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arXiv:2610.31792

Quantum Complexity and Stabilizer-State Expansions: From Pairing to Deformation in Fermionic Systems

S. Momme Hengstenberg, Caroline E. P. Robin

quant-phcond-mat.str-elnucl-thSubmitted Oct 22, 2026• Announced Oct 26, 2026• 0 citations
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Abstract

Understanding how collective phenomena such as pairing and deformation emerge in interacting fermionic systems, and how the correlations underlying them shape the resources required for their simulation, is a central challenge in many-body physics. We investigate various aspects of quantum complexity in a fermionic system exhibiting two main forms of collectivity, specifically deformation and pairing, through entanglement, non-stabilizerness, and fermionic non-Gaussianity. In the deformed phase, high entanglement coexists with low non-stabilizerness, while the superfluid pairing phase, by contrast, exhibits substantial multipartite entanglement and non-stabilizerness, together with maximal fermionic anti-flatness. In a second step we examine how to describe the phase diagram of the system in a classically scalable and symmetry-preserving way, by combining stabilizer-state expansion and variational basis optimization. A physics-guided decomposition of the Hamiltonian identifies the relevant stabilizer states, which can be constructed from graph states. A single stabilizer state captures the deformed phase to a large extent, and, although the number of stabilizer states required to describe the pairing phase grows exponentially with system size, a symmetry-preserving basis rotation yields a high-fidelity approximation using far fewer states. This rotated state can then serve as a practical starting point for quantum refinement.

Comments
23 pages, 16 figures

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Understanding how collective phenomena such as pairing and deformation emerge in interacting fermionic systems, and how the correlations underlying them shape the resources required for their simulation, is a central challenge in many-body physics. We investigate various aspects of quantum complexity in a fermionic system exhibiting two main forms of collectivity, specifically deformation and pairing, through entanglement, non-stabilizerness, and fermionic non-Gaussianity. In the deformed phase, high entanglement coexists with low non-stabilizerness, while the superfluid pairing phase, by contrast, exhibits substantial multipartite entanglement and non-stabilizerness, together with maximal fermionic anti-flatness. In a second step we examine how to describe the phase diagram of the system in a classically scalable and symmetry-preserving way, by combining stabilizer-state expansion and variational basis optimization. A physics-guided decomposition of the Hamiltonian identifies the relevant stabilizer states, which can be constructed from graph states. A single stabilizer state captures the deformed phase to a large extent, and, although the number of stabilizer states required to describe the pairing phase grows exponentially with system size, a symmetry-preserving basis rotation yields a high-fidelity approximation using far fewer states. This rotated state can then serve as a practical starting point for quantum refinement.
We investigate how the stabilizer formalism, in particular highly entangled stabilizer states, can be used to describe the emergence of many-body shape collectivity from individual constituents in a symmetry-preserving and classically efficient way. The method that we adopt is based on determining an optimal separation of the Hamiltonian into a stabilizer component and a residual part inducing nonstabilizerness. The corresponding stabilizer ground state is efficiently prepared using techniques of graph states and stabilizer tableaux. We demonstrate this technique in context of the Lipkin-Meshkov-Glick model, a fully connected spin system presenting a second-order phase transition from spherical to deformed state. The resulting stabilizer ground state is found to capture to a large extent both bipartite and collective multipartite entanglement features of the exact solution in the region of large deformation. We also explore several methods for injecting nonstabilizerness into the system, including adaptive derivative-assembled pseudo-Trotter variational quantum eigensolver and imaginary-time evolution (ITE) techniques. Stabilizer ground states are found to accelerate ITE convergence due to a larger overlap with the exact ground state. While further investigations are required, the present work suggests that collective features may be associated with high but simple large-scale entanglement which can be captured by stabilizer states, while the interplay with single-particle motion may be responsible for inducing nonstabilizerness. This study motivates applications of the proposed approach to more realistic quantum many-body systems, whose stabilizer ground states can be used in combinations with powerful classical many-body techniques and/or quantum methods.
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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.
Stabilizer states constitute an important class of quantum states that can be generated from computational-basis states using Pauli operators and Clifford gates. Although they may exhibit substantial multipartite entanglement, quantum circuits restricted to stabilizer operations can be efficiently simulated classically and therefore cannot, by themselves, provide a quantum computational advantage. Such an advantage requires non-stabilizer resources, commonly referred to as quantum magic. In this paper, we investigate the non-stabilizerness of quantum states arising in a class of spin Hamiltonians by computing their stabilizer Rényi entropies. Using real-space renormalization-group techniques, we obtain a closed-form expression valid in the low-energy, large-distance regime. We analyze how quantum magic evolves under coarse-graining and explore its behavior across different parameter regimes and quantum phases.