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

Quantum Complexity in Fermionic Systems: From Deformation to Pairing

Understanding collective phenomena such as deformation and superfluid pairing in interacting fermionic systems is a central challenge in many‑body physics. In their recent preprint, Sönke Momme Hengstenberg and Caroline E. P. Robin examine how different quantum‑information measures—entanglement entropy, non‑stabilizerness (or "magic"), and fermionic non‑Gaussianity—behave across these phases, and they propose a practical framework for classically scalable simulation using stabilizer‑state expansions.

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1. Contrasting Quantum Resources in Deformed and Paired Phases

The authors study a model that can be tuned between a deformed phase (characterized by a collective shape deformation) and a pairing phase (exhibiting superfluid‑like Cooper pairing). By computing standard entanglement measures they find:

• Deformed phase – the ground state displays relatively high bipartite entanglement, yet its non‑stabilizerness remains low. In other words, the state is close to a stabilizer (Clifford) state, making it amenable to efficient classical description despite the entanglement.
• Pairing phase – the system shows strong multipartite entanglement and a pronounced increase in non‑stabilizerness. Additionally, the fermionic anti‑flatness (a proxy for non‑Gaussianity) reaches its maximum, indicating that the state lies far from the Gaussian manifold.

These observations illustrate that entanglement alone does not capture the full simulation cost; the presence of magic and non‑Gaussianity can dramatically increase the resources required for quantum simulation.

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2. Stabilizer‑State Expansion as a Scalable Ansatz

To address the computational difficulty, the authors adopt a stabilizer‑state expansion strategy. The key steps are:

1. Physics‑guided Hamiltonian decomposition – the Hamiltonian is split into parts that naturally map onto stabilizer states (e.g., graph‑state constructions) and residual terms that generate non‑stabilizer contributions.
2. Construction of a basis of stabilizer states – each stabilizer state is built from a graph state, preserving the symmetries of the original problem (particle‑number, spin, etc.).
3. Variational basis optimization – the coefficients of the stabilizer basis are optimized to minimize the energy, yielding a compact representation of the ground state.

In the deformed phase, a single stabilizer state already captures most of the ground‑state physics, reflecting the low magic content observed earlier. By contrast, the pairing phase formally requires an exponentially growing number of stabilizer states to achieve the same fidelity, because the magic and non‑Gaussianity are extensive.

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3. Symmetry‑Preserving Basis Rotations to Tame the Pairing Phase

Recognizing the exponential blow‑up, the authors introduce a symmetry‑preserving basis rotation. By rotating the stabilizer basis within the subspace that respects the model’s conserved quantities, they obtain a new set of states that align more closely with the true ground state. This rotation dramatically reduces the number of required stabilizer components while maintaining high fidelity (often > 95 %).

The rotated state serves two practical purposes:

• It provides a high‑quality classical approximation that can be used as a starting point for further refinement.
• It offers a compact seed for quantum algorithms (e.g., variational quantum eigensolvers), where the initial state already encodes much of the correlation structure, potentially lowering circuit depth and error accumulation.

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4. Implications and Outlook

The study highlights several broader points for the quantum‑simulation community:

• Resource diagnostics – entanglement, magic, and non‑Gaussianity together give a nuanced picture of simulation difficulty. Systems that are highly entangled but low in magic may still be tractable classically.
• Stabilizer‑centric methods – leveraging graph‑state constructions and stabilizer expansions can yield scalable, symmetry‑respecting approximations, especially in phases with limited magic.
• Hybrid classical‑quantum workflows – a well‑optimized stabilizer‑state ansatz can act as a bridge, delivering a near‑optimal classical description that seeds quantum refinement, thereby reducing the quantum resource overhead.

Future work could explore extending these techniques to larger lattice models, incorporating dynamical (time‑dependent) phenomena, and testing the approach on near‑term quantum hardware. The authors’ combination of quantum‑information diagnostics with physically motivated ansätze offers a promising pathway toward practical simulations of complex fermionic matter.

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Takeaway: By dissecting how entanglement, magic, and non‑Gaussianity differentiate deformation and pairing, and by showing how stabilizer‑state expansions—augmented with symmetry‑preserving rotations—can efficiently capture both phases, the paper provides a concrete roadmap for scalable, hybrid simulation of strongly correlated fermionic systems.