English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

According to the pioneering work of Nielsen and collaborators, the length ofthe minimal geodesic in a geometric realization of a suitable operator spaceprovides a measure of the quantum complexity of an operation. Compared with theoriginal concept of complexity based on the minimal number of gates required toconstruct the desired operation as a product, this geometrical approach amountsto a more concrete and computable definition, but its evaluation is nontrivialin systems with a high-dimensional Hilbert space. The geometrical formulationcan more easily be evaluated by considering the geometry associated with asuitable finite-dimensional group generated by a small number of relevantoperators of the system. In this way, the method has been applied in particularto the harmonic oscillator, which is also of interest in the present paper.However, subtle and previously unrecognized issues of group theory can lead tounforeseen complications, motivating a new formulation that remains on thelevel of the underlying Lie algebras for most of the required steps. Novelinsights about complexity can thereby be found in a low-dimensional setting,with the potential of systematic extensions to higher dimensions as well asinteractions. Specific examples include the quantum complexity of varioustarget unitary operators associated with a harmonic oscillator, invertedharmonic oscillator, and coupled harmonic oscillators. The generality of thisapproach is demonstrated by an application to an anharmonic oscillator with acubic term.

Geometric quantum complexity of bosonic oscillator systems