An Implementation of Quantum Oracles for the Finite Element Method

Abstract. In order to assess potential advantages of quantum algorithms that require quantum oracles as subroutines, the careful evaluation of the overall complexity of the oracles themselves is crucial. This study examines the quantum routines required for the implementation of oracles used in the block-encoding of the [Formula: see text] stiffness and mass matrices, which typically emerge in the finite element analysis of elastic structures. Starting from basic quantum adders, we show how to construct the necessary oracles, which require the calculation of polynomials, square root, and the implementation of conditional operations. We propose quantum subroutines based on fixed-point arithmetic that, given an [Formula: see text]-qubit register, construct the oracle using [Formula: see text] ancilla qubits and have an [Formula: see text] runtime, with [Formula: see text] the order at which we truncate the polynomials and [Formula: see text] the number of iterations in the Newton–Raphson subroutine for the square root, while [Formula: see text] and [Formula: see text] are the number of hypercuboids used to approximate the geometry and the boundary, respectively. Since in practice [Formula: see text] scales as [Formula: see text], for numbers between 0 and [Formula: see text], and assuming that the other parameters are fixed independently of [Formula: see text], this shows that the oracles, while still costly in practice, do not endanger potential polynomial or exponential advantages in [Formula: see text].