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arXiv:2412.18309

Simple Quantum Gradient Descent Without Coherent Oracle Access

Nhat A. Nghiem

Published Dec 24, 2024• 0 citations
arXivHTMLPDF

Abstract

The gradient descent method aims at finding local minima of a given multivariate function by moving along the direction of its gradient, and hence, the algorithm typically involves computing all partial derivatives of a given function, before updating the solution iteratively. In the work of Rebentrost et al. [New Journal of Physics, 21(7):073023, 2019], the authors translated the iterative optimization algorithm into a quantum setting, with some assumptions regarding certain structure of the given function, with oracle or black-box access to some matrix that specifies the structure. Here, we develop an alternative quantum framework for the gradient descent problem, based on the seminal quantum singular value transformation framework. We show that given only classical information of function of interest, it is possible to construct a quantum gradient descent algorithm with a running time logarithmical in the number of variables. In particular, our framework consumes exponentially less qubits than the prior quantum gradient descent algorithm and removes the need for any coherent oracle access to classical information. Thus, our work provides another example demonstrating the power of quantum singular value transformation framework, and in particular, it adds another instance revealing that quantum coherent access is not necessary for quantum computational advantage.

DOI
10.48550/arxiv.2412.18309

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The gradient descent method aims at finding local minima of a given multivariate function by moving along the direction of its gradient, and hence, the algorithm typically involves computing all partial derivatives of a given function, before updating the solution iteratively. In the work of Rebentrost et al. [New Journal of Physics, 21(7):073023, 2019], the authors translated the iterative optimization algorithm into a quantum setting, with some assumptions regarding certain structure of the given function, with oracle or black-box access to some matrix that specifies the structure. Here, we develop an alternative quantum framework for the gradient descent problem, based on the seminal quantum singular value transformation framework. We show that given only classical information of function of interest, it is possible to construct a quantum gradient descent algorithm with a running time logarithmical in the number of variables. In particular, our framework consumes exponentially less qubits than the prior quantum gradient descent algorithm and removes the need for any coherent oracle access to classical information. Thus, our work provides another example demonstrating the power of quantum singular value transformation framework, and in particular, it adds another instance revealing that quantum coherent access is not necessary for quantum computational advantage.
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 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).
Optimization problems are prevalent in various fields, and the gradient-based gradient descent algorithm is a widely adopted optimization method. However, in classical computing, computing the numerical gradient for a function with d variables necessitates at least d + 1 function evaluations, resulting in a computational complexity of O ( d ). As the number of variables increases, the classical gradient estimation methods require substantial resources, ultimately surpassing the capabilities of classical computers. Fortunately, leveraging the principles of superposition and entanglement in quantum mechanics, quantum computers can achieve genuine parallel computing, leading to exponential acceleration over classical algorithms in some cases. In this paper, we propose a novel quantum-based gradient calculation method that requires only a single oracle calculation to obtain the numerical gradient result for a multivariate function. The complexity of this algorithm is just O (1). Building upon this approach, we successfully implemented the quantum gradient descent algorithm and applied it to the variational quantum eigensolver (VQE), creating a pure quantum variational optimization algorithm. Compared with classical gradient-based optimization algorithm, this quantum optimization algorithm has remarkable complexity advantages, providing an efficient solution to optimization problems. The proposed quantum-based method shows promise in enhancing the performance of optimization algorithms, highlighting the potential of quantum computing in this field.
Abstract Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton’s method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.
Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.
Abstract The gradient descent method is central to numerical optimization and is the key ingredient in many machine learning algorithms. It promises to find a local minimum of a function by iteratively moving along the direction of the steepest descent. Since for high-dimensional problems the required computational resources can be prohibitive, it is desirable to investigate quantum versions of the gradient descent, such as the recently proposed (Rebentrost et al. 1 ). Here, we develop this protocol and implement it on a quantum processor with limited resources. A prototypical experiment is shown with a four-qubit nuclear magnetic resonance quantum processor, which demonstrates the iterative optimization process. Experimentally, the final point converged to the local minimum with a fidelity >94%, quantified via full-state tomography. Moreover, our method can be employed to a multidimensional scaling problem, showing the potential to outperform its classical counterparts. Considering the ongoing efforts in quantum information and data science, our work may provide a faster approach to solving high-dimensional optimization problems and a subroutine for future practical quantum computers.