Characterization of Hadamard-type coins among general two-state unitary coins

Investigate whether coins of Hadamard type can be characterized, up to global phase, diagonal unitary transformations, and changes of basis, by an exact-uniformity property for the positive sojourn-time distribution of a two-state quantum walk with a general 2×2 unitary coin.

Background

The paper proves that, within the one-parameter family of real rotation coins, exact uniformity of the return-conditioned positive sojourn-time distribution at time 8 is equivalent to the coin being the Hadamard coin. The authors note that this result does not cover arbitrary 2×2 unitary coin matrices.

The proposed extension must account for transformations that preserve the relevant distribution, namely multiplication by a global phase, diagonal unitary transformations, and changes of basis. The unresolved issue is whether an analogous characterization of Hadamard-type coins holds in the full class of two-state unitary coins.

References

As a future problem, it is natural to extend the result to general unitary coins with two internal states. In this paper, we treated the family of rotation coins. For a general coin given by a $2\times 2$ unitary matrix, it would be natural to investigate whether coins of Hadamard type can be characterized in a similar way, after taking account of the equivalences given by the global phase, diagonal unitary transformations, and changes of basis.

— Hadamard Rigidity of Positive Sojourn Time Distributions for Rotation Coins  (2609.05033 - Tamura et al., 4 Sep 2026) in Section “Conclusion and future problems”