Extend block-encoding lower bounds to adaptive circuits and molecular Hamiltonians

Determine whether the established T-count lower bounds for block encodings of general second-quantized and Kitaev honeycomb Hamiltonians extend to adaptive Clifford+T circuits with mid-circuit measurements and classical feed-forward, and whether the second-quantized lower-bound scaling applies to physically constrained molecular Hamiltonians.

Background

The paper establishes optimal worst-case T-count scaling for block encodings of a broad algebraic family of bounded-coefficient second-quantized Hamiltonians and for bond-dependent Kitaev honeycomb Hamiltonians, within a unitary Clifford+T model that excludes mid-circuit measurements and classical feed-forward. The authors explicitly identify adaptive circuits as outside the scope of their lower bounds.

The second-quantized lower bound relies on the full Θ(n^4)-dimensional coefficient body of the algebraic family. Molecular Hamiltonians generally satisfy additional structural relations among their coefficients, so it is unresolved whether those physically constrained families obey the same asymptotic lower-bound scaling or admit substantially cheaper block encodings.