Structure-Fair Quantum Circuit Complexity: An Auditable Information-Theoretic Lower Bound

Abstract

Quantum circuit complexity is often used to characterize the physical cost of state preparation, but its physical meaning depends on the reference and counting rules; the entropy-removal costs of operations such as reset may be left out of resource accounting. We propose the principle of structural fairness and develop the Reference-Contingent Complexity (RCC) framework, jointly specifying the reference, generation capabilities, and atomic costs. We construct a model family that can approximate arbitrary finite-dimensional pure and mixed states. Within an admissible model fixed in advance, we prove a rigorous lower bound on universal optimal quantum circuit complexity. The target state's smooth one-shot information gap relative to the unbiased structured vacuum (the maximum-entropy state on the reference support) has an entropy-spectrum structure. Calibrated by the atomic control bandwidth and with finite-description corrections included, this gap sets a common cost floor for every admissible successful path. Predeclared final-state measurements and their finite-sample statistics thus yield independently verifiable one-sided complexity lower-bound certificates without reconstructing the generation history. Finally, exact structural allocation relations under changes of observation window and reference motivate a conjecture on the reference covariance of entropy and complexity: a reference can shift the complexity zero point, but cannot remove the burden of generating structure at no cost.