Mathematical Foundations¶

This page captures the minimum canonical equations used across LiDMaS+ workflows. For extended derivations, see LiDMaS+ Math Notes.

Binary Field and Syndrome Map¶

All parity operations are over \( \mathbb{F}_2 \):

\[ a \oplus b = (a+b)\bmod 2 \]

For parity-check matrix \(H\) and error vector \(e\):

\[ s = H e \bmod 2 \]

Surface-code geometry used in paper_04 generators:

\[ n = 2d(d-1), \quad m_X = d^2, \quad m_Z = (d-1)^2 \]

With residual error \(e^{res}=e \oplus c\), logical event indicators are evaluated on canonical logical supports:

\[ \mathcal{L} \in \{0,1\} \]

and aggregated into logical-error statistics across trials.

Pauli mode:

\[ \Pr(e_{X,i}=1)=p,\quad \Pr(e_{Z,i}=1)=p \]

GKP displacement-to-bit mapping (simplified):

\[ \Delta q,\Delta p\sim \mathcal{N}(0,\sigma^2), \quad n_q=\mathrm{round}(\Delta q/\sqrt{\pi}), \quad n_p=\mathrm{round}(\Delta p/\sqrt{\pi}) \]

with parity extraction from integer bins.

For \(N\) trials and \(k\) failures:

\[ \widehat{\mathrm{LER}} = \frac{k}{N} \]

Confidence intervals are computed and exported (Wilson/bootstrap depending on workflow stage).

Finite-size scaling uses:

\[ x=(p-p_c)d^{1/\nu} \]

with fit objective over binned variance/collapse quality.