Differentiable Quantum Computing for Large-scale Linear Control

As industrial models and designs grow increasingly complex, the demand for optimal control of large-scale dynamical systems has significantly increased. However, traditional methods for optimal control incur significant overhead as problem dimensions grow. In this paper, we introduce an end-to-end quantum algorithm for linear-quadratic control with provable speedups. Our algorithm, based on a policy gradient method, incorporates a novel quantum subroutine for solving the matrix Lyapunov equation. Specifically, we build a quantum-assisted differentiable simulator for efficient gradient estimation that is more accurate and robust than classical methods relying on stochastic approximation. Compared to the classical approaches, our method achieves a super-quadratic speedup. To the best of our knowledge, this is the first end-to-end quantum application to linear control problems with provable quantum advantage.

A quantum-classical performance separation in nonconvex optimization

In this paper, we identify a family of nonconvex continuous optimization instances, each $d$-dimensional instance with $2^d$ local minima, to demonstrate a quantum-classical performance separation. Specifically, we prove that the recently proposed Quantum Hamiltonian Descent (QHD) algorithm [Leng et al., arXiv:2303.01471] is able to solve any $d$-dimensional instance from this family using $\widetilde{\mathcal{O}}(d^3)$ quantum queries to the function value and $\widetilde{\mathcal{O}}(d^4)$ additional 1-qubit and 2-qubit elementary quantum gates. On the other side, a comprehensive empirical study suggests that representative state-of-the-art classical optimization algorithms/solvers (including Gurobi) would require a super-polynomial time to solve such optimization instances.

Quantum Hamiltonian Descent

Gradient descent is a fundamental algorithm in both theory and practice for continuous optimization. Identifying its quantum counterpart would be appealing to both theoretical and practical quantum applications. A conventional approach to quantum speedups in optimization relies on the quantum acceleration of intermediate steps of classical algorithms, while keeping the overall algorithmic trajectory and solution quality unchanged. We propose Quantum Hamiltonian Descent (QHD), which is derived from the path integral of dynamical systems referring to the continuous-time limit of classical gradient descent algorithms, as a truly quantum counterpart of classical gradient methods where the contribution from classically-prohibited trajectories can significantly boost QHD's performance for non-convex optimization. Moreover, QHD is described as a Hamiltonian evolution efficiently simulatable on both digital and analog quantum computers. By embedding the dynamics of QHD into the evolution of the so-called Quantum Ising Machine (including D-Wave and others), we empirically observe that the D-Wave-implemented QHD outperforms a selection of state-of-the-art gradient-based classical solvers and the standard quantum adiabatic algorithm, based on the time-to-solution metric, on non-convex constrained quadratic programming instances up to 75 dimensions. Finally, we propose a "three-phase picture" to explain the behavior of QHD, especially its difference from the quantum adiabatic algorithm.

Differentiable Analog Quantum Computing for Optimization and Control

We formulate the first differentiable analog quantum computing framework with a specific parameterization design at the analog signal (pulse) level to better exploit near-term quantum devices via variational methods. We further propose a scalable approach to estimate the gradients of quantum dynamics using a forward pass with Monte Carlo sampling, which leads to a quantum stochastic gradient descent algorithm for scalable gradient-based training in our framework. Applying our framework to quantum optimization and control, we observe a significant advantage of differentiable analog quantum computing against SOTAs based on parameterized digital quantum circuits by orders of magnitude.

Quantum Algorithms for Escaping from Saddle Points

We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a function $f\colon\mathbb{R}^{n}\to\mathbb{R}$, our quantum algorithm outputs an $\epsilon$-approximate second-order stationary point using $\tilde{O}(\log^{2} n/\epsilon^{1.75})$ queries to the quantum evaluation oracle (i.e., the zeroth-order oracle). Compared to the classical state-of-the-art algorithm by Jin et al. with $\tilde{O}(\log^{6} n/\epsilon^{1.75})$ queries to the gradient oracle (i.e., the first-order oracle), our quantum algorithm is polynomially better in terms of $n$ and matches its complexity in terms of $1/\epsilon$. Our quantum algorithm is built upon two techniques: First, we replace the classical perturbations in gradient descent methods by simulating quantum wave equations, which constitutes the polynomial speedup in $n$ for escaping from saddle points. Second, we show how to use a quantum gradient computation algorithm due to Jordan to replace the classical gradient queries by quantum evaluation queries with the same complexity. Finally, we also perform numerical experiments that support our quantum speedup.