Streaming tests for ground states of translation-invariant Hamiltonians

Construct a streaming state-testing protocol for more natural highly entangled states, such as ground states of translation-invariant Hamiltonians, by producing a nearly translation-invariant staircase- or sequential-unitary operator \(H=\sum_{j=1}^m U_j\) with a gapped \(+1\) eigenstate and memory complexity \(O(\log m)\).

Background

The paper develops a one-pass streaming protocol for testing maximally entangled states and certain states obtained from them by constant-bond-dimension matrix product operators. It identifies a broader unresolved direction: testing ground states of translation-invariant Hamiltonians. The proposed route is to construct a sum of nearly translation-invariant staircase circuits with a distinguished gapped eigenstate, which would enable a logarithmic-memory streaming algorithm.

References

Another open question concerns further generalization of the streaming protocol for state testing. It would be interesting to test for more natural states like ground states of translation-invariant Hamiltonians.

— Depth-1 expanders on the unitary group and applications  (2609.01605 - Anshu et al., 1 Sep 2026) in Section 1, Introduction, paragraph “Outlook”