Spectral Arithmetic Process Tomography

Quantum process tomography (QPT) is now the central scaling bottleneck in quantum hardware development. Standard QPT requires O(d⁴ⁿ) measurements for an n-qubit system, making full characterisation beyond 2–3 qubits experimentally infeasible with existing methods. As hardware scales toward and beyond the 105-qubit Willow generation, this bottleneck becomes the defining engineering problem. This paper develops a spectral arithmetic approach to process tomography that breaks the exponential barrier by construction. Rather than estimating a dense Liouville superoperator, a physical error channel is assigned a fixed generator Ω and the noisy gate is tracked on the invariant arithmetic orbit G(α) = exp(αΩ)G₀. The calibration variable is the scalar α, not a d² × d² matrix. Broadband drift is decomposed into orthogonal spectral channels, each evolved by a scalar recursion and stored by Hilbert-space projection. The paper proves: one-dimensional tangent structure; exact composition when [Ω,G₀]=0; BCH-controlled non-commuting residuals (detecting non-Markovian noise without expanding the dense state); gauge-coordinate invariance; covariance separation under orthogonal projections; scalar boundedness; Parseval information conservation; KMS thermal equilibrium of the scalar Kalman filter; and explicit complexity reduction from O(q²d⁸) to O(MK) memory. Connections to the Spectral Computing Programme are established throughout: the arithmetic orbit is the engineering instantiation of the Arithmetic Cylinder; BCH residuals are the Defect Operator Ξ; the physical concentration metric C_phys is the non-birefringence condition of the arithmetic vacuum; and the Kalman steady state is the thermodynamic phase transition at σ = ½. The engineering claim is accepted only where the corresponding calculation is shown. All remaining empirical tasks are explicitly identified.