Computationally Hidden Flipped Set conjecture

Establish that, for a suitable modification algorithm transforming a uniform oracle into a (1-delta)-dense oracle, every T-query quantum algorithm outputting a coordinate i hits a coordinate flipped by the modification with probability at most C T^a delta^b, for absolute constants C, a >= 0, and b > 0.

Background

The paper constructs couplings that transform dense oracle distributions into uniform ones while changing each coordinate with small average probability. Adaptivity creates a difficulty because later query weights can correlate with the coordinates changed by the coupling.

The CHFS conjecture formalizes the required computational hiding property: even after querying the oracle, an efficient quantum algorithm should not be able to identify a coordinate likely to belong to the flipped set. The paper proves weaker variants sufficient for classical preprocessing and bounded quantum preprocessing, while presenting the full conjecture as a route toward the general dense indistinguishability and simulation conjectures.

References

The CHFS conjecture says that, for every efficient quantum algorithm $\mathcal{A}$ that outputs a coordinate $i \in [N]$, the probability that $i \in \mathrm{Flip}_{O, \mathcal{M}}$ is at most $C \cdot Ta \cdot \deltab$ for some absolute constants $C, a \geq 0, b > 0$ (note the constants here are not necessarily the same as those in the dense indistinguishability conjecture).

— Parallel Quantum Advantage with Limited Adaptivity Requires Structure  (2608.20297 - Liu et al., 20 Aug 2026) in Section 5, Computationally Hidden Flipped Set Conjecture