Discrete quantum computation and Lagrange's four-square theorem

García López De Lacalle, Jesús and Gatti Dorpich, Laura Nina (2019). Discrete quantum computation and Lagrange's four-square theorem. "Quantum Information Processing", v. 19 (n. 1); pp. 1-20. ISSN 1573-1332. https://doi.org/10.1007/s11128-019-2528-7.

Description

Title: Discrete quantum computation and Lagrange's four-square theorem
Author/s:
  • García López De Lacalle, Jesús
  • Gatti Dorpich, Laura Nina
Item Type: Article
Título de Revista/Publicación: Quantum Information Processing
Date: 5 December 2019
ISSN: 1573-1332
Volume: 19
Subjects:
Freetext Keywords: Discrete quantum states; p-orthonormal basis extension theorem; Systems of p-orthonormal vectors; Orthogonal lattices
Faculty: E.T.S.I. de Sistemas Informáticos (UPM)
Department: Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones
Creative Commons Licenses: Recognition - No derivative works - Non commercial

Full text

[img]
Preview
PDF - Requires a PDF viewer, such as GSview, Xpdf or Adobe Acrobat Reader
Download (370kB) | Preview

Abstract

We study a problem that arises naturally in the discrete quantum computation model introduced in Gatti and Lacalle (Quantum Inf Process 17:192, 2018). Given an orthonormal system of discrete quantum states of level k(k ∈N ) , can this system be extended to an orthonormal basis of discrete quantum states of the same level? This question turns out to be a difficult problem in number theory with very deep implications. In this article, we focus on the simplest version of the problem, 2-qubit systems with integers (instead of Gaussian integers) as coordinates, but with normalization factor √{p }(p ∈N∗) , instead of √{2k}, being p a prime number. With these simplifications, we prove the following orthogonal version of Lagrange's four-square theorem: Given a prime number p and v1,⋯,vk∈Z4 , 1 ≤k ≤3 , such that ‖ vi‖2=p for all 1 ≤i ≤k and ⟨vi|vj⟩=0 for all 1 ≤i <j ≤k , then there exists a vector v =(x1,x2,x3,x4) ∈Z4 such that ⟨vi|v ⟩=0 for all 1 ≤i ≤k and ‖v‖ 2=x12+x22+x32+x42=p . This means that, in Z4, any system of orthogonal vectors of norm p can be completed to a basis. Besides, we conjecture that the result holds for every integer norm p ≥1 and for every space Zn where n ≡0 mod4 , and that the initial question has a positive answer.

More information

Item ID: 64358
DC Identifier: http://oa.upm.es/64358/
OAI Identifier: oai:oa.upm.es:64358
DOI: 10.1007/s11128-019-2528-7
Official URL: https://link.springer.com/article/10.1007%2Fs11128-019-2528-7
Deposited by: Memoria Investigacion
Deposited on: 26 Jan 2021 07:48
Last Modified: 26 Jan 2021 07:48
  • Logo InvestigaM (UPM)
  • Logo GEOUP4
  • Logo Open Access
  • Open Access
  • Logo Sherpa/Romeo
    Check whether the anglo-saxon journal in which you have published an article allows you to also publish it under open access.
  • Logo Dulcinea
    Check whether the spanish journal in which you have published an article allows you to also publish it under open access.
  • Logo de Recolecta
  • Logo del Observatorio I+D+i UPM
  • Logo de OpenCourseWare UPM