Completeness of the symmetry-preserving A-gate ansatz

Establish whether the Gard et al. symmetry-preserving variational ansatz constructed from A gates spans the targeted particle-number-conserving Hilbert subspace when the number of A gates is equal to the binomial coefficient associated with the system size and target occupation number.

Background

The paper uses a variational ansatz developed by Gard et al. to prepare Schwinger-model states that respect a chosen particle-number symmetry. The ansatz is composed of two-qubit A gates and is intended to generate states within a fixed particle-number sector.

The authors state that the ansatz is conjectured to span the relevant symmetry-respecting Hilbert subspace when the number of A gates is sufficiently large, specifically when it equals the corresponding binomial coefficient. They do not establish this spanning property, so proving or disproving the conjecture is an unresolved question directly relevant to the expressibility and scalability of the VQE state-preparation method.

References

Thus, we can produce states which have half the total sites occupied, and implement a check that ensures after a noisy circuit the state is still in the correct parity sector. The ansatz is built up using iterations of a 2-qubit block of gates (referred to as an A gate), and is conjectured to span the targeted symmetry-respecting Hilbert subspace providing N m A gates are used.

— Symmetry Constrained Quantum Error Mitigation for the Schwinger Model  (2608.21103 - Tomlinson et al., 21 Aug 2026) in Section III.D, “Ansatz,” p. 5

Whether or not it holds, Corollary~\ref{cor:dominance} and Proposition~\ref{prop:ceiling} are unaffected: attainment would only sharpen the inclusion to an equality for that particular $E$, and Strategy~B covers $\Delta_{\mathrm{sym}}$ in either case.

— Q-Edge: Symmetry-Reduced Quantum Simulation of Structured Extreme Dependence  (2609.03706 - Zhang et al., 3 Sep 2026) in Appendix, Section 10.5, Remark 10.19 ("What the bound does and does not assert")

We do not establish that the restricted generator pools can represent every state in these subspaces.

— Classical Active-Space Hybrid Quantum Subspace Expansion (CASH-QSE): Quantum Corrections without Remeasuring the Classically Calculable Energy  (2609.08170 - Izmaylov, 8 Sep 2026) in Section 3.1, paragraph “Exact partition and variational representation”