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 $$(k\in \mathbb {N})$$, 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 $$\sqrt{p}$$ $$(p\in \mathbb {N}^*)$$, instead of $$\sqrt{2^k}$$, 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 $$v_1,\dots ,v_k\in \mathbb {Z}^4$$, $$1\le k\le 3$$, such that $$\Vert v_i\Vert ^2=p$$ for all $$1\le i\le k$$ and $$\langle v_i|v_j\rangle =0$$ for all $$1\le i<j\le k$$, then there exists a vector $$v=(x_1,x_2,x_3,x_4)\in \mathbb {Z}^4$$ such that $$\langle v_i|v\rangle =0$$ for all $$1\le i\le k$$ and $$\begin{aligned} \Vert v\Vert ^2=x_1^2+x_2^2+x_3^2+x_4^2=p. \end{aligned}$$This means that, in $$\mathbb {Z}^4$$, 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 $$p\ge 1$$ and for every space $$\mathbb {Z}^n$$ where $$n\equiv 0\,\text {mod}\,4$$, and that the initial question has a positive answer.