Computational Complexity Bounds for Maxwell's Demon: From Landauer's Principle to Quantum Advantage

Whether quantum computational speedups translate to fundamental thermodynamic energy savings remains unresolved. We establish the first rigorous connection between computational complexity and thermodynamic costs through DEMON(I,E) complexity classes that classify information-processing protocols by information complexity I(n) and erasure energy E(n). Four theorems prove: (1) any g-bit erasure dissipates >= g X k_B T ln 2, generalizing Landauer's principle; (2) measurement disturbance D reduces extractable work to (I-D)k_B T ln 2; (3) Grover-based quantum demons achieve provably optimal Theta(sqrt{N}) thermodynamic advantage over classical search, tight by the BBBV lower bound; and (4) quantum walks on d-dimensional lattices exhibit a dense hierarchy with continuous scaling exponent (d-1)/(2d). Computational validation across six orders of magnitude confirms predictions with <1% error. Experimental predictions include 20.5X energy reduction for trapped-ion Grover search (N=1024, T=1 mK) and 11X effective cooling for superconducting circuit QED heat-bath protocols. This work unifies 158 years of Maxwell demon research, connects algorithmic complexity to physical energy costs, and enables experimental verification of quantum thermodynamic advantages beyond computational speedup.