We present the first analytical study of ECD, focusing on the one-dimensional setting for this first installment. We formalize a stochastic ECD dynamics (sECD) with energy-preserving noise, as well as a quantum analog of the ECD Hamiltonian (qECD), providing the foundation for a quantum algorithm through Hamiltonian simulation in a tractable model where the barrier-crossing mechanism can be computed explicitly. For one-dimensional double-well objectives in the under-guessing regime, we compute the expected dynamical hitting times from a local minimum to the global minimum. We prove that both sECD and qECD exhibit exponential improvements in continuous hitting time relative to their respective gradient-based baselines, stochastic gradient descent (SGD) and quantum tunneling walk (QTW). For objectives with tall barriers, qECD admits a further hitting time improvement over sECD. Mechanistically, ECD sidesteps the exponential cost associated with rare-escape events of SGD from local minima by moving from dissipative to energy-conserving dynamics.
We present the first analytical study of ECD, focusing on the one-dimensional setting for this first installment. We formalize a stochastic ECD dynamics (sECD) with energy-preserving noise, as well as a quantum analog of the ECD Hamiltonian (qECD), providing the foundation for a quantum algorithm through Hamiltonian simulation in a tractable model where the barrier-crossing mechanism can be computed explicitly. For one-dimensional double-well objectives in the under-guessing regime, we compute the expected dynamical hitting times from a local minimum to the global minimum. We prove that both sECD and qECD exhibit exponential improvements in continuous hitting time relative to their respective gradient-based baselines, stochastic gradient descent (SGD) and quantum tunneling walk (QTW). For objectives with tall barriers, qECD admits a further hitting time improvement over sECD. Mechanistically, ECD sidesteps the exponential cost associated with rare-escape events of SGD from local minima by moving from dissipative to energy-conserving dynamics.
Classical algorithms are often not effective for solving nonconvex optimization problems where local minima are separated by high barriers. In this paper, we explore possible quantum speedups for nonconvex optimization by leveraging the global effect of quantum tunneling. Specifically, we introduce a quantum algorithm termed the quantum tunneling walk (QTW) and apply it to nonconvex problems where local minima are approximately global minima. We show that QTW achieves quantum speedup over classical stochastic gradient descents (SGD) when the barriers between different local minima are high but thin and the minima are flat. Based on this observation, we construct a specific double-well landscape, where classical algorithms cannot efficiently hit one target well knowing the other well but QTW can when given proper initial states near the known well. Finally, we corroborate our findings with numerical experiments.
We use an instanton-based analysis to show that Path Integral Stochastic Gradient Descent (PISGD), a classical optimization algorithm instantiated from path integral Monte Carlo (PIMC), can inherit the quantum tunneling mechanism. Quantum tunneling is the phenomenon in which a quantum particle passes through a potential barrier that a classical trajectory cannot cross in the same way. This mechanism can contribute to the performance of quantum optimization algorithms, since a common bottleneck in optimization is that the solution becomes trapped in a local minimum, separated from better solutions by a potential barrier. Classical noise-driven optimizers, such as stochastic gradient descent (SGD), have escape times that scale exponentially with the potential barrier height. For quantum tunneling, the escape time is instead exponential in the bounce action, which equals twice the Agmon distance between the two basins. When the barrier is tall but thin, quantum tunneling can allow escaping local minima faster than such classical methods. Remarkably, this scaling is not exclusive to algorithms with genuine quantum dynamics. For PISGD applied to classical nonconvex global optimization, we adapt Kramers' escape-rate theory to path space and obtain the same exponential scaling as quantum tunneling, in terms of the bounce action rather than the barrier height. We confirm this scaling numerically by implementing PISGD for both separable and nonseparable potentials. That said, PISGD retains only incoherent tunneling, since coherence among multiple tunneling paths is absent. Our analysis therefore does not rule out a potential quantum advantage for quantum-tunneling-based algorithms over classical ones in general.
