arXiv subjects

Jesús Lacalle

Publications and source records attributed to Jesús Lacalle.

3 recordsLinked to original sources

arXiv2026

Effect of isotropic errors on the complexity of Grover's algorithm

Isotropic errors have been shown to be immune to conventional error correction techniques. While general theoretical frameworks have been proposed to model such errors, there have been no studies so far analysing their concrete impact on practical use-cases. Here we explore the effect of isotropic errors on the complexity of Grover's search algorithm through numerical simulations, with an analysis of the impact on the algorithm's performance and success probability. The results provide insights into the robustness of Grover's algorithm against isotropic errors, highlighting potential challenges for implementations on noisy quantum hardware. All results presented here are obtained through numerical simulations using the open-source python library \texttt{isotropic} developed as part of this work. The source code, numerical simulations and documentation for the library are available online at https://www.github.com/lazyoracle/isotropic-error-analysis .

quant-ph↗

arXiv2024

Variance of the sum of independent quantum computing errors

The sum of quantum computing errors is the key element both for the estimation and control of errors in quantum computing and for its statistical study. In this article we analyze the sum of two independent quantum computing errors, $X_1$ and $X_2$, and we obtain the formula of the variance of the sum of these errors: $$ V(X_1+X_2)=V(X_1)+V(X_2)-\frac{V(X_1)V(X_2)}{2}. $$ We conjecture that this result holds true for general quantum computing errors and we prove the formula for independent isotropic quantum computing errors.

quant-ph↗

arXiv2018

Extended Lagrange's four-square theorem

Lagrange's four-square theorem states that every natural number $n$ can be represented as the sum of four integer squares: $n=x_1^2+x_2^2+x_3^2+x_4^2$. Ramanujan generalized Lagrange's result by providing, up to equivalence, all $54$ quadratic forms $ax_1^2+bx_2^2+cx_3^2+dx_4^2$ that represent all positive integers. In this article, we prove the following extension of Lagrange's theorem: given a prime number $p$ and $v_1\in Z^4$, $\dots$, $v_k\in Z^4$, $1\leq k\leq 3$, such that $\|v_i\|^2=p$ for all $1\leq i\leq k$ and $\langle v_i|v_j\rangle=0$ for all $1\leq i<j\leq k$, then there exists $v=(x_1,x_2,x_3,x_4)\in Z^4$ such that $\langle v_i|v\rangle=0$ for all $1\leq i\leq k$ 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 base. We conjecture that the result holds for every norm $p\geq 1$. The problem comes up from the study of a discrete quantum computing model in which the qubits have Gaussian integers as coordinates, except for a normalization factor $\sqrt{2^{-k}}$.

math.NT↗