Quantum Coin Method for Numerical Integration

Abstract

Light transport simulation in rendering is formulated as a numerical integration problem in each pixel, which is commonly estimated by Monte Carlo integration. 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/N). Recently, Johnston [Joh16] introduced quantum super‐sampling (QSS) into rendering as a numerical integration method that can run on quantum computers. QSS breaks the fundamental limitation of the O(1/N) convergence rate of Monte Carlo integration and achieves the faster convergence rate of approximately O(1/N) which is the best possible bound of any quantum algorithms we know today [NW99]. We introduce yet another quantum numerical integration algorithm, quantum coin (QCoin) [AW99], 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 if we were to run light transport simulation in quantum computers in the future.

Real quantum amplitude estimation

“…In [14] they do not exactly match the desired asymptotic complexity, yet the constants involved are much lower. RQAE can be thought of as a generalization of the Quantum Coin algorithm [16,17] and it is based on an iterative strategy, like [14,15]. In particular, RQAE utilizes a set of auxiliary amplitudes which allow to shift in a controlled fashion the amplitude to be retrieved.…”

Quantum Arithmetic for Directly Embedded Arrays

“…The third and last step of the pipeline corresponds to extracting the information that we have stored in the quantum state, namely the read-out of the state that encodes the result of the algorithm. There are multiple techniques for this purpose such as those appearing in [7][8][9][10][11].…”

A general quantum algorithm for numerical integration

“…Shimada et al presented the quantum coin algorithm based on the quantum supersampling algorithm achieving quadratic acceleration 19 , 20 , but the value of the functions is limited to [0,1]. Heinrich raised a quantum integration algorithm with quadratic acceleration for Sobolev-like high-dimensional function 21 .…”