Thermodynamic Signature of Logical Depth in Quantum Circuits

We demonstrate that the internal logical structure of a quantum circuit can leave a distinct thermodynamic signature under progressive decoherence. By comparing deep, conditionally branching circuits with shallow, uniform counterparts—while controlling for overall halting probability and physical resources—we show that branching architectures induce greater entropy flow into the environment. This effect is captured by a logical depth factor \(L_d\), which quantifies entropy accumulation during environmental interactions. We validate our framework through detailed analysis of two 4-branch quantum circuits, demonstrating greater entropy production with \(L_d \approx 1.615\) for conditional versus uniform architectures. An ancilla-based experimental protocol using controlled-phase gates provides a concrete pathway for detecting these thermodynamic signatures on current quantum platforms. Our results establish logical depth as a physically measurable quantity with implications for circuit design, compilation strategies, and verification protocols.