Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2403.03289 · Physical Review A

Efficient representation of Gaussian fermionic pure states in noncomputational bases

Babak Tarighi, Reyhaneh Khasseh, M. A. Rajabpour

Published Jun 14, 2024• 2 citations
arXivHTMLPDF

Abstract

This paper introduces an innovative approach for representing Gaussian fermionic states, pivotal in quantum spin systems and fermionic models, within a range of alternative quantum bases. We focus on transitioning these states from the conventional computational ${\ensuremath{\sigma}}^{z}$ basis to more complex bases, such as $(\ensuremath{\phi},\frac{\ensuremath{\pi}}{2},\ensuremath{\alpha})$, which are essential for accurately calculating critical quantities like formation probabilities and Shannon entropy. Our methodology can be advantageous in quantum quench studies and quantum tomography, where alternative basis representations significantly enhance optimization processes. We present an algorithm that not only simplifies the basis transformation, but also reduces computational complexity, making it feasible to calculate amplitudes of large systems efficiently. Our key contribution is a technique that translates amplitude calculations into the Pfaffian computation of submatrices from an antisymmetric matrix, a process facilitated by understanding domain wall relationships across different bases. As an application, we will determine the formation probabilities for various bases and configurations within the critical transverse field Ising chain, considering both periodic and open boundary conditions. We aim to categorize the configurations and bases by examining the universal constant term that characterizes the scaling of the logarithm of the formation probability in the periodic system, as well as the coefficient of the logarithmic term in the case of open systems. In the open system scenario, this coefficient is influenced by the central charge and the conformal weight of the boundary condition-changing operator. This work is set to expand the toolkit available for researchers in quantum information theory and many-body physics, providing a more efficient and elegant solution for exploring Gaussian fermionic states in nonstandard quantum bases.

DOI
10.1103/physreva.109.062214

Related papers

Explicit Pfaffian formula for amplitudes of fermionic Gaussian pure states in arbitrary Pauli bases

M. A. Rajabpour, M. A. Seifi Mirjafarlou, Reyhaneh Khasseh

The explicit computation of amplitudes for fermionic Gaussian pure states in arbitrary Pauli bases is a long-standing challenge in quantum many-body physics, with significant implications for quantum tomography, experimental studies, and quantum dynamics. These calculations are essential for analyzing complex properties beyond traditional measures, such as formation probabilities, global entanglement, and entropy in nonstandard bases, where exact and computationally efficient methods remain underdeveloped. In addition to these physical applications, having explicit formulas is crucial for optimizing negative log-likelihood functions in quantum tomography, a key task in the NISQ era. In this work, we present an explicit Pfaffian formula (Theorem 1) for determining these amplitudes in arbitrary Pauli bases, utilizing a matrix whose structure reflects the qubit parity. Additionally, we introduce a recursive relation (Theorem 2) that connects amplitudes for systems with varying qubit numbers, enabling scalable computations for large systems. Together, these results provide a versatile framework for studying global entanglement, Shannon-R\'enyi entropies, formation probabilities, and performing efficient quantum tomography, thereby significantly expanding the computational toolkit for analyzing complex quantum systems. Finally, we utilize our formalism to determine the post-measurement entanglement entropy, reflecting how local measurements alter entanglement, and compare the results with conformal field theory predictions.

arXiv:2502.04857•Physical review. B./Physical review. B•Jun 2, 2025•2 citations

Computable fermionic non-Gaussianity from the covariance matrix

Poetri Sonya Tarabunga, Bernhard Jobst, Raúl Morral-Yepes +4 more

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-$α$ 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 Rényi-$α$ 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.

quant-pharXiv:2607.02242•Jul 3, 2026•0 citations

Scaling of the formation probabilities and universal boundary entropies in the quantum XY spin chain

Filiberto Ares, M. A. Rajabpour, Jacopo Viti

Abstract We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse σ z -basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal boundary entropy at the critical point. The latter is a benchmark of the underlying boundary conformal field theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the boundary entropies are computed by exploiting a generalization of the Euler–MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.

arXiv:2004.10606•Journal of Statistical Mechanics Theory and Experiment•Aug 1, 2020•7 citations

