EXACT ALGEBRAIC REDUCTION OF QUANTUM NEURAL CIRCUITS: OBSERVABLE-DEPENDENT RENORMALIZATION AND CURVATURE-BASED RELIABILITY CONTROL

Abstract

The computational cost and operational reliability of parameterized quantum circuits depend on which outputs an information system must preserve. This paper develops an observable-centred workflow linking quantum neural circuits, Lie and Krylov closure, exact algebraic reduction, and a proposed geometric reliability monitor. The dynamical model is a finite-dimensional Markovian quantum system with a declared set of target observables. We distinguish three objects that must not be conflated: the minimal linear observable space, a closed Pauli-coordinate space used for classical simulation, and an associative operator algebra supporting a completely positive trace-preserving reduced quantum model. A direct invariance argument establishes exact preservation of the declared outputs; an explicit sufficient construction specifies when this linear result also admits a physical quantum realization. A structured XX chain with local dephasing illustrates the reduction from an ambient operator space of dimension 4ⁿ to a Pauli-coordinate space of dimension n(2n−1), including 45 rather than 1024 coordinates for five qubits. A separately specified numerical check compares reduced and full evolution, while a generic interacting model demonstrates that the same coordinate-closure procedure need not compress. Quantum Fisher and Berry geometry are introduced with additional sufficiency and smoothness conditions rather than assumed to survive arbitrary output compression. An operational decomposition separates gauge, locally output-insensitive, and output-sensitive directions. Curvature along the output-sensitive directions motivates a candidate monitoring score and a selective regularizer, with an explicit distinction between first-order sensitivity and dynamical instability. The contribution is a technically delimited route from observable preservation to reliability assessment for quantum information-processing components. The numerical study tests only observable preservation and closure size; it does not validate the proposed reliability controller, establish attack detection, or demonstrate improved generalization. No numerical improvement is asserted for the curvature-based mechanism.