Establish conditional computational hardness for scalar OPI

Establish, under standard computational complexity assumptions, a hardness result for scalar optimal polynomial intersection (scalar OPI) comparable to the oracle-based quantum advantage established for folded OPI.

Background

The paper proves a quantum–classical query-complexity separation for approximate optimization only in an oracle model based on balanced folded OPI and separately callable block-membership oracles. Scalar OPI, by contrast, is the standard formulation in which each acceptance test concerns a single polynomial evaluation.

The authors explicitly identify proving hardness for scalar OPI under standard computational assumptions as unresolved. Such a result would remove the reliance on an oracle and establish a comparable separation in a non-oracular computational setting.

References

The corresponding question of proving hardness for scalar OPI, based on standard computational assumptions, as studied in Refs., remains open.

— A provable quantum advantage for approximate optimization via decoded quantum interferometry  (2610.02145 - Kramer et al., 1 Oct 2026) in Section 1, Introduction; also Section 6, Discussion

Can polynomial-time postprocessing suffice with similar space and pass bounds? Alternatively, can one establish computational hardness?

— Exponential quantum advantages for decoded quantum interferometry in the streaming setting  (2610.01902 - Wu et al., 1 Oct 2026) in Section 1, subsection “Open problems and AI disclosure”