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American Journal of Physics

From Schrödinger’s equation to the quantum search algorithm

Lov K. Grover

Published Jul 1, 2001• 171 citations
Publisher

Abstract

The quantum search algorithm is a technique for searching N possibilities in only O(N) steps. Although the algorithm itself is widely known, not so well known is the series of steps that first led to it; these are quite different from any of the generally known forms of the algorithm. This paper describes these steps, which start by discretizing Schrödinger’s equation. This paper also provides a self-contained introduction to quantum computing algorithms from a new perspective.

DOI
10.1119/1.1359518

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The quantum search algorithm is a technique for searching N possibilities in only O(N) steps. Although the algorithm itself is widely known, not so well known is the series of steps that first led to it; these are quite different from any of the generally known forms of the algorithm. This paper describes these steps, which start by discretizing Schrödinger’s equation. This paper also provides a self-contained introduction to quantum computing algorithms from a new perspective.
As quantum computers become available to the general public, the need has arisen to train a cohort of quantum programmers, many of whom have been developing classical computer programs for most of their careers. While currently available quantum computers have less than 100 qubits, quantum computing hardware is widely expected to grow in terms of qubit count, quality, and connectivity. This review aims at explaining the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional. We give an introduction to quantum computing algorithms and their implementation on real quantum hardware. We survey 20 different quantum algorithms, attempting to describe each in a succinct and self-contained fashion. We show how these algorithms can be implemented on IBM’s quantum computer, and in each case, we discuss the results of the implementation with respect to differences between the simulator and the actual hardware runs. This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations.
Quantum search is a quantum mechanical technique for searching N possibilities in only √ N steps.There are several different perspectives from which one can get to the algorithm -Schrodinger's equation, antenna array, rotation in a two-dimensional Hilbert space, just to name a few.This paper gives a fresh perspective on the algorithm in terms of a resonance phenomenon which is implemented through classical coupled oscillators.Consider N oscillators, one of which is of a different resonant frequency.We could identify which one this is by measuring the oscillation frequency of each oscillator, a procedure that would take about N cycles.We show how, by coupling the oscillators together in a very simple way, it is possible to identify the different one in only √ N cycles.In case there are multiple oscillators of a different frequency, we can estimate the number of these in a time which is the square-root of that required by the sampling method.An extension of this technique to the quantum case leads to the quantum search & some novel algorithms.
Abstract Quantum search is a quantum mechanical technique for searching N possibilities in only √ N steps. There are several different perspectives from which one can get to the algorithm - Schro¨dinger’s equation, antenna array, rotation in a two-dimensional Hilbert space, just to name a few. This paper gives a fresh perspective on the algorithm in terms of classical coupled oscillators. Consider N oscillators, one of which is of a different resonant frequency. We could identify which one this is by measuring the oscillation frequency of each oscillator, a procedure that would take about N cycles. We show how, by coupling the oscillators together in a very simple way, it is possible to identify the different one in only √ N cycles. In case there are multiple oscillators of a different frequency, we can estimate the number of these in a time which is the square-root of that required by the sampling method. An extension of this technique to the quantum case leads to the quantum search and some novel algorithms.
In the sub-field of quantum algorithms, physicists and computer scientist take classical computing algorithms and principles and see if there is a more efficient or faster approach implementable on a quantum computer, i.e. a ”quantum advantage”. We take random walks, a widely applicable group of classical algorithms, and move them into the quantum computing paradigm. Additionally, an introduction to a popular quantum search algorithm called Grover’s search is included to guide the reader to the development of a quantum search algorithm using quantum random walks. To close the gap between algorithm and hardware, we will look at using neutral-atom (also known as cold-atom) quantum computing to physically implement two types of quantum random walks and Grover’s algorithm.
This paper is a gentle but rigorous introduction to quantum computing intended for discrete mathematicians. Starting from a small set of assumptions on the behavior of quantum computing devices, we analyze their main characteristics, stressing the differences with classical computers, and finally describe two well-known algorithms (Simon's algorithm and Grover's algorithm) using the formalism developed in previous sections. This paper does not touch on the physics of the devices, and therefore does not require any notion of quantum mechanics. Numerical examples based on an implementation of Grover's algorithm using open-source software are provided.