Full optimization of the memory-restricted N-replica Pauli dual program

Determine whether optimizing the full N-replica dual program for memory-restricted Pauli shadow tomography yields a stronger lower bound than the restricted feasible-point ansatz used in the paper.

Background

The supplemental analysis derives a lower bound for memory-restricted estimation of the full Pauli family by restricting the dual variables to a particular linear span of Pauli-monomial commutants. This restriction produces an exponential obstruction but does not match the known tight memory-free sample-complexity scaling.

The authors note that the loss may result from the restricted ansatz and leave unresolved whether solving the complete N-replica dual program would improve the lower bound.

References

The loss arises from the restricted feasible-point ansatz in Eq.~eq:feasible_Gamma and Eq.~eq:feasible_Z and optimizing the full $N$-replica dual program in Eq.~eq:n-replica-Pauli-dual may yield a stronger bound, which we leave for future work.

— Pauli-resolved virtual distillation  (2609.24132 - Chen et al., 21 Sep 2026) in Supplemental Material, Appendix D, subsection “The N-replica memory restricted strategy”