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arXiv:2609.28233

Certifying fermionic Gaussian states (and a little more) with optimal precision dependence

Ninnat Dangniam, Laphas Premcharoen, Metrasit Sripech, Thiparat Chotibut

quant-phcond-mat.stat-mechSubmitted Sep 23, 2026• Announced Sep 24, 2026• 0 citations
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Abstract

Fermionic Gaussian states are the workhorse reference states for qubit-based quantum simulation of fermionic systems, yet existing certification protocols either proceed via fidelity estimation, leading to suboptimal sample complexity in the target precision $ε$, only apply to Haar-typical states, or require adaptive measurements on a large fraction of qubits. We give an adaptive protocol that certifies any $d$-mode pure fermionic Gaussian state using $O(d^2ε^{-1})$ copies, single-qubit measurements with only one qubit measured adaptively per copy, and $O(d^3)$ classical processing time per copy. The $ε^{-1}$ dependence is optimal for the certification problem, even among strategies using entangled measurements. The sample complexity is controlled by the spectral gap of a $d\leftrightarrow d-2$ down-up walk---a Markov chain studied in the theory of high-dimensional expanders, equivalent to a two-site Glauber dynamics in statistical physics---relating certification efficiency to relaxation time to equilibrium. The worst-case bound is tight for this protocol and is attained by physically relevant states, including ground states of the fully dimerized Su-Schrieffer-Heeger (SSH) chain and BCS pair states. In contrast, numerics up to $d=14$ suggest that $O(d ε^{-1})$ copies suffice to certify Haar-typical Gaussian states, a factor of $d$ improvement over the worst-case bound. Finally, because the bound depends only on the target's computational-basis distribution, the protocol extends to efficiently phase-dressed states---states obtained by injecting an efficiently computable diagonal phase to Gaussian states---with the same sample complexity. The class includes a continuous family of four-mode non-Gaussian magic states for matchgate computation.

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29 pages, 2 figures, comments welcome

