Exponential planted independent space conjecture

Establish that, for cubic polynomials over F₂ in n variables and every constant ε > 0, no algorithm running in time o(2ⁿ) can distinguish a uniformly random cubic polynomial from one drawn from the distribution of cubic polynomials with a planted h-dimensional independent space when h ≤ (1/2 − ε)n.

Background

A planted independent space is a linear subspace on which the cubic component of a polynomial vanishes. Knowledge of such a space enables substantially faster simulation of the associated degree-3 IQP circuit, whereas the proposed protocol requires the space to remain hidden from an adversary. The conjecture formalizes the claimed exponential hardness of distinguishing planted instances from uniformly random cubic polynomials below the n/2-dimensional threshold.

References

There is no o(2{n})-time algorithm that distinguishes a random cubic polynomial from a random cubic polynomial with a planted h-dimensional independent space, where h ≤ (1/2 − ε)n for any constant ε > 0.

— Verifiable quantum advantage based on polynomials with planted structures  (2609.39918 - Bläser et al., 30 Sep 2026) in Conjecture 1, Section 1, subsection “Quantum circuits from polynomials with planted independent spaces”

We note that in fact we believe the \PIS\ problem to be exponentially hard even for quantum computers. This is relevant to our envisioned application of planted independent spaces for classically certifiable randomness. We leave a more detailed analysis to future work, however.

— Verifiable quantum advantage based on polynomials with planted structures  (2609.39918 - Bläser et al., 30 Sep 2026) in Section 2, subsection “Statement of independent space problems”

One of the central open questions raised by our work is whether our approach can be extended to achieve polynomial-time verification of quantum advantage, that is, an exponential gap between verification and simulation runtimes.

— Verifiable quantum advantage based on polynomials with planted structures  (2609.39918 - Bläser et al., 30 Sep 2026) in Section 1, subsection “Discussion and outlook,” paragraph “Towards efficient verification”

We leave it to future work to fully assess the feasibility and security of this approach.

— Verifiable quantum advantage based on polynomials with planted structures  (2609.39918 - Bläser et al., 30 Sep 2026) in Section 1, subsection “Discussion and outlook,” paragraph “Towards efficient verification”