Stabilizer Statistical Mechanics: A Framework for Efficient Quantification and Classification of Magic States

Abstract

The partition function is statistical mechanics' answer to an exponentially large spectrum, distilling it into a single analytic object whose temperature dependence resolves the full structure of the underlying ensemble. We show that magic, the resource separating universal quantum computation from classically simulable stabilizer dynamics, admits precisely such a description. Mapping the Pauli spectrum of a quantum state onto the energy levels of a fictitious many-body system, the Pauli gas, we construct its canonical partition function, the stabilizer partition function, and from its associated free energy a magic monotone that we call the stabilizer work. Both these objects are analytic functions of an inverse-temperature-like parameter and are efficiently estimable via Bell sampling. The framework is analytically tractable. We derive exact ensemble-averaged partition functions for Haar-random, $\nu$-compressible, and pseudomagic states, together with concentration guarantees. We show that for every value of its parameter, the stabilizer work is a faithful, Subadditive magic monotone, while remaining efficiently accessible on quantum hardware and admitting an operational interpretation. Unlike measures that probe a single moment of the Pauli distribution, the stabilizer work is intrinsically moment-generating. As the temperature is tuned from high to low, it interpolates continuously between the stabilizer 2-R\'enyi entropy and the stabilizer nullity, revealing two previously disconnected monotones as limiting cases of a single object. We demonstrate the framework on low-rank stabilizer simulation, resource interconversion, molecular ground states, and quantum many-body systems. A thermodynamics of magic is therefore not merely an analogy but a working toolkit, opening a statistical-mechanical route to magic properties of quantum systems that no single measure can access.