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Fermionic Gaussian states are the workhorse reference states for qubit-based quantum simulation of fermionic systems, yet existing certification protocols either proceed via fidelity estimation, leading to suboptimal sample complexity in the target precision εεε, only apply to Haar-typical states, or require adaptive measurements on a large fraction of qubits. We give an adaptive protocol that certifies any ddd-mode pure fermionic Gaussian state using O(d2ε−1)O(d^2ε^{-1})O(d2ε−1) copies, single-qubit measurements with only one qubit measured adaptively per copy, and O(d3)O(d^3)O(d3) classical processing time per copy. The ε−1ε^{-1}ε−1 dependence is optimal for the certification problem, even among strategies using entangled measurements. The sample complexity is controlled by the spectral gap of a d↔d−2d\leftrightarrow d-2d↔d−2 down-up walk---a Markov chain studied in the theory of high-dimensional expanders, equivalent to a two-site Glauber dynamics in statistical physics---relating certification efficiency to relaxation time to equilibrium. The worst-case bound is tight for this protocol and is attained by physically relevant states, including ground states of the fully dimerized Su-Schrieffer-Heeger (SSH) chain and BCS pair states. In contrast, numerics up to d=14d=14d=14 suggest that O(dε−1)O(d ε^{-1})O(dε−1) copies suffice to certify Haar-typical Gaussian states, a factor of ddd improvement over the worst-case bound. Finally, because the bound depends only on the target's computational-basis distribution, the protocol extends to efficiently phase-dressed states---states obtained by injecting an efficiently computable diagonal phase to Gaussian states---with the same sample complexity. The class includes a continuous family of four-mode non-Gaussian magic states for matchgate computation.
Deciding whether an unknown quantum state either belongs to, or is far from, a given family of states is a natural question of quantum information theory. In many such cases, there is a natural 2-copy test which always accepts whenever the state indeed belongs to the target family; it is often much more difficult, however, to bound the probability of acceptance for states that are far from the family. Here we develop a simple new technique for upper bounding this probability for general families, and therefore upper bounding the sample-complexity of the decision problem itself. As a particularly striking example of this framework, we show that deciding whether an unknown pure nnn-mode state is fermionic Gaussian, or at least ε\varepsilonε-far in trace distance from all Gaussian states, can be accomplished using the optimal number Θ(ε−2)Θ(\varepsilon^{-2})Θ(ε−2) of copies. Remarkably, our analysis applies to the well-known Bell sampling procedure for testing fermionic Gaussianity, which we therefore show to be optimal, even compared to protocols making arbitrary collective and adaptive measurements. As a second example, we show that the analogous decision problem for the family of Slater determinant states can also be solved with mode-independent sample-complexity.
We prove that learning an unknown mixed fermionic Gaussian state on mmm modes to trace distance εεε requires Ω(m3/ε2)Ω(m^3/ε^2)Ω(m3/ε2) copies when measurements act on one copy at a time, even with arbitrary POVMs, fresh ancillas and classical adaptivity, but without quantum memory between copies. The bound holds for 0<ε≤1/36000<ε\le1/36000<ε≤1/3600 and separates this model from the known collective rate Θ(m2/ε2)Θ(m^2/ε^2)Θ(m2/ε2). It follows from a uniform single-copy Fisher-information budget and a dimension-independent comparison between trace distance and Gaussian parameters. On covariance matrices of operator norm at most 1−c1-c1−c, we prove a Frobenius-to-trace-norm continuity bound with constant [2c(2−c)]−1/2[2c(2-c)]^{-1/2}[2c(2−c)]−1/2; matchgate shadows then attain O(m3/(cε2))O(m^3/(cε^2))O(m3/(cε2)) copies. This also gives explicit error certificates for thermal free-fermion states. For a class of mixed passive bosonic Gaussian states with total mean photon number at most one, a reduction to bounded-block qudit tomography gives a single-copy lower bound Ω(m3/(ε2log⁡(m/ε)))Ω(m^3/(ε^2\sqrt{\log(m/ε)}))Ω(m3/(ε2log(m/ε)​)), versus collective complexity Θ(m2/ε2)Θ(m^2/ε^2)Θ(m2/ε2). These separations answer the mixed-state measurement-resource question posed by Chen et al. A two-mode example has optimal two-copy Fisher trace 555, versus the separable ceiling 444. An IBM replication gave a pre-registered witness lower bound 4.124.124.12 at one-sided 95%95\%95% shot-noise confidence; its interpretation assumes the prescribed preparations and a common measurement channel and does not bound preparation systematics.
In this work, we consider the fundamental task of quantum state certification: given copies of an unknown quantum state ρρρ, test whether it matches some target state σσσ or is εεε-far from it. For certifying ddd-dimensional states, Θ(d/ε2)Θ(d/ε^2)Θ(d/ε2) copies of ρρρ are known to be necessary and sufficient. However, the algorithm achieving this complexity makes fully entangled measurements over all O(d/ε2)O(d/ε^2)O(d/ε2) copies of ρρρ. Often, one is interested in certifying states to a high precision; this makes such joint measurements intractable even for low-dimensional states. Thus, we study whether one can obtain optimal rates for quantum state certification and related testing problems while only performing measurements on ttt copies at once, for some 1<t≪d/ε21 < t \ll d/ε^21<t≪d/ε2. While it is well-understood how to use intermediate entanglement to achieve optimal quantum state learning, the only protocol known to achieve optimal testing is the one using fully entangled measurements. Our main result is