Variance of the sum of independent quantum computing errors

TL;DR

This paper derives a formula for the variance of the sum of two independent quantum computing errors, providing insights into error estimation and control in quantum computing systems.

Contribution

It introduces a specific variance formula for the sum of independent quantum errors and proves it for isotropic errors, suggesting broader applicability.

Findings

Variance of sum: V(X1+X2)=V(X1)+V(X2)-V(X1)V(X2)/2

Proven for independent isotropic errors

Conjectured to hold for general errors

Abstract

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.