Measurement-dependent reduction of open dynamics in the quantum phase estimation algorithm: the visible subspace and the damped Fejér kernel
For the quantum phase estimation (QPE) algorithm we study the minimal space of observables that is sufficient to predict the entire output distribution under open dynamics of the counting register. Building on the general Krylov theory of reduction, we compute the structure of this space explicitly for the full Fourier measurement. The outcome probabilities are determined by cyclic sums of the state matrix. If the adjoint generator is diagonal on the matrix units, the spectral classes inside the residue classes of the index difference modulo the register dimension $M$ give a constructive basis, an exact dimension formula, and attainable bounds: the dimension of the visible space lies between $M$ and $M^2-M+1$. The class includes independent qubit dephasing, and an analytically solvable reference model of Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) type yields a damped Fejér kernel; two random-unitary channels with the same Ramsey contrast show that higher harmonics are distinguishable in the QPE distribution. For approximate reduction we use a finite-time Gramian and an a posteriori residual estimate, with a quadratic error order for a reducing unperturbed subspace and a compatible initial state. Numerical calculations check the consistency of the representations and the behaviour of the error under non-Toeplitz perturbations.