Optimality of logarithmic-error complexity for contractive linear differential equations

Determine whether the O(λT log(1/ϵ)) query complexity for block-encoding the final-time propagator e^{-HT} in the contractive case is optimal.

Background

The paper reviews quantum algorithms for solving linear differential equations and notes that, in the semi-contractive case, prior work obtained an O(λT log(1/ϵ)) query bound for block-encoding the propagator e{-HT}. The paper explicitly reports that the optimality of this dependence had only been conjectured, rather than established by a matching lower bound.

The MQSP results provide general upper bounds for arbitrary time-dependent nonnormal matrices, but the quoted discussion does not establish a lower bound proving optimality of the contractive-case propagator complexity. Consequently, whether the logarithmic precision dependence and overall scaling are optimal remains unresolved in the stated context.

References

This was improved using different kernel functions [ACL26, JLM+25] to block-encode the time-evolution Be[e−HT ]using O(λT log 1ϵ ) queries [LS25] to the block-encoding ofH, which was conjectured to be optimal.

— Multivariate Quantum Signal Processing  (2610.01125 - Low, 1 Oct 2026) in Section 1.2.3, discussion of prior work on linear differential equations

The approximate construction has the stated logarithmic upper bound, with no matching logarithmic lower bound asserted here.

— Multivariate Quantum Signal Processing  (2610.01125 - Low, 1 Oct 2026) in Section 6.1.2, discussion following the weighted SOS phase-amplification construction