Implementacija kvantnih algoritama

master's thesis

• Title (english): Implementation of quantum algorithms
• Author: Matej Kovačević
• Mentor: Hrvoje Buljan (mentor)
• Committee member: Hrvoje Buljan (predsjednik povjerenstva)
• Committee member: Davor Horvatić (član povjerenstva)
• Committee member: Marko Tomislav Cvitaš (član povjerenstva)
• Committee member: Tamara Nikšić (član povjerenstva)
• Committee member: Petar Žugec (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Physics); Zagreb
• Defense date and country: 2024-11-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Physics

Abstract

Kvantno računanje prvo se pojavilo s idejom lakšeg simuliranja prirodnih kvantnih procesa za što treba ogromna količina klasičnih resursa. Od početaka do danas to područje je formalizirano u vidu kvantne teorije informacije i pokazalo se da postoje neki problemi koje bi takav pristup mogao riješiti efikasnije od bilo kojeg poznatog klasičnog algoritma. Najpoznatiji takav primjer je Shorov algoritam za faktoriziranje cijelih brojeva koji je privukao veliku pažnju javnosti zbog potencijalne primjene na razbijanje kriptografije. U ovom radu dan je teorijski uvod u kvantnu teoriju informacije i opisani su principi kojima kvantno računanje stječe eventualnu prednost nad klasičnim. Ukratko je dan osvrt na postojeća hardverska rješenja i na tehnike kojima se može poboljšati njihova pouzdanost. Objašnjen je Deutschev problem i pokazano je kako implementirati Deutschev algoritam formalizmom kvantnih krugova koristeći Qiskit paket u Pythonu. Detaljno je objašnjen Harrow - Hassidim - Lloyd (HHL) algoritam za rješavanje sustava linearnih jednadžbi i sve njegove subrutine koji je zatim primijenjen na konkretan linearni sustav dimenzija 2 × 2 i pokazana je implementacija svih kvantnih logičkih vrata potrebnih za to. Pripremljeni kvantni krug izvršen je uzorkovanjem, simuliranjem na klasičnom računalu i na stvarnom IBM-ovom kvantnom računalu. Usporedbom rezultata klasične simulacije i stvarnog mjerenja na kvantnom računalu dobivamo uvid u izazove koji stoje pred kvantnim računanjem prije nego ono postane pouzdana metoda računanja na velikim skalama. Klasičnom simulacijom pokazana je poanta HHL algoritma, a izvedbom na stvarnom kvantnom računalu vidljiva je eksperimentalna nesavršenost realnih qubita i njihova podložnost smetnjama i dekoherenciji.

Abstract (english)

Quantum computing first emerged with the idea of facilitating the simulation of natural quantum processes, which requires an immense amount of classical resources. From its beginnings until today, this field has been formalized as quantum information theory, and it has been shown that there are some problems that such an approach could solve more efficiently than any known classical algorithm. The most famous example of this is Shor’s algorithm for factoring integers, which attracted significant public attention due to its potential application in breaking cryptography. In this paper, a theoretical introduction to quantum information theory is provided, and the principles by which quantum computing could gain an advantage over classical computing are described. A brief overview is given of existing hardware solutions and techniques that can improve their reliability. Deustch’s problem is explained, along with a demonstration of how to implement Deutsch’s algorithm using the formalism of quantum circuits with the Qiskit package in Python. The Harrow–Hassidim–Lloyd (HHL) algorithm for solving systems of linear equations is explained in detail, including all its subroutines, and is then applied to a specific 2×2 linear system, with a step-by-step implementation of all the quantum logic gates needed. The prepared quantum circuit was executed through sampling, simulation on a classical computer, and on a real IBM quantum computer. By comparing the results from classical simulation and actual measurement on a quantum computer, we gain insight into the challenges that quantum computing faces before it can become a reliable method for large-scale computation. The classical simulation demonstrates the essence of the HHL algorithm, while the execution on a real quantum computer reveals the experimental imperfections of real qubits and their susceptibility to interference and decoherence.

• Keywords: kvantno računanje; qubit; superpozicija; isprepletenost; kvantna logička vrata; kvantni paralelizam; kvantni algoritam; kvantni krug
• Keywords (english): quantum computing; qubit; superposition; entanglement; quantum logic gates; quantum paralelism; quantum algorithm; quantum circuit
• Language: croatian
• URN:NBN: urn:nbn:hr:217:460998
• Study programme: Title: The University Integrated Undergraduate and Graduate Programme in Physics; Research Physics; specializations in: Research Physics; Course: Research Physics; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: sveučilišni magistar fizike (sveučilišni magistar fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-01-31 13:35:44