Optimizing rodeo projection

TL;DR

This paper improves the rodeo quantum algorithm by replacing random time selection with a deterministic approach, significantly reducing fluctuations and enhancing suppression efficiency in projecting states onto fixed energy levels.

Contribution

It demonstrates that choosing times deterministically over multiple scales avoids large fluctuations and greatly improves suppression compared to the original random method.

Findings

Deterministic time selection reduces suppression fluctuations.

Average suppression improves by orders of magnitude.

Exponential time scales optimize projection efficiency.

Abstract

The rodeo algorithm has been proposed recently as an efficient method in quantum computing for projection of a given initial state onto a state of fixed energy for systems with discrete spectra. In the initial formulation of the rodeo algorithm these times were chosen randomly via a Gaussian distribution with fixed RMS times. In this paper it is shown that such a random approach for choosing times suffers from exponentially large fluctuations in the suppression of unwanted components: as the number of iterations gets large, the distribution of suppression factors obtained from random selection approaches a log-normal distribution leading to remarkably large fluctuations. We note that by choosing times intentionally rather than randomly such fluctuations can be avoided and strict upper bounds on the suppression can be obtained. Moreover, the average suppression using fixed computational cost can be reduced by many orders of magnitude relative to the random algorithm. A key to doing this is to choose times that vary over exponentially many times scales, starting from a modest maximum scale and going down to time scales exponentially smaller.