Reducing the dependence on L in the second prior quantum SCO algorithm

Determine whether the second quantum algorithm of Sidford and Zhang (2024) for quantum stochastic convex optimization—whose query complexity is currently stated as \~O(d^{5/8} ((L + sigma_V) R / epsilon)^{3/2})—can be modified to achieve a similar reduction in query complexity as their first algorithm (which can be reduced to \~O(d^{3/2} sigma_V R / epsilon)), thereby removing the explicit dependence on L in the second algorithm’s bound.

Background

The authors present a new quantum algorithm for SCO under a quantum variance-bounded SGO (QVSGO) achieving ~O(d R sigma_V / epsilon) queries. They compare this to two prior quantum algorithms of Sidford and Zhang (2024) with complexities ~O(d^{3/2}(L+sigma_V)R/epsilon) and ~O(d^{5/8}((L+sigma_V)R/epsilon)^{3/2}).

They note that the first prior algorithm’s dependence on L can be removed (reduced to depend only on sigma_V) with minor modifications, but they explicitly state uncertainty about whether a similar reduction is possible for the second prior algorithm.