Fermionic Magic Resources of Quantum Many-Body Systems

TL;DR

The paper introduces fermionic antiflatness (FAF), a computationally efficient measure of fermionic non-Gaussianity that vanishes on fermionic Gaussian states and is invariant under Gaussian unitaries. By leveraging the fermionic commutant and the covariance matrix of Majorana modes, FAF captures the extent to which a state deviates from Gaussianity, with a clear physical interpretation in terms of Majorana correlations. The authors systematically explore FAF across simple, typical, RMPS, and circuit-generated states, and examine its behavior in equilibrium models (TFIM and ANNNI) and in out-of-equilibrium dynamics, including ergodic evolution and dynamics under random circuits. They uncover universal scaling near critical points, boundary-condition dependent subleading terms, and rapid FAF growth in ergodic settings, highlighting FAF as a meaningful resource describing quantum state complexity beyond entanglement and nonstabilizerness. These results establish FAF as a practical diagnostic for quantum complexity in many-body systems and motivate future work on mixed states, experimental verification, and connections to other non-Gaussianity measures.

Abstract

Understanding the computational complexity of quantum states is a central challenge in quantum many-body physics. In qubit systems, fermionic Gaussian states can be efficiently simulated on classical computers and hence can be employed as a natural baseline for evaluating quantum complexity. In this work, we develop a framework for quantifying fermionic magic resources, also referred to as fermionic non-Gaussianity, which constitutes an essential resource for universal quantum computation. We leverage the algebraic structure of the fermionic commutant to define the fermionic antiflatness (FAF)-an efficiently computable and experimentally accessible measure of non-Gaussianity, with a clear physical interpretation in terms of Majorana fermion correlation functions. Studying systems in equilibrium, we show that FAF detects phase transitions, reveals universal features of critical points, and uncovers special solvable points in many-body systems. Extending the analysis to out-of-equilibrium settings, we demonstrate that fermionic magic resources become more abundant in highly excited eigenstates of many-body systems. We further investigate the growth and saturation of FAF under ergodic many-body dynamics, highlighting the roles of conservation laws and locality in constraining the increase of non-Gaussianity during unitary evolution. This work provides a framework for probing quantum many-body complexity from the perspective of fermionic Gaussian states and opens up new directions for investigating fermionic magic resources in many-body systems. Our results establish fermionic non-Gaussianity, alongside entanglement and non-stabilizerness, as a resource relevant not only to foundational studies but also to experimental platforms aiming to achieve quantum advantage.

Figures (11)

• Figure 1: Resources characterizing the computational complexity of quantum many-body states $\ket{\Psi_{a,b}}$ of $N$ qubits. (a) Entanglement is a central tool for analyzing many-body phenomena and has recently been complemented by measures of non-stabilizerness (non-stab.). This work introduces a framework for studying fermionic magic resources, proposing the fermionic antiflatness (FAF), $\mathcal{F}_k$, as a diagnostic in many-body systems. (b) Hilbert space $\mathcal{H}_2^{\otimes N}$ for $N$ qubits, showing the classes of computationally tractable states: tensor-network states with limited entanglement, $S_{\mathrm{ent}}=O(1)$, stabilizer states with vanishing SRE, $\mathcal{M}_q=0$, and fermionic Gaussian states with vanishing FAF, $\mathcal{F}_k=0$.
• Figure 2: Fermionic antiflatness $\mathcal{F}_k$ in random MPS. (a) The difference $\Delta \mathcal{F}_1$, Eq. \ref{['eq:dF']}, between FAF of RMPS and FAF of typical state $\mathcal{F}^{\mathrm{typ}}_k$ as a function of bond dimension $\chi$ in $\mathbb{Z}_2$-symmetric RMPS (i.e., with fixed $\mathcal{P} =1$) in a system of $N$ qubits. (b) At large $N$, $\Delta \mathcal{F}_k/N$ decays according to a power-law $\Delta \mathcal{F}_k/N \propto \chi^{-\beta}$, for both RMPS with and without the $\mathbb{Z}_2$ symmetry for $k=1,2$. The exponent $\beta=2.00(5)$ for $k=1$ and $\beta=3.7(3)$ for $k=2$. For clarity of presentation, data for $k=1$ are rescaled by a factor $80$, while data for $k=2$ are rescaled by factors $1/10$ and $1/80$, respectively, for RMPS with and without the $\mathbb{Z}_2$ symmetry.
• Figure 3: Fermionic antiflatness $\mathcal{F}_k$ growth in random quantum circuit dynamics acting on $N$ qubits. (a) $\mathcal{F}_k$ (shown for $k = 1, 2$) increases rapidly under the dynamics of a $\mathbb{Z}_2$-symmetric circuit of depth $t$. (b) The difference $\Delta \mathcal{F}_1$ decays exponentially in time $t$, signaling the saturation of FAF toward its Haar-random value $\mathcal{F}^{\mathrm{typ}}_k$. (c) The rescaled gap $\Delta \mathcal{F}_k/N$ collapses across system sizes onto a universal curve decaying as $\propto e^{-\alpha_k t}$, with rates $\alpha_1 = 0.45(2)$ and $\alpha_2 = 0.73(3)$. For clarity, data for $k = 1$ (without $\mathbb{Z}_2$ symmetry) are rescaled by a factor $1/10$, and for $k = 2$ (with $\mathbb{Z}_2$ symmetry) by $1/100$. (d) The saturation time $t_{\mathrm{sat}}$, defined by the condition $\Delta \mathcal{F}_k = \epsilon$ (with $\epsilon = 1$), scales logarithmically with system size: $t_{\mathrm{sat}} \propto \log(N)$.
• Figure 4: Fermionic antiflatness $\mathcal{F}_k$ in the impurity model Eq. \ref{['eq:imp1']}. (a) The FAF (shown for $k=1,2$) is enhanced in the vicinity of the ferromagnetic–paramagnetic transition, and the slope of the $\mathcal{F}_k(h_z)$ curves increases near the critical point $h_z = h_c = 1$, indicated by the red dashed line. The transition belongs to the Ising universality class with critical exponent $\nu = 1$, as shown by the collapse of the Binder cumulant $\mathcal{B}$ in the inset. (b) The absolute value of the derivative $|\mathcal{F}'_k|$ with respect to $h_z$ exhibits a maximum at field $h = h_m$. The inset shows the scaling collapse of $|\mathcal{F}'_1| - F_1$ near the transition. (c) The value of $|\mathcal{F}'_{k=1}|$ at the maximum, denoted $F_1$, increases logarithmically with $N$. (d) The location $h_m$ of the maximum converges to $h_c$, with $\delta h = |h_m - h_c| \propto N^{-\beta_h}$ and $\beta_h = 0.98(4)$.
• Figure 5: Fermionic antiflatness $\mathcal{F}_k$ in ANNNI model Eq. \ref{['eq:ani']} with OBC. (a) The leading term $D_k$ in the FAF system size dependence (shown here for $k=1,2$) is enhanced in the vicinity of the ferromagnetic-paramagnetic transition. The latter belongs to the Ising universality class with the exponent $\nu=1$, as shown in the inset by the collapse of the Binder around the critical point $h_z=h_c=0.4367(1)$ for $\lambda=0.3$. (b) The subleading term $f_k$ admits a maximum at field $h=h_m$, and $f_k(h_m)$ is increasing logarithmically with system size $N$, as presented in (c), while its position is converging to $h_c$, as shown by $\delta h = |h_m-h_c|\propto N^{-\beta_h}$ (with $\beta_h=0.99(5)$), shown in panel (d). The inset in (b) shows the collapse of $f_k-f_k(h_m)$ at the transition.