N^2\kappa^2/h_x^2\epsilon)$ queries to a phase oracle to find an $\epsilon$-approximate local optimum, where $\kappa$ is the condition number of the Hessian matrix and $h_x$ is the discretization spacing. In contrast, we show that methods based on gradient descent require $O(N^{3/2}(1/\epsilon)^{\kappa \log(3)/4})$ queries. This corresponds to an exponential separation between the query upper bounds for the benchmark problem. Our third algorithm can find the global optimum of $f$ by preparing a classical low-temperature thermal state via simulation of the classical Liouvillian operator associated with the Nosé Hamiltonian. We use results from the quantum thermodynamics literature to bound the thermalization time for the discrete system. Additionally, we analyze barren plateau effects that commonly plague quantum optimization algorithms and observe that our approach is vastly less sensitive to this problem than standard gradient-based optimization. Our results suggests that these dynamical optimization approaches may be far more scalable for future quantum machine learning, optimization and variational experiments than was widely believed.

Abstract: We introduce new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, broadening the scope of quantum advantage beyond traditional tunneling-based explanations. By establishing a rigorous correspondence between the spectral properties of Schrödinger operators and classical Langevin diffusion, we identify regimes where a real-space adiabatic algorithm (RsAA) can significantly outperform classical stochastic gradient-based methods. Leveraging this connection and novel non-asymptotic versions of well-known semi-classical results, we provide polynomial (in the dimension $d$) runtime bounds for RsAA on rotated block-separable functions, and offer theoretical evidence that black-box classical algorithms require exponential time for optimization in these settings, thereby generalizing and formalizing prior work by Leng et al (arXiv:2311.00811). While we can design certain specialized, structure-aware classical algorithms that can efficiently optimize these functions, we further show that the quantum ground state exhibits remarkable robustness to nontrivial perturbations. Leveraging recent advances in the study of the hypercontractivity of Schr\"{o}dinger operators, we construct new families of nonconvex functions for which RsAA achieves polynomial-time optimization, whereas both off-the-shelf and structure-aware classical algorithms incur exponential computational costs. These results provide new insight into quantum-classical separations in nonconvex optimization and highlight tractable and general pathways to advantage in this setting.