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- Title
Discrete quantum computation and Lagrange's four-square theorem.
- Authors
Lacalle, J.; Gatti, L. N.
- 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 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 , ⋯ , v k ∈ Z 4 , 1 ≤ k ≤ 3 , such that ‖ v i ‖ 2 = p for all 1 ≤ i ≤ k and ⟨ v i | v j ⟩ = 0 for all 1 ≤ i < j ≤ k , then there exists a vector v = (x 1 , x 2 , x 3 , x 4) ∈ Z 4 such that ⟨ v i | v ⟩ = 0 for all 1 ≤ i ≤ k and ‖ v ‖ 2 = x 1 2 + x 2 2 + x 3 2 + x 4 2 = p. This means that, in 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 ≥ 1 and for every space Z n where n ≡ 0 mod 4 , and that the initial question has a positive answer.
- Subjects
QUANTUM computing; PRIME number theorem; GAUSSIAN integers; QUANTUM states; PRIME numbers
- Publication
Quantum Information Processing, 2020, Vol 19, Issue 1, p1
- ISSN
1570-0755
- Publication type
Article
- DOI
10.1007/s11128-019-2528-7