Quantum error mitigation on noisy devices is limited not only by residual bias but also by the shot noise and calibration errors introduced by the mitigation procedure itself. We derive finite-shot mean-square-error boundaries for probabilistic error cancellation (PEC), Clifford data regression (CDR), and no mitigation for noisy Pauli-observable estimates. Exact PEC removes the target bias under an exact noise inverse at the price of a quasi-probability variance overhead, whereas population linear CDR can have smaller target-shot variance but retains a calibration floor when the training and target noise responses do not match. This competition yields a finite CDR-dominant operating window whose upper endpoint scales as
BPEC=CDR(p)∝1/(δ12p), where
δ1 is the first-order CDR calibration mismatch. We further prove a target-response projection theorem showing that response-blind affine CDR removes the first-order bias only when the target noise response is affine in the ideal target value; otherwise a nonzero projection error gives an irreducible local calibration floor. The same mean-square-error formulation extends to second-order calibration, commuting Pauli Hamiltonians, finite CDR training shots, and residual PEC model bias. A closed-form two-qubit calculation and QAOA simulations support the predicted no-mitigation, CDR-dominant, and PEC-dominant regimes.
We present a canonical mathematical framework based on an open correlational configuration space equipped with admissible functionals, projection operators, and second-order functional response. The formalism is defined without assuming fundamental degrees of freedom, spacetime structure, or dynamical laws. A central role is played by the projected Hessian of correlational functionals, whose spectral properties determine stability, persistence, and the existence of effective descriptions. Structural complexity functionals are introduced to define minimality conditions, leading to a canonical structural condition that selects admissible regimes. Quantum, geometric, dissipative, and critical structures arise as distinct regions in a unified regime space characterized by spectral, projectional, and non-closure properties. Consistency constraints and no-go results are formulated, and reductions to standard quantum and geometric frameworks are obtained as limiting cases. The framework is mathematically closed at the structural level and admits further development only through explicit constructions and applications.
We introduce the Spectral Autodiff Kernel Expansion (SAKE), a differentiable computational framework for transporting nonlinear spectroscopic response between neighboring quantum dynamical models. Rather than recomputing multidimensional spectra independently for each Hamiltonian or Liouvillian, SAKE constructs local transport expansions about a reference model by combining forward-mode automatic differentiation with Duhamel transport theory. Automatic differentiation generates first-, second-, and third-order derivatives of the parameter-dependent Liouvillian, which are assembled into a pathway transport operator that maps the nonlinear response of a reference model onto neighboring systems. The framework is validated for a four-level excitonic dimer possessing an
su(2)×su(2) symmetry by comparing second- and third-order transported pathway operators with exact projected transport matrices obtained from direct calculations. The third-order expansion accurately reproduces the projected transport operator and its associated pathway mixing. Beyond providing an efficient computational strategy, the transport operator reveals how coherent and dissipative perturbations redistribute amplitude among double-sided Feynman pathways, exposing mechanistic information that is not directly apparent from the nonlinear spectrum. SAKE thereby establishes a differentiable computational framework for nonlinear spectroscopy that supports efficient local parameter exploration, sensitivity analysis, and future inverse-design applications.