Unconditional query lower bounds for orbit state preparation

Establish an unconditional query lower bound for orbit state preparation when the moves are provided only through oracle access, without relying on an unresolved query lower bound for orbit membership.

Background

The paper's only general lower bound reduces orbit equivalence to orbit state preparation, but this reduction is vacuous when the invariant already separates orbits. In a black-box model of the moves, prior work gives a conditional lower bound based on a query lower bound for orbit membership.

The paper identifies two unresolved issues: the underlying orbit-membership query lower bound remains open, and no unconditional query lower bound for orbit state preparation is known in the oracle-access model. These limit the current understanding of the intrinsic complexity of coherent orbit-state generation.

References

Lutomirski's lower bound in a black-box model of the moves\S6 is conditional on a query lower bound for orbit membership that is itself open\S6. With the moves given as an oracle, no unconditional query lower bound for orbit state preparation is known.

— Path-Finding, Orbit State Preparation, and the Security of Invariant Quantum Money  (2609.39774 - Schmiedel et al., 30 Sep 2026) in Section 10, Conclusion, paragraph “Lower bounds for orbit state preparation”