Quantum Telescoping Part III: Query Complexity and Lower Bounds

This paper establishes structural lower bounds on query complexity and refinement cost forquantum telescoping schemes. Building on the channel-level telescoping framework developed inParts I and II, we show that the decay rate of telescoping increments directly constrains the minimumnumber of oracle queries required to achieve a given simulation accuracy within the class ofrefinement-structured algorithms. We prove general lower bounds relating power-law telescopingorder to polynomial query complexity, and show that exponential telescoping is necessaryto achieve logarithmic dependence on the error tolerance. These results explain, from a structuralperspective, why quantum signal processing (QSP) and related eigenvalue-transformationmethods attain near-optimal ε-dependence in standard block-encoding models [3, 7, 8], whileproduct-formula and randomized product-formula methods remain algebraically convergent inε [10, 11]. Our analysis applies to unitary channels, CPTP maps, and block-encoded oraclemodels, and provides a unifying framework relating algorithmic structure to query complexity.We clarify the relationship between our results and known information-theoretic lower bounds,positing our theorems as structural constraints on algorithmic approaches rather than fundamentaloracle lower bounds.