Quantum Telescoping Part IV: Measurement-Efficient Verification of Telescoping Order

This paper addresses the problem of verifying telescoping order in quantum simulation usinglimited measurements. Building on the telescoping framework and lower bounds developed inParts I–III, we study how increment decay rates can be certified without full process tomography. We show that power-law and exponential telescoping can be distinguished via a witness-familyseparation test using Ntotal = O ( (log n - log (d²/δ) ) /ε²) measurements for d2 Pauli witness observables, n telescoping levels, error tolerance ε, and confidence 1−δ, where d = 2m is the dimension for mqubits. We further analyze three structural barriers: (i) the refinement cost axiom underlyingquery lower bounds, (ii) the apparent obstruction to exponential telescoping in open quantumsystems, and (iii) intrinsic limits imposed by time-dependent Hamiltonians. Our results clarifywhich aspects of telescoping behavior are observable, which are algorithmic, and which reflectfundamental physical constraints.