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.
Fermionic Gaussian states form a central class of classically tractable quantum states, while fermionic non-Gaussianity provides the resource required to go beyond free-fermion dynamics. A key challenge is to quantify this resource through monotones that are both mathematically rigorous and experimentally accessible. Here, we show that the fermionic entropy, defined through the squared Frobenius norm of the correlation matrix, is a strong pure-state Gaussian monotone. Its simple closed-form expression also makes it directly measurable: we show that the associated fermionic purity can be unbiasedly estimated up to additive error $\varepsilon$ using $O(\varepsilon^{-2})$ two-copy measurements, independently of the system size. Moreover, we prove that the fermionic entropy obeys asymptotic continuity and, as a direct consequence, establish its operational meaning as the upper bound to the asymptotic rate of non-Gaussianity distillation. We further derive a linear sample complexity bound for tolerant testing of fermionic Gaussian states, providing a quadratic improvement over the state of the art. As a further application of our results, we study unitary designs generated by Matchgate circuits supplemented with Majorana-local non-Gaussian gates. We prove that a linear number of such gates is necessary even to achieve an approximate state $2$-design with error below $0.4\%$. Combined with known nearly linear upper bounds for relative-error designs, this determines the optimal doping level, up to logarithmic factors, across all relevant design notions and reveals the extensive non-Gaussianity cost required to generate Haar-like quantum dynamics in this architecture.
The density of states (DoS) encodes the thermodynamic and spectral properties of quantum many-body systems, yet its reconstruction becomes intractable for Hilbert spaces too large to diagonalize. Classically, the kernel polynomial method (KPM) addresses this by combining stochastic trace estimation with a smoothing kernel. Here we show that the Rodeo algorithm---one of the simplest eigenvalue-location protocols for near-term quantum hardware---provides a direct quantum analogue of this approach. Averaging the Rodeo response over Haar-random input states yields the DoS convolved with a spectral kernel fixed entirely by the distribution of evolution times: the random states play the role of stochastic trace estimation, and the temporal sampling distribution that of the damping kernel. The construction requires only the standard single-ancilla circuit, and quantum typicality suppresses the statistical error as the Hilbert-space dimension grows. We derive the estimator and its uncertainties, establish an explicit dictionary between signal-processing window functions and quantum reconstruction kernels, and validate the method on the one-dimensional transverse-field Ising and spin-1 models.
Magic and entanglement are independent quantum resources, yet their exact relation in many-body dynamics has remained elusive. We uncover two structural principles. First, at any stabilizer state, the curvature of the second stabilizer R\'enyi entropy under an arbitrary Hermitian generator equals the quantum Fisher information up to a fixed normalization, creating a bidirectional bridge between computational and metrological resources. Second, for commuting Ising evolution on any forest graph, a Clifford pruning circuit yields the full stabilizer-R\'enyi family at arbitrary size and in any spatial embedding, thereby furnishing a graph-theoretic construction of families with finite magic density and vanishing entanglement density in the thermodynamic limit. We solve two paradigmatic one-dimensional realizations central to quantum simulation -- an Ising quench and a kicked Floquet chain -- exactly for arbitrary system size and directly in the thermodynamic limit, revealing finite magic density with vanishing entanglement density, distinct magic and entanglement revival periods, and Clifford points with zero magic but finite bipartite entanglement. The same tangent geometry fixes initial growth, perturbative revival lifting, and stability of thermodynamic magic minima. Large-scale Pauli-basis matrix-product-state calculations verify all predictions, and the tangent bridge yields a concrete protocol for detecting magic through established quantum-Fisher-information measurements.
Magic plays a dual role in quantum computation: it promotes stabilizer dynamics from efficient classical simulability to universality, but it presents a central challenge for fault tolerance, since non-stabilizer operations are harder to protect against noise. Magic state distillation addresses this issue; however, existing protocols typically assume prior structure in the input, such as proximity to the target or a specified noise model. Here we introduce universal magic state concentration: a fixed stabilizer protocol that converts a few copies of an unknown pure non-stabilizer qubit state into an exact target magic state. Motivated by the obstruction to exact $T$-state concentration, we show that $\mathrm{CCZ}$ states behave fundamentally differently. Six input copies are necessary and sufficient to distill one exact $\mathrm{CCZ}$ state, with an optimal success probability determined by the linearized order-three stabilizer R\'enyi entropy $M^{\mathrm{lin}}_3$. Beyond this, we show that $M^{\mathrm{lin}}_3$ governs the optimal state dependence of any protocol up to nine input copies, and we showcase an eight-copy protocol with improved success probability. Furthermore, block repetition of our protocols yields asymptotic distillation rates that achieve optimal scaling up to logarithmic factors. As a corollary, any unknown pure qubit magic state suffices for universal quantum computation via exact $\mathrm{CCZ}$ injection. Together, these results identify the stabilizer R\'enyi entropy as a fundamental operational quantity in magic state distillation.
Quantum circuits under sufficiently weak monitoring purify on a timescale $T_P$ exponentially long in the system size. This slowness underlies a universal purification dynamics, whose quantitative description has so far required the replica trick, with a delicate analytic continuation. We show that monitored Clifford circuits on $L$ qudits of prime dimension $q$ bypass this construction entirely: in the scaling limit at fixed $x = t/T_P(L)$, purification reduces to the Markovian decay of the density-matrix rank, an exactly solvable death process descending from infinity. We compute the full scaling functions in compact form: all R\'enyi entropies collapse onto a universal curve $\langle S(x) \rangle$. Exact stabilizer simulations at $q=2,3,5$ confirm the predictions, with no fitting parameter for the global model and $T_P$ as the only fitted scale for local brick-wall circuits. Also, the replica problem amounts to a tilted version of the same Markov process, in agreement with exact computations from the Clifford commutant. Finally, the quantization of the rank leaves two hallmarks that distinguish Clifford dynamics from generic monitored circuits: the entropy fluctuations saturate at short scaled times $x\to0$ to an $O(1)$ variance, instead of vanishing, and observables develop a temporal modulation periodic in $\log_q x$, which cannot be captured by the replica approach.
Beatrice Magni, Federico Gerbino, X. Turkeshi et al.· 1 citation
In this review we discuss semi-classical methods that are traditionally used to describe many-body systems in physics, but may also be used to describe partitions of integers in analytic number theory. Specifically, we explore the connection between the methods of statistical mechanics and number partitions. Though the two fields appear very different, their fundamental issues bear a close resemblance. In the former case it is the distribution of a given amount of energy among the particles in an ensemble at a given temperature with well defined properties, while in the latter case it is the way an integer is partitioned into other integers, with or without restrictions. We begin with a discussion of the single-particle quantum density of states, also called the level density, in which we illustrate the connection between the density of states and the classical periodic orbits through the semiclassical trace formula. This is then extended to many particle systems. We show that the asymptotic number partition is reproduced by the average (smooth) part of the level density at discrete integer values of the argument. In the especially interesting case of distinct square partitions, pronounced oscillations are well reproduced by the periodic orbit theory in terms of a few orbits characterised by Pythagorean number triples. We speculate on the connection to Fermat's theorem as to why such regular oscillations (though vanishing asymptotically) exist only in this special case. Finally, we discuss some new results for integer partitions of primes, both unrestricted and distinct.