Linear-depth implementation without an auxiliary qubit

Improve the WDB-CG circuit for arbitrary spin-tree state preparation to achieve at least linear depth and eliminate its auxiliary qubit.

Background

The paper extends the restricted weight-distribution block (WDB) circuit to a full WDB unitary by combining ordinary and negative-logic WDB modules controlled by an auxiliary qubit. This construction enables preparation of general spin-tree states through WDB-CG circuits, but the auxiliary control prevents the parallelization available for the restricted WDB circuit.

As a result, the proposed WDB-CG implementation has depth of order n squared rather than the linear depth achieved in the restricted case. The authors explicitly ask whether the circuit can be improved to linear depth and whether the auxiliary qubit can be removed, regarding these goals as potentially related.

References

As a result, our circuit can only be implemented in depth ${\mathcal O}(n2)$. Could we improve such a circuit to at least linear depth? And, could we get rid of the auxiliary qubit? The two questions could be related. We would like to investigate them in the future.

— Deterministic Preparation of Arbitrary Spin Eigenfunctions  (2608.22892 - Tao et al., 24 Aug 2026) in Section 4, subsection “WDB-CG Circuits”; reiterated in Section 7, “Summary and Outlook”

Could we adjust our WDB-CG circuits to solve QMC and/or EPR on this hierarchy of graphs, at least approximately? This sounds like a very promising direction to pursue.

— Deterministic Preparation of Arbitrary Spin Eigenfunctions  (2608.22892 - Tao et al., 24 Aug 2026) in Section 7, “Summary and Outlook”

Could we collect all the spin eigenstates produced from branching paths, and achieve the whole Schur transformation? We believe that this could be done in principle. In fact, the quantum algorithm proposed in for the Schur transformation implicitly used branching paths. Could we integrate the concrete circuits in this paper into the procedure in, and get a more transparent and practical realization of the Schur transformation?

— Deterministic Preparation of Arbitrary Spin Eigenfunctions  (2608.22892 - Tao et al., 24 Aug 2026) in Section 7, “Summary and Outlook”