A Quantum Algorithm for Ray Casting using an Orthographic Camera Quantum computing has the potential to provide solutions to many problems which are challenging or out of reach of classical computers. There are several problems in rendering which are amenable to being solved in quantum computers, but these have yet to be demonstrated in practice. This work takes a first step in applying quantum computing to one of the most fundamental operations in rendering: ray casting. This technique computes visibility between two points in a 3D model of the world which is described by a collection of geometric primitives. The algorithm returns, for a given ray, which primitive it intersects closest to its origin. Without a spatial acceleration structure, the classical complexity for this operation is  O(N) . In this paper, we propose an implementation of Groveru0027s Algorithm (a quantum search algorithm) for ray casting. This provides a quadratic speed up allowing for visibility evaluation for unstructured primitives in  O(sqrtN) . However, due to technological limitations associated with current quantum computers, in this work the geometrical setup is limited to rectangles and parallel rays (orthographic projection).

Span programs and quantum space complexity While quantum computers hold the promise of significant computational speedups, the limited size of early quantum machines motivates the study of spacebounded quantum computation. We relate the quantum space complexity of computing a function f with onesided error to the logarithm of its span program size, a classical quantity that is wellstudied in attempts to prove formula size lower  the :59,more natural bounded error model, we show that the amount of space needed for a unitary quantum algorithm to compute f with bounded (twosided) error is lower bounded by the logarithm of its approximate span program size. Approximate span programs were introduced in the field of quantum algorithms but not studied classically. However, the approximate span program size of a function is a natural generalization of its span program  :130,no nontrivial lower bound is known on the span program size (or approximate span program size) of any concrete function, a number of lower bounds are known on the monotone span program size. We show that the approximate monotone span program size of f is a lower bound on the space needed by quantum algorithms of a particular form, called monotone phase estimation algorithms, to compute f. We then give the first nontrivial lower bound on the approximate span program size of an explicit function.