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arXiv:2609.15739

Pauli spectrum and nonstabilizerness of random fermionic Gaussian states

Xhek Turkeshi, Piotr Sierant, Poetri Sonya Tarabunga

quant-phcond-mat.stat-mechSubmitted Sep 14, 2026• Announced Sep 15, 2026• 0 citations
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Abstract

We characterize the magic of random fermionic Gaussian states through their Pauli spectrum and their stabilizer entropies. For the Haar ensemble of Majorana Gaussian states we derive closed finite-$N$ expressions for the average stabilizer purities and reconstruct the Pauli spectrum exactly, as a mixture of products of beta-distributed variables resolved by Majorana weight. The stabilizer entropies, and their filtered versions, freeze at Rényi index $q_c=2$: for $q>2$ the magic density is $1/(q-1)$, controlled by rare weight-two Majorana strings. Resolving these contributions requires exponentially many characteristic samples, making direct sampling estimates of the stabilizer entropies inefficient in this frozen regime. For Haar-random Gaussian states at fixed filling, we express the averaged stabilizer purities at positive integer $q$ as coefficient integrals over $\mathrm{SU}(2q)$, with dimension independent of $N$, and evaluate the $q=2$ case exactly. We establish that the frozen density describes typical states in the Majorana and half-filled number-conserving ensembles and relate these ensembles to the corresponding $\mathrm{SYK}_2$ ground states. Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction $κ<κ_\star^{\rm G}\simeq0.7613$ of their modes in the thermodynamic limit, remaining magical even beyond the $2/3$ threshold for Haar-random states in the full Hilbert space.

