Fourier Sectors of Shifted Overlaps under a Compact Abelian Symmetry

A compact abelian group acts unitarily on a finite-dimensional space. This paper proves that the overlap of two states, one shifted by a group element, is a finite Fourier series whose coefficients are positive semi-definite kernels indexed by the isotypic components of the conjugation representation; the coefficient of the trivial character is the inner product of the twirled states, and the sum of the squared moduli of the other coefficients is the variance of the overlap over the group. For the circle the series is a trigonometric polynomial whose degree is bounded by the total charge, so it is determined exactly by its values at finitely many equally spaced shifts. The same decomposition is applied to a reset channel, a fixed unitary that absorbs encoded inputs one at a time into a hidden register and discards the input between steps. Its readout under a common shift of the inputs is a finite Fourier series whose zero mode is the readout of the jointly twirled input; this need not equal the readout of inputs twirled one at a time, and an explicit Clifford channel separates the two. The degree of the readout is at most the charge bound times the number of steps, and for every charge bound an explicit channel attains it. Every coefficient can be estimated from overlaps measured at finitely many shifts, with a number of runs quadratic in the inverse error and logarithmic in the number of coefficients, and a charge sector of an encoded state can be read at the amplitude level by exact phase estimation of the charge followed by a Hadamard test. Sector kernels of product encodings and Fourier sectors of Clifford channels with Pauli readout are shown to be classically computable in polynomial time. The paper claims no quantum advantage. A final section carries the Fourier argument to the signature of a piecewise-linear path, read as a vector in a Fock space, and decomposes the signature kernel under rotation of the increments into harmonics that are themselves kernels.