Optimal dependence on additional normalization and re-encoding costs

Determine the optimal dependence of quantum query complexity on additional normalization and re-encoding costs for general Riccati-equation inputs beyond the explicitly constructed oracle families.

Background

The paper establishes product-form query lower bounds for CARE, DARE, DRE, and finite Riccati recursion on explicitly specified oracle families. These bounds capture the multiplicative effects of generalized spectral singularity and either algebraic output normalization or dynamic initialization conditioning under fixed output conventions.

The analysis also notes that general implementations may incur additional costs from normalization changes and re-encoding steps. The optimal dependence of query complexity on these extra costs for arbitrary Riccati inputs is not resolved by the stated lower bounds or upper bounds, leaving a concrete complexity-theoretic problem.