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We characterize the magic of random fermionic Gaussian states through their Pauli spectrum and their stabilizer entropies. For the Haar ensemble of Majorana Gaussian states we derive closed finite-NNN expressions for the average stabilizer purities and reconstruct the Pauli spectrum exactly, as a mixture of products of beta-distributed variables resolved by Majorana weight. The stabilizer entropies, and their filtered versions, freeze at Rényi index qc=2q_c=2qc​=2: for q>2q>2q>2 the magic density is 1/(q−1)1/(q-1)1/(q−1), controlled by rare weight-two Majorana strings. Resolving these contributions requires exponentially many characteristic samples, making direct sampling estimates of the stabilizer entropies inefficient in this frozen regime. For Haar-random Gaussian states at fixed filling, we express the averaged stabilizer purities at positive integer qqq as coefficient integrals over SU(2q)\mathrm{SU}(2q)SU(2q), with dimension independent of NNN, and evaluate the q=2q=2q=2 case exactly. We establish that the frozen density describes typical states in the Majorana and half-filled number-conserving ensembles and relate these ensembles to the corresponding SYK2\mathrm{SYK}_2SYK2​ ground states. Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction κ<κ⋆G≃0.7613κ<κ_\star^{\rm G}\simeq0.7613κ<κ⋆G​≃0.7613 of their modes in the thermodynamic limit, remaining magical even beyond the 2/32/32/3 threshold for Haar-random states in the full Hilbert space.
Entanglement and magic are fundamental resources that capture the complexity of quantum many-body systems. Non-local magic isolates the irreducible nonstabilizerness intrinsically tied to entanglement. However, evaluating this quantity generally requires a prohibitive minimization over the full Hilbert space, making it computationally inaccessible beyond a few qubits. Here, we overcome this bottleneck by establishing a closed-form expression for the non-local stabilizer entropies of fermionic Gaussian states over local Gaussian unitaries, which we prove at Rényi index α=2α=2α=2 for arbitrary subsystem size, and which can be evaluated in polynomial time directly from the eigenvalues of the reduced Majorana covariance matrix. We apply this framework to characterize fermionic non-local magic across diverse physical regimes: we derive an exact Page-like curve for typical random states, reveal logarithmic scaling at the quantum critical point of the XY model, and establish a quasiparticle picture for magic generation during out-of-equilibrium quantum quenches. Crucially, because our result relies solely on two-point correlation functions, it provides a scalable route for the experimental estimation of fermionic non-local magic in large-scale quantum processors via fermionic shadow tomography.
An important question of quantum information is to characterize genuinely quantum (beyond-Clifford) resources necessary for universal quantum computing. Here, we use the Pauli spectrum to quantify how ``magic,'' beyond Clifford, typical many-qubit states are. We first present a phenomenological picture of the Pauli spectrum based on quantum typicality, and then we confirm it for Haar random states. We then introduce filtered stabilizer entropy, a magic measure that can resolve the difference between typical and atypical states. We proceed with the numerical study of the Pauli spectrum of states created by random circuits as well as for eigenstates of chaotic Hamiltonians. We find that in both cases, the Pauli spectrum approaches the one of Haar random states, up to exponentially suppressed tails. We discuss how the Pauli spectrum changes when ergodicity is broken due to disorder. Our results underscore the difference between typical and atypical states from the point of view of quantum information.
The non-stabilizerness, or "magic", of a quantum state is a common metric for the hardness of simulation on a classical computer, and is often quantified by the stabilizer Rényi entropy (SRE). We show how random superpositions of classically simulable states saturate the SRE to the symmetry-restricted Haar-random maximum. For any large-NNN random pure-state ensemble satisfying the conditions of Schur's lemma, the average SRE of the mmm-component superposition ⟨M2(m)⟩=N−log⁡2[⟨W(m)⟩]\langle M_2(m)\rangle=N-\log_2[\langle W(m)\rangle]⟨M2​(m)⟩=N−log2​[⟨W(m)⟩] is governed by an average witness ⟨W(m)⟩\langle W(m)\rangle⟨W(m)⟩ that universally decomposes into a Haar floor and an ensemble-specific excess. As the number of components mmm increases, the excess is suppressed, resulting in a logarithmic scaling of the SRE before plateauing at the Haar-valued maximum. We derive a general closed form for ⟨W(m)⟩\langle W(m)\rangle⟨W(m)⟩ in the thermodynamic limit, and numerically verify our result for random stabilizer states and random Gaussian states.
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, ννν-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-Rényi 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.
Nonlocal magic quantifies the irreducible nonstabilizerness of a bipartite quantum state after optimizing over local basis changes. We study nonlocal magic for pure fermionic Gaussian states, and derive a simple closed-form entanglement spectrum bound in terms of the singular values of the subsystem-restricted covariance matrix. We benchmark our result against simulated annealing over local Gaussian unitary transformations, which supports optimality along the full local Gaussian orbit. For states drawn from the Gaussian Haar ensemble, we show that the average nonlocal magic is extensive and determine its thermodynamic limit using random matrix theory for the appropriate circular unitary ensemble. We also study Gaussian ground states, focusing on the Kitaev chain, and find that nonlocal magic is suppressed deep in both trivial and topological phases and peaks near the critical points. Finally, we investigate Gaussian evolution via random circuits and in quenches with the XY chain. For random circuits, we find that nonlocal magic grows diffusively, while in the XY chain the XX limit reveals a striking separation between nonlocal magic and entanglement.
We introduce an efficient method to quantify nonstabilizerness in fermionic Gaussian states, overcoming the long-standing challenge posed by their extensive entanglement. Using a perfect sampling scheme based on an underlying determinantal point process, we compute the Stabilizer Renyi Entropies (SREs) for systems with hundreds of qubits. Benchmarking on random Gaussian states with and without particle conservation, we reveal an extensive leading behavior equal to that of Haar random states, with logarithmic subleading corrections. We support these findings with analytical calculations for a set of related quantities, the participation entropies in the computational (or Fock) basis, for which we derive an exact formula. We also investigate the time evolution of non-stabilizerness in a random unitary circuit with Gaussian gates, observing that it converges in a time that scales logarithmically with the system size. Applying the sampling algorithm to a two-dimensional free-fermionic topological model, we uncover a sharp transition in non-stabilizerness at the phase boundaries, highlighting the power of our approach in exploring different phases of quantum many-body systems, even in higher dimensions.