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GapP-measure/dimension of BQP/qpoly

Determine the GapP-measure and resource-bounded dimension (e.g., GapP-strong dimension) of the nonuniform quantum class BQP/qpoly; in particular, establish whether mu_{GapP}(BQP/qpoly)=0 or provide an exact dimension characterization.

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Background

The paper shows BQP/poly has GapP-strong dimension 0 via acceptance-probability martingales. Extending these methods to quantum advice (qpoly) is nontrivial and not resolved.

A classification for BQP/qpoly under GapP-measure or dimension would illuminate the power of quantum advice in the resource-bounded measure framework.

References

Another interesting direction is determining the measure or dimension of $BQP/qpoly$. While we know $BQP/poly$ has $GapP$-strong dimension 0, extending this to quantum advice remains open.

— Counting Martingales for Measure and Dimension in Complexity Classes (2508.07619 - Hitchcock et al., 11 Aug 2025) in Section 6.2, Quantum Circuit Complexity