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.
Fermionic non-Gaussianity, or fermionic magic, is a key resource underlying the computational complexity of fermionic quantum systems, yet tractable and operationally meaningful ways to quantify it remain limited. We address this challenge by developing a convex resource theory of fermionic non-Gaussianity and introducing two families of computable quantities for pure fermionic states, both derived from the Williamson normal form of the covariance matrix. The first family, occupation number entropies, is defined as the Tsallis-$\alpha$ entropy of the occupation numbers. We prove that one member of this family is monotonic under Gaussian protocols, establishing it as a computable convex resource monotone. It consequently lower bounds the number of non-Gaussian gates needed for state preparation. The second family, natural-orbital participation entropies, is given by the R\'enyi-$\alpha$ entropy of the squared amplitudes of the state in the natural-orbital basis, defined by the eigenvectors of the covariance matrix. They quantify state compressibility in this basis and thus upper bound the classical simulation cost in an orthonormal Gaussian basis. We analyze both families for stabilizer and translation-invariant states, where they simplify and reveal additional structure. We further study representative examples, including random SWAP-doped matchgate circuits and the bond-modulated XXZ model, highlighting the role of non-Gaussianity in many-body phenomena. Our work establishes a resource-theoretic framework for computable fermionic non-Gaussianity that unifies notions arising across quantum information, condensed-matter physics, and quantum chemistry, opening new directions for studying the complexity of quantum many-body systems and providing practical tools to assess the classical simulability of fermionic states relevant for quantum advantage.
P. S. Tarabunga, B. Jobst, Ra'ul Morral-Yepes et al.· 4 citations· ⚡1
Pure-state entanglement rests on a single algebraic backbone: majorization of the Schmidt spectrum governs state conversion under local operations and classical communication, and constrains entanglement monotones. Here we establish a corresponding majorization law for fermionic non-Gaussianity, the resource that elevates free fermions to universal quantum computation. Under any fermionic Gaussian protocol with pure state outcomes, the Williamson spectrum of a pure state's Majorana covariance matrix is weakly majorized by its ensemble average. This spectral law mirrors that of entanglement theory. It turns computable non-Gaussianity quantifiers such as fermionic antiflatness and occupation entropies into strong monotones for fermionic non-Gaussianity, and delivers necessary conditions and converse bounds on state conversion under Gaussian protocols. When fermion parity is conserved, no catalyst can remove a majorization obstruction---unless it carries parity coherence---and asymptotic interconversion is irreversible already for pure states. All relevant quantities are accessible from two-point Majorana correlators, turning the theory developed here into experimentally observable properties of quantum matter, testable on present-day quantum devices.
X. Turkeshi, P. Sierant, P. S. Tarabunga· 0 citations
We prove an exponential variant of the Gaussian de Finetti theorem: the subsystems of permutation-invariant, free-fermionic Gaussian states are well-approximated by convex combinations of almost-i.i.d. states that are Gaussian on subsets of their parts. Our result provides an error bound between the original state and its approximants that decays exponentially in the number of unconstrained parts, becoming super-exponential when the subsystem under consideration is small. The dimensional penalty of our bound is polylogarithmic in the local Hilbert space dimension, an exponential improvement over the standard de Finetti theorem of [Nat. Phys. 3, 645-649]. In the fully i.i.d. limit, our bound recovers the Gaussian de Finetti theorem of [arXiv:2603.12392]. Previous works considered Gaussian-symmetric states, which are supported on the trivial irrep of the tensor matchgate representation. We extend these to a broader class of Gaussian-invariant states containing, for example, i.i.d. copies of single-replica mixed Gaussian states. We show that Gaussian-invariant states are precisely the partial traces of Gaussian-symmetric states on locally enlarged replicas, and always admit a purification into a larger Gaussian-symmetric state. This extends de Finetti theorems to the full set of Gaussian-invariant states, with only a polynomial overhead in the dimensional penalty of the error bound.
Jędrzej Burkat, Michal Studzi'nski, Sergii Strelchuk· 0 citations
Multimode quantum processes are generally difficult to learn, due to the large dimensionality and complex entanglement structure beyond the Gaussian class. Here, we show that the fundamental obstruction is not non-Gaussianity itself, but the buildup of irreducible multimode non-Gaussian correlations. We establish a tractability frontier for bosonic unitary learning: a general $m$-mode unitary with input energy at most $E$ per mode requires at least $\Omega(E^{2m})$ channel uses, whereas two broad non-Gaussian families---$t$-doped Gaussian unitaries and Gaussian-entanglable unitaries---can be learned with resources polynomial in $m$. The latter can exhibit both extensive non-Gaussianity and strong multimode entanglement. Our forward-only protocols use coherent-state probes, Gaussian operations, local heterodyne detection, and classical post-processing to identify the global Gaussian mixing and reduce the remaining task to single- or few-mode learning. The analysis also yields a multimode quantum Darmois--Skitovich theorem showing that mode-spreading passive networks preserve product structure only for Gaussian input states, an almost-sure activation theorem for non-Gaussian processes showing that non-Gaussian unitaries yield non-Gaussian outputs for almost all coherent input states, and a method for learning unitaries from uncalibrated coherent probes. Our results identify that complexity of learning arises from irreducible mixing of non-Gaussianity and entanglement, rather than either resource alone.
Recent demonstrations of quantum computational advantage have been driven largely by sampling problems. A prominent model, boson sampling, involves sampling from the output distribution of a linear optical network. However, its classical hardness hinges on two plausible yet less-studied conjectures: the average-case hardness of approximating Gaussian permanents, and the permanent anti-concentration conjecture (PACC). The PACC is a purely mathematical assertion regarding the distributional properties of random Gaussian matrices. While the typical magnitude of the permanent has been established for discrete random matrices, the complex Gaussian case, which governs transition amplitudes in linear optical networks, has remained open. Here, we establish a weak anti-concentration bound by upper-bounding the probability that a random Gaussian permanent is superexponentially smaller than its standard deviation. Tightening this bound to an inverse-polynomial fraction would prove the original PACC. As a corollary, we establish the typical magnitude of Gaussian permanents, on par with Tao and Vu's seminal result for Bernoulli matrices. Combined with the Aaronson-Arkhipov framework, our result implies that classically simulating boson sampling to within a superexponentially small total variation distance would collapse the polynomial hierarchy, assuming the remaining conjectures hold.
Fei Meng, Bin Cheng, Jianan Li et al.· 0 citations
Quantum evolution is generally expected to drive a quantum many-body system toward equilibrium. This expectation is often justified by the Hilbert-space ergodicity of generic quantum dynamics, namely, the idea that pure-state evolution explores Hilbert space uniformly up to physical constraints. Such a statement can be made rigorous by requiring the associated state ensemble to form the Haar-random ensemble, or its more structured generalization, the Scrooge ensemble. In this Letter, we report the emergence of exact Hilbert-space ergodicity in a continuously monitored quantum many-body system. For any target density matrix $\sigma$, we construct a continuously monitored system for which we rigorously prove that the Scrooge ensemble of $\sigma$ is the unique late-time equilibrium distribution of quantum trajectories. Remarkably, this requires only that the jump operators in the monitoring form a deformed unitary 1-design, a seemingly much weaker condition than full ergodicity. We numerically demonstrate our predictions by simulating continuously monitored systems whose equilibrium states are thermal states. Our results establish a rigorous mechanism for the emergence of Hilbert-space ergodicity and provide a practical route for its investigation on quantum devices.