Write a PREreview

Decoherence and the exponential scaling of Hilbert space are fundamental challenges for scalable quantum technologies. This work introduces Difference-Based Variational Reconstruction (DVR), a novel function-space control paradigm that addresses these challenges by enabling feedback directly in the operator domain. By expanding the density matrix in a complete orthonormal operator basis, DVR represents the quantum system's state evolution through observable-mode amplitudes. We demonstrate that this DVR coefficient hierarchy provides a complete coordinate system for quantum states in terms of observable moments, including all orders of correlations. We rigorously prove that the evolution of these coefficients under a Lindblad master equation is equivalent to Heisenberg-picture expectation dynamics. Furthermore, we show that the time trajectory of these coefficients defines a path in observable-moment space, which can be derived from a variational principle involving a real-valued action functional: \[ S[c]=\int_0^t \sum_\alpha \left(\frac{dc_\alpha}{ds} - \sum_\beta M_{\alpha\beta} c_\beta(s)\right)^2 ds. \] This real-action path integral formulation offers a powerful description of dissipative quantum dynamics, unifying the Heisenberg picture and statistical-mechanical path integral approaches within a single operator-algebraic framework. This unification provides a profound physical interpretation of DVR, establishing it not merely as a numerical method, but as a coordinate system on the full quantum observable hierarchy, enabling direct control and diagnostics compatible with the inherent complexities of open quantum systems