Partial Stabilizer Learning under Fixed Commuting Constraints and the Absence of a Copy-Rate Discount

Abstract

We consider an unknown pure stabilizer state of \(n\) qudits with a fixed prime local dimension \(p\), together with prescribed commuting Pauli observables. Rather than learning the whole state, the learner must recover only complementary stabilizers that commute with the prescribed observables, together with their eigenvalues. Once the prescribed measurement is performed, this information and the observed outcome determine the corresponding conditional stabilizer state. We compare the number \(k\) of copies available to the learner with the number \(n\) of qudits and refer to their ratio \(k/n\) as the copy rate. Suppose that the dimension \(m\) of the requested complementary stabilizer subspace satisfies \(m=βn+o(n)\) for a fixed \(0<β\leq1\). If the copy rate converges to a value below one, the optimal exact recovery probabilities and verification scores converge to zero, both in the worst case and under the uniform prior. Conversely, if the number \(k\) of copies minus the number \(n\) of qudits tends to positive infinity, the entire stabilizer state can be identified with probability tending to one, and the requested complementary information can then be extracted. Thus, restricting the learning target to the stabilizer information needed after the prescribed measurement yields no copy-rate discount.