Gradient Descent is an optimization algorithm used for finding the minima of a function. It moves along the function with steps proportional to the negative of the gradient. Quantum version of this algorithm has the advantage of exponential speedup for problems with high dimensional inputs. On the other hand, Factorization Machines (FMs) are modeling classes with important advantages in comparison with other modeling ones such as SVMs. They, in contrast to SVMs, use factorized parameters in order to model all interactions between variables. Accordingly, unlike SVMs, they can estimate interactions in problems with large sparsity. In this work, we show that Factorization Machine can be implemented on quantum computers, using quantum gradient descent to learn the parameters of a FM. The parameters we have chosen to learn via quantum gradient descent is a bottleneck in learning parameters of a FM. Our proposed idea results in time computational complexity O(k polylog n) where k and n are the dimensions of the parameter matrix. By doing so and under some constraints on the dimensions, an exponential speedup over classical algorithms can be obtained.
Gradient Descent is an optimization algorithm used for finding the minima of a function. It moves along the function with steps proportional to the negative of the gradient. Quantum version of this algorithm has the advantage of exponential speedup for problems with high dimensional inputs. On the other hand, Factorization Machines (FMs) are modeling classes with important advantages in comparison with other modeling ones such as SVMs. They, in contrast to SVMs, use factorized parameters in order to model all interactions between variables. Accordingly, unlike SVMs, they can estimate interactions in problems with large sparsity. In this work, we show that Factorization Machine can be implemented on quantum computers, using quantum gradient descent to learn the parameters of a FM. The parameters we have chosen to learn via quantum gradient descent is a bottleneck in learning parameters of a FM. Our proposed idea results in time computational complexity O(k polylog n) where k and n are the dimensions of the parameter matrix. By doing so and under some constraints on the dimensions, an exponential speedup over classical algorithms can be obtained.
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.
Quantum Machine Learning (QML) is rapidly emerging as a transformative paradigm for various computational tasks, including machine learning. Traditional machine learning algorithms often struggle with high-dimensional data and complex non-linear relationships, limiting their effectiveness in certain applications. This paper proposes a novel quantum machine learning optimization algorithm, dubbed the "Quantum-Enhanced Gradient Descent" (QED) algorithm, designed to address these challenges. We explore the potential of quantum computing to accelerate gradient descent and enhance the exploration of solution spaces, ultimately leading to improved model performance. The core mechanism involves leveraging quantum properties to perform more efficient and nuanced parameter adjustments, specifically focusing on non-linear optimization. This paper provides a detailed theoretical foundation and preliminary experimental validation demonstrating the effectiveness of QED in optimizing machine learning models, particularly concerning high-dimensional data and complex non-linear dependencies.
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).
Quantum computing has been proven to excel in factorization issues and unordered search problems due to its capability of quantum parallelism. This unique feature allows exponential speed-up in solving certain problems. However, this advantage does not apply universally, and challenges arise when combining classical and quantum computing to achieve acceleration in computation speed. This paper aims to address these challenges by exploring the current state of quantum machine learning and benchmarking the performance of quantum and classical algorithms in terms of accuracy. Specifically, we conducted experiments with three datasets for binary classification, implementing Support Vector Machine (SVM) and Quantum SVM (QSVM) algorithms. Our findings suggest that the QSVM algorithm outperforms classical SVM on complex datasets, and the performance gap between quantum and classical models increases with dataset complexity, as simple models tend to overfit with complex datasets. While there is still a long way to go in terms of developing quantum hardware with sufficient resources, quantum machine learning holds great potential in areas such as unsupervised learning and generative models. Moving forward, more efforts are needed to explore new quantum learning models that can leverage the power of quantum mechanics to overcome the limitations of classical machine learning.
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.