Lower bounds beyond the Knill–Laflamme regime

Determine whether an oracle-distillation protocol whose pre-query subspaces violate the Knill–Laflamme conditions can distill noisy oracles into ideal oracles with query complexity below the lower bounds established for protocols satisfying those conditions.

Background

The paper proves query-complexity lower bounds for oracle-distillation protocols that retain correctability of bounded-weight Z errors at every query, formalized through the Knill–Laflamme conditions on pre-query subspaces. These bounds apply broadly, including to Grover oracles and certain families of Boolean oracles, but they do not cover protocols that abandon this error-correction condition.

The unresolved issue is whether violating the pre-query Knill–Laflamme conditions could enable a fundamentally more query-efficient distillation strategy. Resolving this would clarify whether the paper’s lower bounds reflect an unconditional limitation of oracle distillation or only a limitation within the class of protocols whose intermediate states preserve the specified error-correction structure.

References

The first is whether a protocol whose pre-query subspaces violate the KLC~eq:KLC_prequery can distill noisy oracles into ideal ones with query complexity below the bounds above.

eq:KLC_prequery:

Pt,fZaZbPt,f=Ct,fa,bPt,f,a,b∈{0,1}n,∣a∣,∣b∣≤r,P_{t,f} Z^a Z^b P_{t,f} = C^{a,b}_{t,f} P_{t,f},\quad a,b\in \{0,1\}^n, |a|,|b|\leq r,

— Oracle Distillation  (2609.31596 - Shen et al., 25 Sep 2026) in Section 4, immediately following Theorem 4.2 in Section 4 (Optimality of Boolean Oracle Distillation)

The second is to prove an unconditional lower bound for a concrete distillation task, for example distilling the family of i.i.d.-depolarizing Boolean oracles $\tilde{\mathfrak{O}_F{\mathrm{iid}$ into the family of ideal Boolean oracles $\mathfrak{O}_F$ with precision $\epsilon$.

— Oracle Distillation  (2609.31596 - Shen et al., 25 Sep 2026) in Section 4, immediately following Theorem 4.2 in Section 4 (Optimality of Boolean Oracle Distillation)

It therefore remains open whether a better construction of query states can close the $2\alpha$ gap in the exponent in the $r=\lfloor \alpha n \rfloor$ regime.

— Oracle Distillation  (2609.31596 - Shen et al., 25 Sep 2026) in Section 10 (Summary and Outlook), first paragraph