Grover's Algorithm with Diffusion and Amplitude Steering

Abstract

We review the basic theoretical underpinnings of the Grover algorithm, providing a rigorous and well motivated derivation. We then present a generalization of Grover's algorithm that searches an arbitrary subspace of the multi-dimensional Hilbert space using a diffusion operation and an amplitude amplification procedure that has been biased by unitary steering operators. We also outline a generalized Grover's algorithm that takes into account higher level correlations that could exist between database elements. In the traditional Grover algorithm, the Hadamard gate selects a uniform sample of computational basis elements when performing the phase selection and diffusion. In contrast, steered operators bias the selection process, thereby providing more flexibility in selecting the target state. Our method is a generalization of the recently proposal pattern matching algorithm of Hiroyuki et al..

Variational Amplitude Amplification for Solving QUBO Problems

“…The quantum strategy for finding the optimal solutions to C(X) investigated in this study is amplitude amplification [4][5][6][7][8][9], which is the generalization of Grover's algorithm [1]. The full algorithm is shown below in Algorithm 1, which is almost identical to Grover's algorithm except for the replacement of Grover's oracle U G with our cost oracle U c .…”

“…Originally proposed by Grover in 1996 [1], and later shown to be optimal [2,3], the combination of his oracle U G and 'diffusion' U s operators is able to drive a quantum system to a superposition state where one (or multiple) basis state(s) has nearly 100% probability of being measured. Since then, many researchers have contributed to the study of U G and U s [4][5][6][7][8][9], seeking to better understand how the fundamental nature of amplitude amplification is dependent on these two operators. Similarly, the aim of this study is to further extend the capabilities of amplitude amplification as a means for solving combinatorial optimization problems using gate-based quantum computers.…”

Gaussian Amplitude Amplification for Quantum Pathfinding

“…Qubits allow for incredible parallelism in computations via superposition states, but reliably pulling out a single answer via measurements is often difficult. In 1996, Grover demonstrated one of the first mechanisms overcoming this weakness [ 1 ], later shown to be optimal [ 2 , 3 ], and has since been refined into a broader technique in quantum algorithms known as ‘amplitude amplification’ [ 4 , 5 , 6 , 7 , 8 , 9 , 10 ]. In this study, we seek to extend the capabilities of amplitude amplification as a means of pathfinding on a directed graph with weighted edges.…”