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Determine the P-dimension of BQP

Determine the polynomial-time dimension of BQP by establishing the exact value of dim_P(BQP) or proving bounds that resolve whether it is zero or positive.

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Background

Using acceptance-probability martingales based on GapP, the paper shows BQP has GapP-strong dimension 0. Nevertheless, the P-dimension of BQP is not known.

A resolution would bridge resource-bounded dimension theory with quantum complexity, indicating how classical time-bounded martingales interact with quantum decision classes.

References

For example, we show that $BPP$ has $$-dimension 0 and $BQP$ has $GapP$-dimension 0, whereas the $P$-dimensions of these classes remain open.

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