Gaussian Amplitude Amplification for Quantum Pathfinding

Abstract

We study an oracle operation, along with its circuit design, which combined with the Grover diffusion operator boosts the probability of finding the minimum or maximum solutions on a weighted directed graph. We focus on the geometry of sequentially connected bipartite graphs, which naturally gives rise to solution spaces describable by Gaussian distributions. We then demonstrate how an oracle that encodes these distributions can be used to solve for the optimal path via amplitude amplification. And finally, we explore the degree to which this algorithm is capable of solving cases that are generated using randomized weights, as well as a theoretical application for solving the Traveling Salesman problem.

Variational Amplitude Amplification for Solving QUBO Problems

“…Indeed, the histogram is approximately Gaussian; importantly, however, it has imperfections resulting from the randomized weights. At large enough problem sizes (around N ≥ 20), these imperfections have minimal impact on a problem's aptitude for amplitude amplification, which is one result from our previous study [19]. Similarly, another recent study [26] has demonstrated that, in addition to symmetric Gaussians, solution space distributions for both skewed Gaussians and exponential profiles lead to successful amplitude amplifications.…”

“…This study is a continuation of our previous work [19], in which we demonstrated an oracle design which was capable of encoding and solving a weighted directed graph problem. The motivation for this oracle was to address a common criticism of U G [18,[20][21][22][23], namely, that the circuit construction of oracles too often hardcodes the solution it aims to find, negating the use of quantum entirely.…”

“…Finding the optimal p s value for boosting X min or X max is non-trivial, and was a major focus of our previous study [19] as well as this one. In general, the scale of p s needed for finding the optimal solution can be obtained using Equation ( 14) below, which scales the numerical range of a problem [C(X min ) , C(X max )] to exactly [x , x + 2π].…”

“…In order to use Algorithm 1 to find the optimal solution to a given cost function, we must construct a cost oracle U c which encodes the weighted information and connectivity of the problem. In our previous study, we referred to this operation as a 'phase oracle' U P [19], and it has similarly been called a 'subdivided phase oracle' SPO [25,26] or 'non-Boolean oracle' [27]. Although how one constructs U c is problem-specific, the general strategy is to primarily use two quantum gates, as shown below in Equations ( 7) and (8).…”

“…The second and more challenging tradeoff when using U c is that the success of amplitude amplification is largely dependant on the correct choice of a single free parameter p s [19]. This scalar parameter is multiplied into every phase gate for the construction of U c (P(θ • p s ) and CP(θ • p s )), and is responsible for transforming the numeric scale of a given optimization problem to values which form a range of approximately 2π.…”