Stochastic Gradient Descent (SGD) and its variants underpin modern machine learning by enabling efficient optimization of large-scale models. However, their local search nature limits exploration in complex landscapes. In this paper, we introduce Stochastic Quantum Hamiltonian Descent (SQHD), a quantum optimization algorithm that integrates the computational efficiency of stochastic gradient methods with the global exploration power of quantum dynamics. We propose a Lindbladian dynamics as the quantum analogue of continuous-time SGD. We further propose a discrete-time gate-based algorithm that approximates these dynamics while avoiding direct Lindbladian simulation, enabling practical implementation on near-term quantum devices. We rigorously prove the convergence of SQHD for convex and smooth objectives. Numerical experiments demonstrate that SQHD also exhibits advantages in non-convex optimization. All these results highlight its potential for quantum-enhanced machine learning.
This work investigates the possibility of quantum speedups for continuous optimization through quantum Hamiltonian simulation. We establish the first rigorous query complexity bounds for unconstrained convex optimization via digital quantum annealing, based on the non-adiabatic Quantum Hamiltonian Descent (QHD) framework. In the process, we derive rigorous resource estimates for digital quantum simulation of Schrödinger operators depending only on input simulation parameters, given black-box evaluation access to separable, Lipschitz continuous potential b(t) f(x). We apply these simulation bounds to assess the complexity of high-dimensional convex optimization. Our annealing schedule achieves arbitrarily fast convergence rates in the evolution time, with computational time determined solely by the cost of discretization. We show that a G-Lipschitz convex function can be optimized to an error of ε with 𝒪̃(d^{1.5} G² R²/ε²) queries, given a starting point that is Euclidean distance R from optimal. Under reasonable assumptions such as the complexity of simulating Schrödinger operators, we show that Ω̃(d/ε²) queries are necessary. This suggests QHD does not offer improvements over classical methods in the noiseless zeroth order setting. However, we show that the QHD algorithm can tolerate 𝒪̃(ε³ /d^{1.5} G² R²) noise in function evaluation, and as a result, provides a super-quadratic query advantage over the best existing noise-tolerant classical algorithms in the high-dimensional setting. We leverage this to design a quantum algorithm for stochastic convex optimization that offers a super-quadratic speedup over all known classical and quantum algorithms in this regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization obtained through a dynamical algorithm.
We present new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, expanding the scope of quantum advantage beyond traditional tunneling-based explanations. As our main building-block, we demonstrate a rigorous correspondence between the spectral properties of Schrödinger operators and the mixing times of classical Langevin diffusion. This correspondence motivates a mechanism for separation on functions with unique global minimum: while quantum algorithms operate on the original potential, classical diffusions correspond to a Schrödinger operators with a WKB potential having nearly degenerate global minima. We formalize these ideas by proving that a real-space adiabatic quantum algorithm (RsAA) achieves provably polynomial-time optimization for broad families of nonconvex functions. First, for block-separable functions, we show that RsAA maintains polynomial runtime while known off-the-shelf algorithms require exponential time and structure-aware algorithms exhibit arbitrarily large polynomial runtimes. These results leverage novel non-asymptotic results in semiclassical analysis. Second, we use recent advances in the theory of intrinsic hypercontractivity to demonstrate polynomial runtimes for RsAA on appropriately perturbed strongly convex functions that lack global structure, while off-the-shelf algorithms remain exponentially bottlenecked. In contrast to prior works based on quantum tunneling, these separations do not depend on the geometry of barriers between local minima. Our theoretical claims about classical algorithm runtimes are supported by rigorous analysis and comprehensive numerical benchmarking. These findings establish a rigorous theoretical foundation for quantum advantage in continuous optimization and open new research directions connecting quantum algorithms, stochastic processes, and semiclassical analysis.
This paper studies the continuous-time limit of a quantum analog of the classical Stochastic Gradient Descent (SGD) algorithm by formulating the problem in the Wigner description of quantum mechanics, thereby offering novelty in the description of the corresponding Stochastic Differential Equation (SDE) versus more obscure approaches regarding intuition, such as Lindblad formulations of quantum SGD algorithms. The Wigner formulation is a natural bridge between classical and quantum probabilistic algorithms, which, to our knowledge, has not been applied yet to the study of quantum analogs of SGD. We focus on the particular case of Quantum SGD in its continuous-time limit under a harmonic potential for the optimization of a quadratic function. We present both theoretical and computational studies of its convergence for the aforementioned benchmark problem. We prove, for this benchmark, the independence of the mixing time (convergence rate) from the phase-space dimension by studying the eigenvalues of the steadystate inverse covariance matrix in any dimension.