A Technique for Quantum Numerical Differentiation and Integration

Numerical algorithms to estimate derivatives and integrals from a set of function samples are essential tools for scientific computing. As the field of quantum computing is rapidly growing, several successful quantum calculus algorithms have been identified. However, these algorithms typically require a known function as input, which severely limits their applicability. This work focuses on developing quantum extensions of popular numerical techniques such as central difference differentiation and trapezoidal integration. The algorithms are spectral in nature and exploit properties of the highly efficient quantum Fourier transform (QFT). Thus, these algorithms are dubbed the QFT-based Derivative (QFTD) and the QFT-based Integral (QFTI). Following a theoretical description of the two algorithms, numerical tests are conducted to illustrate their performance when applied to different types of functions. Results show that, for a quantum circuit given N input samples and evaluated using M shots, the QFTD algorithm is capable of accurately estimating a derivative with error given by max[O(N ^-2), O(M ^-1/2)], while the QFTI’s error is given by max[O[(N ^-1), O(M ^-1/2)]. As both algorithms rely on the QFT, they exhibit computational complexities of O(log₂N). These algorithms also extend naturally to higher dimensions, returning simultaneous estimates of all partial derivatives or differential elements with unchanged errors and complexities.