Perturbative Stability of Topologically Ordered Systems with Local Defects

master's thesis

• Title (croatian): Perturbativna stabilnost topološki uređenih sustava s lokalnim defektima
• Author: Ivana Kurečić
• Mentor: Tobias Osborne (mentor)
• Mentor: Davor Horvatić (mentor)
• Committee member: Davor Horvatić (predsjednik povjerenstva)
• Committee member: Mihael Makek (član povjerenstva)
• Committee member: Ivica Smolić (član povjerenstva)
• Committee member: Nikola Poljak (član povjerenstva)
• Committee member: Tomislav Marketin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Physics); Zagreb
• Defense date and country: 2016-09-09, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Physics

Abstract

Quantum spin systems have been providing modern physics with ever new challenges to tackle in the past several decades; from the theoretical postulation of topological quantum order, a new chapter was revealed in the understanding of phase transitions in quantum systems. The novel properties of topologically ordered systems had caught the interest of quantum information scientists, and research into their potential in the field of quantum computing soon began. Nowadays, this utility of topologically ordered systems is thoroughly researched in the field of topological quantum computing. Systems that exhibit topological order present a valuable resource for quantum computing – by use of the global observables that remain robust when such a system is acted on by a local perturbation. Under these conditions, the information stored in the system remains stable, providing a unique modus of circumventing the problems that plague standard quantum computing schemes. To improve on the applicability of such systems in fault-tolerant quantum computation, there exists a need to understand the exact behavior of the encoded information under perturbation. To this end, this thesis presents critical mathematical methods used to research the dynamics of local quantum systems — the quasi-adiabatic continuation, the Lieb-Robinson bounds, and the self-energy expansion—and utilizes them to further the understanding of the effect of weak local perturbations on the information stored in a topologically ordered system. In particular, the influence of perturbation on topologically ordered systems with local puncture defects is examined, and general bounds on the stability of topological order in such systems are derived.

Abstract (croatian)

Kvantni spinski sustavi donose uvijek nove izazove u modernoj fizici proteklih nekoliko desetljeća; teorijski opis topološkog kvantnog uređenja je otkrio novo poglavlje u razumijevanju faznih prijelaza u kvantnim sustavima. Novoopažena svojstva topološki uređenih sustava zaokupila su pozornost i znanstvenika iz polja kvantne teorije informacija, čime je započelo istraživanje njihovoga potencijala u kvantnom računarstvu. Od nedavno, korisnost topološki uređenih sustava detaljno se istražuje u području topološkog kvantnog računarstva. Sustavi u kojima se pojavljuje topološko uređenje imaju vrijedne mogućnosti za kvantno računarstvo – korištenjem globalnih opservabli koje ostaju robustne kada se na sustav djeluje lokalnom perturbacijom. Pod ovim uvjetima, informacije pohranjene u sustavu ostaju stabilne, što pruža jedinstven način zaobilaženja problema koji zaokupljaju standardne sheme korištene u kvantnom računarstvu. U svrhu bolje primjenjivosti ovih sustava u kvantnom računarstvu otpornom na pogreške, potrebno je razumjeti točno ponašanje pohranjenih informacija pod utjecajem vanjske perturbacije. Ovaj rad predstavlja važne matematičke metode koje se koriste pri istraživanju dinamike lokalnih kvantnih sustava — kvazi-adijabatsko produljenje, Lieb-Robinsonove granice, i razvoj vlastite energije — te prikazuje njihovu primjenu pri unaprjeđivanju razumijevanja utjecaja slabih lokalnih perturbacija na informacije pohranjene u topološki uređenim sustavima. Specifično, istražen je utjecaj perturbacije na topološki uređene sustave s lokalnim defektima tipa rupe te su izvedene opće granice za stabilnost topološkog uređenja u takvim sustavima.

• Keywords: quantum spin system; topologically ordered system
• Keywords (croatian): kvantni spinski sustav; topološki uređenih sustav
• Language: english
• URN:NBN: urn:nbn:hr:217:414231
• Study programme: Title: Physics; specializations in: Research; Course: Research; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra fizike (magistar/magistra fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-04-04 19:18:05