We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles, and demonstrate the first dimension-independent query complexities (up to poly-logarithmic factors) for zeroth-order convex optimization in both smooth and nonsmooth settings. Interestingly, only using noisy function evaluation oracles, we match the first-order query complexities of classical gradient descent, thereby exhibiting exponential separation between quantum and classical zeroth-order optimization. We then generalize these algorithms to work in non-Euclidean settings by using quantum (sub)gradient estimation to instantiate mirror descent and its variants, including dual averaging and mirror prox. By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games. We identify a parameter regime in which our zero-sum games algorithm is faster than any existing classical or quantum approach.
We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles, and demonstrate the first dimension-independent query complexities (up to poly-logarithmic factors) for zeroth-order convex optimization in both smooth and nonsmooth settings. Interestingly, only using noisy function evaluation oracles, we match the first-order query complexities of classical gradient descent, thereby exhibiting exponential separation between quantum and classical zeroth-order optimization. We then generalize these algorithms to work in non-Euclidean settings by using quantum (sub)gradient estimation to instantiate mirror descent and its variants, including dual averaging and mirror prox. By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games. We identify a parameter regime in which our zero-sum games algorithm is faster than any existing classical or quantum approach.
Recent results suggest that quantum computers possess the potential to speed up nonconvex optimization problems. However, a crucial factor for the implementation of quantum optimization algorithms is their robustness against experimental and statistical noises. In this paper, we systematically study quantum algorithms for finding an ε-approximate second-order stationary point (ε-SOSP) of a d-dimensional nonconvex function, a fundamental problem in nonconvex optimization, with noisy zeroth- or first-order oracles as inputs. We first prove that, up to noise of O(ε10/d5), accelerated perturbed gradient descent with quantum gradient estimation takes O(logd/ε1.75) quantum queries to find an ε-SOSP. We then prove that perturbed gradient descent is robust to the noise of O(ε6/d4) and O(ε/d0.5+ζ) for ζ>0 on the zeroth- and first-order oracles, respectively, which provides a quantum algorithm with poly-logarithmic query complexity. We then propose a stochastic gradient descent algorithm using quantum mean estimation on the Gaussian smoothing of noisy oracles, which is robust to O(ε1.5/d) and O(ε/d) noise on the zeroth- and first-order oracles, respectively. The quantum algorithm takes O(d2.5/ε3.5) and O(d2/ε3) queries to the two oracles, giving a polynomial speedup over the classical counterparts. Moreover, we characterize the domains where quantum algorithms can find an ε-SOSP with poly-logarithmic, polynomial, or exponential number of queries in d, or the problem is information-theoretically unsolvable even by an infinite number of queries. In addition, we prove an Ω(ε−12/7) lower bound in ε for any randomized classical and quantum algorithm to find an ε-SOSP using either noisy zeroth- or first-order oracles.
We consider a generic framework of optimization algorithms based on gradient descent. We develop a quantum algorithm that computes the gradient of a multi-variate realvalued function f : ℝd → ℝ by evaluating it at only a logarithmic number of times in superposition. Our algorithm is an improved version of Jordan's gradient computation algorithm [28], providing an approximation of the gradient ▽f with quadratically better dependence on the evaluation accuracy of f, for an important class of smooth functions. Furthermore, we show that objective functions arising from variational quantum circuits usually satisfy the necessary smoothness conditions, hence our algorithm provides a quadratic improvement in the complexity of computing their gradient. We also show that in a continuous phase-query model, our gradient computation algorithm has optimal query complexity up to poly-logarithmic factors, for a particular class of smooth functions. Moreover, we show that for low-degree multivariate polynomials our algorithm can provide exponential speedups compared to Jordan's algorithm in terms of the dimension d. One of the technical challenges in applying our gradient computation procedure for quantum optimization problems is the need to convert between a probability oracle (which is common in quantum optimization procedures) and a phase oracle (which is common in quantum algorithms) of the objective function f. We provide efficient subroutines to perform this delicate interconversion between the two types of oracles incurring only a logarithmic overhead, which might be of independent interest. Finally, using these tools we improve the runtime of prior approaches for training quantum auto-encoders, variational quantum eigensolvers (VQE), and quantum approximate optimization algorithms (QAOA).
We consider a generic framework of optimization algorithms based on gradient descent. We develop a quantum algorithm that computes the gradient of a multi-variate real-valued function f : Rd → R by evaluating it at only a logarithmic number of times in superposition. Our algorithm is an improved version of Jordan's gradient computation algorithm [28], providing an approximation of the gradient ▽f with quadratically better dependence on the evaluation accuracy of f, for an important class of smooth functions. Furthermore, we show that objective functions arising from variational quantum circuits usually satisfy the necessary smoothness conditions, hence our algorithm provides a quadratic improvement in the complexity of computing their gradient. We also show that in a continuous phase-query model, our gradient computation algorithm has optimal query complexity up to poly-logarithmic factors, for a particular class of smooth functions. Moreover, we show that for low-degree multivariate polynomials our algorithm can provide exponential speedups compared to Jordan's algorithm in terms of the dimension d.One of the technical challenges in applying our gradient computation procedure for quantum optimization problems is the need to convert between a probability oracle (which is common in quantum optimization procedures) and a phase oracle (which is common in quantum algorithms) of the objective function f. We provide efficient subroutines to perform this delicate interconversion between the two types of oracles incurring only a logarithmic overhead, which might be of independent interest. Finally, using these tools we improve the runtime of prior approaches for training quantum auto-encoders, variational quantum eigensolvers (VQE), and quantum approximate optimization algorithms (QAOA).
We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function f:Rn→R and its (sub)gradient. Our goal is to find an ε-approximate minimum of f starting from a point that is distance at most R from the true minimum. If f is G-Lipschitz, then the classic gradient descent algorithm solves this problem with O((GR/ε)2) queries. Importantly, the number of queries is independent of the dimension n and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension n. In this paper we reprove the randomized lower bound of Ω((GR/ε)2) using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using O(GR/ε) quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need Ω((GR/ε)2) queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.
We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function f:ℝⁿ → ℝ and its (sub)gradient. Our goal is to find an ε-approximate minimum of f starting from a point that is distance at most R from the true minimum. If f is G-Lipschitz, then the classic gradient descent algorithm solves this problem with O((GR/ε)²) queries. Importantly, the number of queries is independent of the dimension n and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension n. In this paper we reprove the randomized lower bound of Ω((GR/ε)²) using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using O(GR/ε) quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need Ω((GR/ε)²) queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.
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.