Numerical calculus algorithms which estimate derivatives and integrals from data series acquired either via measurements or by sampling functions are essential in scientific computing. To date, a few quantum algorithms have been developed to perform calculus operations based on closed form functional inputs; yet, in many practical applications, field variables are numerically described via series of samples rather than closed form expressions. This paper presents the theoretical development and the gate-level circuit implementation of novel quantum algorithms for numerical differentiation and indefinite integration with a prescribed integration constant. The methodology relies on a spectral approach that leverages the computational efficiency of the quantum Fourier transform and the parallel computing capability afforded by quantum superposition to evaluate outputs at all domain points simultaneously. The differentiation approach is also extended to enable gradient estimation, and post-processing procedures are presented to recover sign information. The primary output of the proposed algorithms are quantum state vectors directly proportional to the numerical derivative or integral of the given data; therefore, the correctly signed results are made available to proceeding quantum computations. This result lays the foundation for the proposed algorithms to serve as core subroutines in applied quantum computing operations such as image processing, data analysis, and machine learning.
Numerical calculus algorithms which estimate derivatives and integrals from data series acquired either via measurements or by sampling functions are essential in scientific computing. To date, a few quantum algorithms have been developed to perform calculus operations based on closed form functional inputs; yet, in many practical applications, field variables are numerically described via series of samples rather than closed form expressions. This paper presents the theoretical development and the gate-level circuit implementation of novel quantum algorithms for numerical differentiation and indefinite integration with a prescribed integration constant. The methodology relies on a spectral approach that leverages the computational efficiency of the quantum Fourier transform and the parallel computing capability afforded by quantum superposition to evaluate outputs at all domain points simultaneously. The differentiation approach is also extended to enable gradient estimation, and post-processing procedures are presented to recover sign information. The primary output of the proposed algorithms are quantum state vectors directly proportional to the numerical derivative or integral of the given data; therefore, the correctly signed results are made available to proceeding quantum computations. This result lays the foundation for the proposed algorithms to serve as core subroutines in applied quantum computing operations such as image processing, data analysis, and machine learning.
Quantum computing is a promising emerging field in computational science, attracting significant international research interest. Data representation is crucial to the success of machine learning models, and covariance analysis to determine relationships between pairs of random variables is an essential step in exploratory data analysis. Here we introduce a novel quantum algorithm for covariance approximation and its circuit implementation for discrete-value vectors. The algorithm leverages vital components of quantum computing, including amplitude encoding, the Grover diffusion algorithm for amplitude amplification and the quantum Fourier transform. As a core innovation, we utilize two quantum states and their properties of superposition product states to estimate the covariance between input vectors. Evaluating the quantum algorithm’s performance over five discrete-value distributions, we found high levels of agreement between the classical and quantum computations, with maximum average errors averaging 0.153. Furthermore, the analysis revealed additional applications of the Grover diffusion operator to solve problems.
In large-scale scientific computing and engineering modeling, calculus problems are widely used as basic mathematical tools in physical simulation, control systems, image processing and other fields. However, their analytical and numerical solutions in high-dimensional and complex functions often face bottlenecks such as high computational complexity and slow convergence. To this end, this paper proposes a calculus acceleration solution method based on a quantum computing framework. First, this paper constructs a quantum state encoding mechanism suitable for function integration and differentiation operations. Then, Variational Quantum Algorithm (VQA) is applied to solve the optimal solution path in structured calculus expressions, and combined with Quantum Fourier Transform (QFT) to improve function boundary processing and analysis speed. Finally, a quantum-classical hybrid iterator is designed through a hybrid architecture with classical numerical methods. Experimental results show that in high-dimensional integration scenarios, compared with the classical Monte Carlo method, Quantum Amplitude Estimation (QAE) can reduce the number of queries from thousands to hundreds with the same error accuracy (such as); in the simulation experiment of ordinary differential equations, the simulated VQA shows a higher convergence efficiency with the same number of iterations. The above results fully demonstrate the acceleration potential and solution accuracy advantages of quantum algorithms in high dimensions and complex boundary conditions, and provide new ideas for the application of calculus problems in large-scale scientific and engineering calculations.
While quantum computers are naturally well-suited to implementing linear operations, it is less clear how to implement nonlinear operations on quantum computers. However, nonlinear subroutines may prove key to a range of applications of quantum computing from solving nonlinear equations to data processing and quantum machine learning. Here we develop a series of basic subroutines for implementing nonlinear transformations of input quantum states. Our algorithms are framed around the concept of a weighted state, a mathematical entity describing the output of an operational procedure involving both quantum circuits and classical post-processing.
A novel family of Cosine series Quantum Sampling (QCoSamp) operators appropriate for quantum computing is described. The development of quantum algorithms, analogous to classical algorithms, we apply to the harmonic analysis of signals. We show quantum sampling through measurements of a quantum system, and after operators of the family are applied, allow for input signal mapping with a Fourier series representation. Technical methodologies employed, facilitating the implementation of each QCoSamp algorithm to a quantum computer and application to the field of signal and image processing we also described. Keywords: quantum computing, quantum information theory, quantum operator, quantum sampling, Fourier sine-cosine series, signal processing, image processing
While quantum computers are naturally well suited to implementing linear operations, it is less clear how to implement nonlinear operations on quantum computers. However, nonlinear subroutines may prove key to a range of applications of quantum computing from solving nonlinear equations to data processing and quantum machine learning. Here we develop a series of basic subroutines for implementing nonlinear transformations of input quantum states. Our algorithms are framed around the concept of a weighted state, a mathematical entity describing the output of an operational procedure involving both quantum circuits and classical postprocessing.
Canonical quantization provides a systematic procedure for constructing quantum models from classical Hamiltonians. Here, we apply this principle to a fundamental computational primitive of machine learning: the neuron. Specifically, by viewing a neuron as a composition of an energy function and an activation function, we quantize this model by replacing the energy function with a quantum Hamiltonian and applying the activation function to it through matrix functional calculus. This results in an activation observable that can be measured on an input quantum state. We investigate the use of these quantized neurons for function approximation, where the objective is to learn an unknown observable from labeled quantum data. For this purpose, we develop hybrid quantum-classical algorithms for training and evaluation, including procedures for measuring the activation observable and estimating gradients of the squared loss error. Our algorithms for gradient estimation rely on basic primitives like classical random sampling, the Hadamard test, and Hamiltonian simulation, and those for measuring an activation observable rely on quantum algorithms known as the power of one qumode and Schroedingerization. Numerical experiments demonstrate that our quantized neurons exhibit enhanced expressive capabilities relative to corresponding classical neurons on representative learning tasks. Our work establishes canonical quantization as a principled framework for constructing quantum machine learning primitives and provides a foundation for developing neural architectures tailored to quantum data.