Discrete quantum computation and Lagrange's four-square theorem

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.