a smooth copy complexity upper bound for state certification as a function of ttt, which achieves a near-optimal rate at t=d2t = d^2t=d2. In the high-precision regime, i.e., for ε<1dε< \frac{1}{\sqrt{d}}ε<d​1​, this is a strict improvement over the entanglement used by the aforementioned optimal protocol. We also extend our techniques to develop new algorithms for the related tasks of mixedness testing and purity estimation, and show tradeoffs achieving the optimal rates for these problems at t=d2t = d^2t=d2 as well. Our algorithms are based on novel reductions from testing to learning and leverage recent advances in quantum state tomography in a non-black-box fashion. We complement our testing upper bounds with a (nearly) matching lower bound that holds in the high-precision regime, implying that one must take t≥Ω(d2)t \geq Ω(d^2)t≥Ω(d2) to achieve the optimal rate for certification.
Fermionic non-Gaussianity is an essential resource for unlocking the full computational power of fermionic quantum platforms. In this work we develop monotones and efficient quantum algorithms for fermionic non-Gaussianity, all built on the eigenvalue structure of the operator Λ=∑j=12nγj⊗γjΛ= \sum_{j=1}^{2n}γ_j\otimesγ_jΛ=∑j=12n​γj​⊗γj​ defined on two copies of an nnn-mode fermionic state, accessible via Bell sampling. In particular, we introduce the \emph{bridge degree} of even pure states, a novel non-Gaussianity monotone defined as the largest eigenvalue of ΛΛΛ whose eigenspace is populated by two copies of the state. Our key technical result is that the bridge degree is non-increasing under post-selected Gaussian protocols, which yields no-go theorems for Gaussian conversion beyond the reach of previously known monotones and shows that the resource theory of fermionic non-Gaussianity is irreversible in the exact-conversion setting. Beyond this, the bridge degree exhibits several further features: it (i) is easy to compute, (ii) is efficiently witnessed through Bell sampling, (iii) lower-bounds the non-Gaussian gate complexity of state preparation, (iv) controls the non-Gaussian gate complexity of producing quantum state designs, and (v) naturally extends to mixed states via the Choi--Jamiołkowski isomorphism. We further develop an approximate variant together with an efficiently measurable lower bound, yielding an experimentally certifiable lower bound on the non-Gaussian cost of approximately preparing any state, based directly on Bell-sampling data. Finally, the same eigenvalue structure underlies two Bell-sampling-based algorithmic primitives, both with polynomial sample complexity: a two-copy Gaussianity test with perfect completeness, optimal among two-copy tests sharing this property, and a test for the state 222-design property of matchgate-invariant ensembles.
We study the problem of property testing quantum states: given a pure state ∣ψ⟩|ψ\rangle∣ψ⟩ that either (A) belongs to some class CCC of pure states, or (B) is ε\varepsilonε-far in trace distance from all states in CCC, determine which is the case with high probability. It is known that there exists classes CCC for which the sample complexity of property testing necessarily scales with the size of the system. In this work, we identify classes for which testing only requires a size-independent number of samples. We prove a tight sample complexity of Θ(1/ε2)Θ(1/\varepsilon^2)Θ(1/ε2) for the property testing of five important classes of Gaussian quantum states: 1) Fermionic Gaussian states, 2) Slater determinants, 3) bosonic Gaussian states, 4) zero-mean bosonic Gaussian states, and 5) bosonic coherent states. While property testers for some of these classes have been studied previously, those analyses only gave upper bounds scaling polynomially in the number of modes. In each case, our tester is the projection onto the ``top'' irrep of two or three copies. We get to constant sample complexity by lower bounding how quickly the rejection probability increases with distance from the class; the main technical component involves bounding the spectral gap of symmetric extension of the test projector. We then establish optimality by proving matching lower bounds based on binary state discrimination for each class. Our testers and optimality claims also extend to the tolerant setting, wherein ψψψ may be O(ε)O(\varepsilon)O(ε)-close to CCC in case (A), rather than lying exactly inside.
Fermionic Gaussian states describe free fermions and underlie the mean-field picture of matter, from metals to superconductors; they are also efficiently simulable on classical computers. Departures from Gaussianity -- the correlations produced by interactions -- are therefore what make a fermionic system hard to simulate classically and useful for quantum computation, analogous to the role of magic in stabilizer-based quantum computation. Yet detecting and quantifying such non-Gaussianity at scale has remained challenging. Here we introduce practical tests and witnesses of fermionic non-Gaussianity built on fermionic antiflatness, a measure derived from the two-point covariance matrix. We estimate it with two protocols -- a two-copy Bell measurement and a single-copy scheme using commuting Majorana bilinears -- that determine whether a state is Gaussian or far from it at lower measurement cost than existing approaches, using only operations native to fault-tolerant hardware. For mixed states, a purity-corrected witness certifies non-Gaussianity and remains robust under strong noise; running it on the IQM quantum processor, we find that noise can both reduce and enhance non-Gaussianity. Finally, we show that preparing pseudorandom fermionic states requires extensive non-Gaussianity. Together, these tools enable the study and certification of non-Gaussian fermionic resources on present-day quantum devices.