Quantum Functional Information through the Evolution of Random Circuits

We introduce Quantum Functional Information (QFI), a new metric to quantify the rarity and utility of quantum states and circuits. Unlike standard measures such as fidelity or entropy, QFI captures the balance between functionality and rarity within the Hilbert space. We validate QFI through two approaches: random circuit sampling and evolutionary algorithms guided by fidelity or QFI objectives. Random sampling shows that states with near-perfect fidelity are less informational than slightly suboptimal but rarer states, while correlations reveal that high fidelity typically requires fewer gates and shallower circuits. Evolutionary results demonstrate that fidelity-only optimization converges quickly but reduces diversity and robustness, whereas QFI optimization sustains exploration, generates richer structures, and favors robust circuits with high fidelity. These findings position QFI as a practical and interpretable tool for circuit design, benchmarking, variational quantum algorithms, and exploring emergent patterns in quantum systems.