Efficient implementation of the rank-support POVM

Determine whether the POVM that tests whether a tensor power of a stabilizer state lies in the span of tensor powers of rank-deficient stabilizer states can be implemented efficiently.

Background

The learning algorithm uses a reversible affine-rank test to reject stabilizer states whose computational-basis support has dimension less than n. The authors note that a seemingly more direct POVM could instead test membership in the span of tensor powers of rank-deficient stabilizer states, but they do not establish an efficient implementation for that measurement.

References

It might seem simpler to instead employ the POVM which tests if $\ket{S}{\otimes t}$ lies within the span of the $t$-fold tensor powers of the stabilizer states of rank less than $n$; it is however unclear if that POVM can be implemented efficiently.

— Sample-optimal learning of stabilizer states  (2609.10974 - Chang et al., 10 Sep 2026) in Section 2.1, footnote following the discussion of the affine-rank test