Quantum Coin Method for Numerical Integration

Abstract: Monte Carlo integration approximates an integral of a black-box function by taking the average of many evaluations (i.e., samples) of the function (integrand). For $N$ queries of the integrand, Monte Carlo integration achieves the estimation error of $O(1/ \sqrt{N})$. Recently, Johnston introduced quantum supersampling (QSS) into rendering as a numerical integration method that can run on quantum computers. QSS breaks the fundamental limitation of the $O(1/ \sqrt{N})$ convergence rate of Monte Carlo integration and achieves the faster convergence rate. We introduce yet another quantum numerical integration algorithm, quantum coin (QCoin), and provide numerical experiments that are unprecedented in the fields of both quantum computing and rendering. We show that QCoin's convergence rate is equivalent to QSS's. We additionally show that QCoin is fundamentally more robust under the presence of noise in actual quantum computers due to its simpler quantum circuit and the use of fewer qubits. Considering various aspects of quantum computers, we discuss how QCoin can be a more practical alternative to QSS in quantum computers in the future.