Quantum advantage for spectral discrimination and related graph problems

Determine whether a quantum algorithm based on implementing the coined quantum walk operator U_G can achieve a genuine computational advantage over the polynomial-time classical algorithm for spectral discrimination and related graph problems.

Background

The paper implies a classical polynomial-time method for distinguishing strongly regular graphs of prime order using the quantum walk characteristic polynomial. It also notes that U_G can be implemented with O(n) quantum gates, suggesting practical quantum realizations.

The outstanding question is whether any related graph problems admit a provable quantum speedup over the available classical polynomial-time algorithms when leveraging quantum walk–based techniques.

References

Finally, since $U_G$ admits an implementation using $O(n)$ quantum gates, the spectral discrimination problem for prime-order SRGs lies within the reach of near-term quantum hardware; whether a genuine quantum advantage over the polynomial-time classical algorithm derived here is achievable for related graph problems remains an appealing open question at the interface of quantum computing and algebraic combinatorics.

— The Quantum Walk Characteristic Polynomial Distinguishes All Strongly Regular Graphs of Prime Orde  (2604.01507 - Roldan, 2 Apr 2026) in Section 7 (Concluding Remarks)