Discrete quantum computation and Lagrange’s four-square theorem

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 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.