Z-only corrections in d-regular graphs (conjecture)

Prove the conjecture that, for every d-regular graph G, the local Pauli corrections required by the phase quantum walk (PQW) graph state distribution protocol are Z-type only at every node; equivalently, establish that the X-correction parity f_v, defined as the XOR of near-side outcomes over all edges incident to vertex v, satisfies f_v = ⊕_{e∋v} s_{near(e,v)} = 0 for all vertices v.

Background

Empirically and through specific proofs, the authors observe that certain cyclic and regular graphs (e.g., C4, C5, K4) require only Z corrections in the PQW protocol. This suggests a structural cancellation mechanism tied to vertex degree and graph regularity.

They conjecture a general principle: in any d-regular graph, X-type corrections cancel at every node, leaving only Z corrections. A proof—possibly via the cycle-space decomposition of the graph—would significantly simplify correction rules for broad classes of networks and solidify the protocol's generality.

References

The $Z$-only observation for $C_n$ and $K_4$ is conjectured to hold for all $d$-regular graphs via degree-parity cancellation but is not yet proved. Formally: \emph{for any $d$-regular graph $G$, the correction at every node is $Z$-only, i.e.\ $f_v = \bigoplus_{e\ni v} s_{\mathrm{near}(e,v)} = 0$ for all $v$.} A proof via the cycle-space decomposition of $G$ is the most immediate open problem.

— The Phase Quantum Walk: A Unified Framework for Graph State Distribution in Quantum Networks  (2604.02169 - Dutta, 2 Apr 2026) in Item (i), Open Problems, Section 9 (Discussion)