Measurement-dependent reduction of open dynamics in the quantum phase estimation algorithm: the visible subspace and the damped Fejér kernel

Abstract

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.

The paper asks which counting-register observables suffice to predict the full quantum phase estimation (QPE) output distribution under open dynamics before the inverse quantum Fourier transform. It specializes known observable-Krylov reduction to the Fourier measurement, seeking an exact linear model valid for every initial state and all times. The visible space is $\mathcal K=\operatorname{span}\{G^rE_k\}$, where $G$ is the adjoint generator. QPE probabilities are discrete Fourier transforms of cyclic sums of density-matrix entries. When $G$ is diagonal on matrix units, grouping eigenvalues within each residue class $j-\ell\bmod M$ gives an explicit basis and the exact dimension $\dim\mathcal K=\sum_r|\Lambda_r|$, with attainable bounds $M\leq\dim\mathcal K\leq M^2-M+1$. The proof uses Fourier decomposition and Vandermonde interpolation. A solvable GKLS dephasing model yields a damped Fejér kernel and dimensions $M$, $2M-2$, or $2M-1$, depending on damping and detuning. Two random-unitary channels with identical first-harmonic Ramsey response have different QPE distributions. For approximate reduction, a finite-time Gramian supplies optimal trajectory-projection subspaces. Duhamel estimates give probability-error bounds controlled by the invariance residual; errors are quadratic in perturbation strength when the unperturbed subspace is reducing and the initial state is compatible. Numerical checks reproduce the analytic model through $M=64$ and test non-Toeplitz perturbations at $M=8,16,32$. The dimension formulas concern universal linear observable models, not necessarily physically admissible reduced quantum dynamics. Closed-form results require matrix-unit diagonality; the numerical studies are deterministic consistency checks, and the uniform residual bounds can be substantially looser than state-specific errors.