Quantum Telescoping Part IX: Telescoping for Inverse Problems (Parameter Recovery from Quantum Channels)

This paper develops a telescoping framework for inverse problems in quantum dynamics, focusingon parameter recovery from quantum channels. We study families of channels Φ (θ) θ∈Θgenerated by unknown Hamiltonian or Lindbladian parameters and analyze how telescopingrefinement controls the resolution of parameter estimation. We show that telescoping orderbounds the achievable rate of inverse convergence and establish lower bounds linking parameteridentifiability to channel distinguishability. In particular, we prove that exponential telescopingof forward simulation is necessary to achieve logarithmic resolution in inverse problems, whilepower-law telescoping implies polynomial sample and query complexity. These results providefundamental limits on parameter estimation accuracy as a function of computational resources, unifying quantum simulation, quantum metrology, and inverse problems under a single telescopingperspective. We establish concrete complexity bounds for Hamiltonian learning andLindbladian tomography, showing that the latter is intrinsically harder due to the absence ofexponential telescoping in dissipative dynamics. Numerical simulations validate the predictedscaling laws for both power-law and exponential telescoping regimes.