Characterization of state complexity in adaptive quantum circuits

Preparing long-range entangled states is critical for quantum information processing and many-body physics, but typically requires impractically deep circuits. Recent progress on adaptive circuits, which incorporate mid-circuit measurements and classical feedback, shows that adaptivity can greatly reduce depth and even prepare arbitrary states in constant depth. This advantage, however, comes with a trade-off between depth and ancillary qubits, motivating the study of adaptive state complexity: the minimal depth and number of ancillas needed to prepare a given state. In this work, we investigate adaptive state complexity for generic mixed states and general circuit architectures. Our approach analyzes the growth of correlation range in adaptive circuits and shows that insufficient depth or ancilla resources strictly limit the formation of long-range correlations. This yields rigorous bounds on approximate state and gate complexity given the correlation value. We demonstrate the generality and applicability of our approach with representative examples, including permutation-invariant states, Gibbs states of many-body Hamiltonians, and multi-qubit gates like the Toffoli gate.