Variance of the sum of independent quantum computing errors
Jesús Lacalle, Luis Miguel Pozo Coronado
TL;DR
The paper addresses how independent quantum computing errors accumulate by modeling $n$-qubit states as unit-sphere points on $S^d$ and errors as random variables on $S^d$, introducing a variance-based measure of error size. It derives the exact variance of the sum of two independent errors, $V(X_1+X_2)=V(X_1)+V(X_2)-\frac{V(X_1)V(X_2)}{2}$, proving it for independent isotropic errors and proposing a general conjecture for arbitrary errors. The study extends to multiple errors, showing that with equal per-error variance $\sigma$, the accumulated variance is $V(E_1+\\cdots+E_k)=2-2(1-\\frac{\\sigma}{2})^k$, and discusses how the distribution of accumulated errors tends toward a uniform on $S^d$ with increasing $k$, implying potential information loss. The results offer a framework to analyze error accumulation in quantum computing and yield threshold-like insights relating per-step variance to overall feasibility, notably in conjunction with fidelity considerations and error-correction strategies.
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.