Quantum Computational Resources and Conformal Field Theory: Unifying Spins, Bosons, and Fermions

Ryota Matsuda, Masahiro Hoshino, Yuto Ashida

Characterizing a quantum state through the lens of quantum resources provides an information-theoretic perspective on many-body systems. While quantum entanglement serves as the paradigmatic example of a quantum resource, recent studies have shown that quantum magic, a resource for universal quantum computation, captures aspects of many-body states complementary to those described by entanglement. For instance, in spin systems, conformal field theory (CFT) analysis of the stabilizer Rényi entropy has revealed universal features of nonstabilizerness qualitatively distinct from entanglement. In bosonic and fermionic systems, however, a comparable formulation for their computational resource, non-Gaussianity, has yet to be established. In this work, we introduce a unified measure, the magic Rényi entropy (MRE), to quantify computational resources in spins, bosons, and fermions on an equal footing. We show that the MRE is a resource monotone under stabilizer and Gaussian protocols involving measurements and feedforward operations. The MRE reveals common universal aspects of nonstabilizerness and non-Gaussianity in critical many-body states. In particular, our CFT analysis shows that the universal contribution to the MRE appears as the size-independent term determined by the Affleck-Ludwig boundary entropy. We find that non-Gaussianity can continuously renormalize this universal contribution or drive a boundary transition through bulk-induced boundary renormalization-group flows. As a concrete example, we present a CFT analysis of non-Gaussianity in interacting spinless fermions described by the Tomonaga-Luttinger liquid, showing boundary transitions at the Luttinger parameters $K=1/3$ and $K=3$. Our field-theoretical predictions are confirmed by numerical calculations. These results provide a unified field-theoretical understanding of many-body magic across spins, bosons, and fermions.

quant-pharXiv:2607.05343•Jul 7, 2026•0 citations

Classical simulation of free-fermionic dynamics and quantum chemistry with magic input

Changhun Oh, Michał Oszmaniec, Oliver Reardon-Smith +1 more

Establishing the precise computational boundary between classically tractable fermionic systems and those capable of genuine quantum advantage is a central challenge in quantum simulation. While injecting non-Gaussian ``magic" inputs into free-fermion circuits is widely expected to generate intractable complexity, we identify a physically motivated intermediate regime. We prove that for block-product paired non-Gaussian fermionic states, essential quantum simulation primitives -- transition amplitudes, overlaps, and arbitrary-weight number correlators -- can be efficiently approximated to additive error under free-fermionic dynamics. This tractability stems from an algebraic reduction that compresses exponentially large multiparticle interference into a single coefficient of a multivariate Pfaffian polynomial. Because these classical estimators match the intrinsic $O(1/\sqrt{K})$ statistical uncertainty of quantum hardware utilizing $K$ measurement shots, they constitute a practical benchmark. Building on this foundation, we construct an additive-error estimator for high-weight Wilson observables in the noninteracting quench of recent trapped-ion experiments, providing a rigorous classical benchmark. Extending this to quantum chemistry, we demonstrate that core overlap-based subroutines for antisymmetrized products of strongly orthogonal geminals admit efficient additive-error Pfaffian-kernel estimators. Ultimately, these results sharpen the boundary of quantum advantage, establishing that the paired-electron scaffold is dequantized and clarifying where quantum resources are indispensable.

