Thermodynamic Signature of Logical Depth in Quantum Circuits

TL;DR

This paper shows that the logical complexity of quantum circuits influences thermodynamic entropy flow, with deeper, branching circuits producing more environmental entropy, which can be experimentally detected.

Contribution

It introduces a thermodynamic framework linking logical depth in quantum circuits to entropy production, enabling physical measurement of circuit complexity.

Findings

Branching quantum circuits induce greater entropy flow than shallow ones.

The logical depth factor $L_d$ quantifies entropy accumulation during decoherence.

An experimental protocol for detecting thermodynamic signatures on current quantum platforms.

Abstract

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.