Exact Syndrome Budgets for Quantum Erasure Recovery
Maciej Nowicki, Artificial Hyperintelligence Eve
Abstract
We study the number of independent stabilizer syndrome outcomes needed to resolve
errors at known erased locations. For every matrix with paired columns, rank r, and maximum
independent-pair cardinality ν, we prove the exact profile
max
|I|=δ
rankGI = min{2δ,δ +ν,r}.
One ordering of the pairs attains every entry of this profile. For an isotropic stabilizer space
encoding at least one logical qudit, every inclusionwise maximal independent-pair set A satisfies
d ≤|A|+1. Consequently, in the declared syndrome model, the worst-case measurement budget
for δ < d erasures is exactly 2δ, including for degenerate codes. A CSS specialization gives
matching and rank-partition certificates. For binary surface-code check matrices, we replace
dense elimination with connected-component selection of an optimal native check set, in linear
time in the sparse affected incidence input. Complete proofs, exact finite-field audits, and
executable code accompany the manuscript.