quant-pharXiv:2604.26813•Apr 30, 2026•0 citations

Computable measures of fermionic non-Gaussianity from the covariance matrix

Poetri Sonya Tarabunga, B. Jobst, Raúl Morral-Yepes +4 more

Fermionic non-Gaussianity, also known as fermionic magic, is a key resource underlying the computational complexity of fermionic quantum systems, yet quantifying it in a computationally tractable and operationally grounded manner remains a central challenge. In this work, we address this gap by establishing a convex resource theory of fermionic non-Gaussianity and introducing two families of computable measures for pure fermionic states, both built from the Williamson normal form of the covariance matrix. The first family, the occupation number entropies, is defined as the Tsallis-$\alpha$ entropy of the occupation numbers, obtained as shifted and rescaled Williamson eigenvalues of the covariance matrix. We prove that one member of this family yields a monotone under Gaussian protocols, thereby validating it as a computable convex resource monotone for fermionic non‑Gaussianity. As a consequence, it provides a lower bound on the number of non-Gaussian gates required for state preparation. The second family, the natural-orbital participation entropy, is the R\'enyi-$\alpha$ entropy of the squared magnitude of the state's amplitudes in the natural-orbital basis, defined by the eigenvectors of the covariance matrix. Operationally, they quantify the compressibility of the state in the natural-orbital basis, and hence upper bound the classical simulation cost in an orthonormal Gaussian basis. We analyze the two measures in two important settings, for stabilizer and translation-invariant states, where they admit simplified forms and additional structural insights. We investigate both families through several representative examples, including random SWAP-doped matchgate circuits and the bond-modulated XXZ model, showcasing 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.

•Jul 3, 2026•0 citations
This paper introduces an innovative approach for representing Gaussian fermionic states, pivotal in quantum spin systems and fermionic models, within a range of alternative quantum bases. We focus on transitioning these states from the conventional computational \ensuremathσz{\ensuremath{\sigma}}^{z}\ensuremathσz basis to more complex bases, such as (\ensuremathϕ,\ensuremathπ2,\ensuremathα)(\ensuremath{\phi},\frac{\ensuremath{\pi}}{2},\ensuremath{\alpha})(\ensuremathϕ,2\ensuremathπ​,\ensuremathα), which are essential for accurately calculating critical quantities like formation probabilities and Shannon entropy. Our methodology can be advantageous in quantum quench studies and quantum tomography, where alternative basis representations significantly enhance optimization processes. We present an algorithm that not only simplifies the basis transformation, but also reduces computational complexity, making it feasible to calculate amplitudes of large systems efficiently. Our key contribution is a technique that translates amplitude calculations into the Pfaffian computation of submatrices from an antisymmetric matrix, a process facilitated by understanding domain wall relationships across different bases. As an application, we will determine the formation probabilities for various bases and configurations within the critical transverse field Ising chain, considering both periodic and open boundary conditions. We aim to categorize the configurations and bases by examining the universal constant term that characterizes the scaling of the logarithm of the formation probability in the periodic system, as well as the coefficient of the logarithmic term in the case of open systems. In the open system scenario, this coefficient is influenced by the central charge and the conformal weight of the boundary condition-changing operator. This work is set to expand the toolkit available for researchers in quantum information theory and many-body physics, providing a more efficient and elegant solution for exploring Gaussian fermionic states in nonstandard quantum bases.
The explicit computation of amplitudes for fermionic Gaussian pure states in arbitrary Pauli bases is a long-standing challenge in quantum many-body physics, with significant implications for quantum tomography, experimental studies, and quantum dynamics. These calculations are essential for analyzing complex properties beyond traditional measures, such as formation probabilities, global entanglement, and entropy in nonstandard bases, where exact and computationally efficient methods remain underdeveloped. In addition to these physical applications, having explicit formulas is crucial for optimizing negative log-likelihood functions in quantum tomography, a key task in the NISQ era. In this work, we present an explicit Pfaffian formula (Theorem 1) for determining these amplitudes in arbitrary Pauli bases, utilizing a matrix whose structure reflects the qubit parity. Additionally, we introduce a recursive relation (Theorem 2) that connects amplitudes for systems with varying qubit numbers, enabling scalable computations for large systems. Together, these results provide a versatile framework for studying global entanglement, Shannon-R\'enyi entropies, formation probabilities, and performing efficient quantum tomography, thereby significantly expanding the computational toolkit for analyzing complex quantum systems. Finally, we utilize our formalism to determine the post-measurement entanglement entropy, reflecting how local measurements alter entanglement, and compare the results with conformal field theory predictions.
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-ααα 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 Rényi-ααα 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.
Abstract We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse σ z -basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal boundary entropy at the critical point. The latter is a benchmark of the underlying boundary conformal field theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the boundary entropies are computed by exploiting a generalization of the Euler–MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.
Characterizing a quantum state through the lens of quantum resources provides an information-theoretic perspective on many-body systems. While quantum entanglement serves as the paradigmatic example of a quantum resource, recent studies have shown that quantum magic, a resource for universal quantum computation, captures aspects of many-body states complementary to those described by entanglement. For instance, in spin systems, conformal field theory (CFT) analysis of the stabilizer Rényi entropy has revealed universal features of nonstabilizerness qualitatively distinct from entanglement. In bosonic and fermionic systems, however, a comparable formulation for their computational resource, non-Gaussianity, has yet to be established. In this work, we introduce a unified measure, the magic Rényi entropy (MRE), to quantify computational resources in spins, bosons, and fermions on an equal footing. We show that the MRE is a resource monotone under stabilizer and Gaussian protocols involving measurements and feedforward operations. The MRE reveals common universal aspects of nonstabilizerness and non-Gaussianity in critical many-body states. In particular, our CFT analysis shows that the universal contribution to the MRE appears as the size-independent term determined by the Affleck-Ludwig boundary entropy. We find that non-Gaussianity can continuously renormalize this universal contribution or drive a boundary transition through bulk-induced boundary renormalization-group flows. As a concrete example, we present a CFT analysis of non-Gaussianity in interacting spinless fermions described by the Tomonaga-Luttinger liquid, showing boundary transitions at the Luttinger parameters K=1/3K=1/3K=1/3 and K=3K=3K=3. Our field-theoretical predictions are confirmed by numerical calculations. These results provide a unified field-theoretical understanding of many-body magic across spins, bosons, and fermions.
Establishing the precise computational boundary between classically tractable fermionic systems and those capable of genuine quantum advantage is a central challenge in quantum simulation. While injecting non-Gaussian ``magic" inputs into free-fermion circuits is widely expected to generate intractable complexity, we identify a physically motivated intermediate regime. We prove that for block-product paired non-Gaussian fermionic states, essential quantum simulation primitives -- transition amplitudes, overlaps, and arbitrary-weight number correlators -- can be efficiently approximated to additive error under free-fermionic dynamics. This tractability stems from an algebraic reduction that compresses exponentially large multiparticle interference into a single coefficient of a multivariate Pfaffian polynomial. Because these classical estimators match the intrinsic O(1/K)O(1/\sqrt{K})O(1/K​) statistical uncertainty of quantum hardware utilizing KKK measurement shots, they constitute a practical benchmark. Building on this foundation, we construct an additive-error estimator for high-weight Wilson observables in the noninteracting quench of recent trapped-ion experiments, providing a rigorous classical benchmark. Extending this to quantum chemistry, we demonstrate that core overlap-based subroutines for antisymmetrized products of strongly orthogonal geminals admit efficient additive-error Pfaffian-kernel estimators. Ultimately, these results sharpen the boundary of quantum advantage, establishing that the paired-electron scaffold is dequantized and clarifying where quantum resources are indispensable.
Fermionic non-Gaussianity, also known as fermionic magic, is a key resource underlying the computational complexity of fermionic quantum systems, yet quantifying it in a computationally tractable and operationally grounded manner remains a central challenge. In this work, we address this gap by establishing a convex resource theory of fermionic non-Gaussianity and introducing two families of computable measures for pure fermionic states, both built from the Williamson normal form of the covariance matrix. The first family, the occupation number entropies, is defined as the Tsallis-α\alphaα entropy of the occupation numbers, obtained as shifted and rescaled Williamson eigenvalues of the covariance matrix. We prove that one member of this family yields a monotone under Gaussian protocols, thereby validating it as a computable convex resource monotone for fermionic non‑Gaussianity. As a consequence, it provides a lower bound on the number of non-Gaussian gates required for state preparation. The second family, the natural-orbital participation entropy, is the R\'enyi-α\alphaα entropy of the squared magnitude of the state's amplitudes in the natural-orbital basis, defined by the eigenvectors of the covariance matrix. Operationally, they quantify the compressibility of the state in the natural-orbital basis, and hence upper bound the classical simulation cost in an orthonormal Gaussian basis. We analyze the two measures in two important settings, for stabilizer and translation-invariant states, where they admit simplified forms and additional structural insights. We investigate both families through several representative examples, including random SWAP-doped matchgate circuits and the bond-modulated XXZ model, showcasing 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.