Non-Stabilizer Complexity & Entanglement Transitions

• The paper demonstrates that stabilizer Rényi entropy (SRE) precisely quantifies non-stabilizer resources, marking distinct complexity transitions in frustrated systems.
• It reveals that non-local magic resources cause sharp jumps in SRE, distinguishing true quantum computational hardness from conventional entanglement measures.
• The framework unifies analyses across spin chains, random circuits, and lattice gauge theories, offering a robust diagnostic for quantum phase transitions and simulation limits.

Complexity and Entanglement Transitions via Non-Stabilizer Resources provide a unified framework for understanding the emergence of quantum complexity in many-body systems, especially in regimes where conventional entanglement measures are insufficient. Central to this development is the stabilizer Rényi entropy (SRE), also known as "magic," which rigorously quantifies the non-stabilizer (non-Clifford) resource content of quantum states. Non-stabilizer resources are crucial for the onset of true quantum computational hardness, underpinning both the complexity of simulation and the nature of quantum phase transitions in numerous models ranging from frustrated spin chains and lattice gauge theories to random circuits and strongly interacting fermion systems.

1. Fundamental Definitions and Resource Quantifiers

The main complexity diagnostic is the stabilizer Rényi-$\alpha$ entropy: $$M_\alpha(|\psi\rangle) = \frac{1}{1-\alpha} \log_2 \left[2^{-N} \sum_{P \in \mathcal P_N} |\langle\psi|P|\psi\rangle|^{2\alpha} \right]$$ for a pure $N$-qubit state $|\psi\rangle$, with $P$ running over the $4^N$ $N$-qubit Pauli strings. For $\alpha=2$, $M_2$ gives an operationally tractable monotone that is strictly zero for stabilizer states and positive otherwise, reflecting the presence of non-Clifford (magic) resources (Odavić et al., 2022, Haug et al., 2024).

Related measures include:

• Mana: $M(\rho) = \log_2 \|\rho\|_W$ using the discrete Wigner norm, widely used for resource theory of magic (Ahmadi et al., 2024).
• Robustness of Magic: minimized decomposition distance to convex hull of stabilizer states.
• Non-stabilizerness Entanglement Entropy (NsEE): the minimal bipartite entanglement entropy achievable by Clifford unitaries, isolating entanglement that cannot be removed by free (Clifford) operations (Huang et al., 2024).

2. Complexity Transitions in Spin Chains and Frustrated Systems

Complexity transitions associated with non-stabilizer resources are sharply exposed in topologically frustrated quantum spin chains. At classical frustration points (e.g., antiferromagnetic Ising coupling on a ring with odd length), the ground state is a delocalized W-state with nontrivial SRE scaling as $3\log_2 L$ for a chain of length $L$—a resource with no counterpart in GHZ (unfrustrated) states, which always have zero SRE (Odavić et al., 2022). As one tunes away from the classical point, local quantum correlations dress the W-state, and the SRE decomposes as: $$M_2(J=+1, L, \lambda) = M_2(J=-1, L, \lambda) + M_2^W(L)$$ enumerating local and non-local (W-state) contributions separately.

Non-local magic resources drive complexity transitions not seen in entanglement entropy, which remains continuous even as SRE exhibits a jump—an indicator of a "pure magic transition" (Catalano et al., 2024). This signals fundamentally new quantum resource behavior at frustrated points, making such systems central for studies of non-stabilizer-induced complexity.

3. Magic–Entanglement Interplay and Diagnosing Quantum Phases

In generic spin models (e.g., XXZ, XY, Cluster Ising), magic quantifiers such as $M_2$ sharply distinguish between trivial, critical, and nontrivial quantum phases. Alongside SRE, quantities such as entanglement spectrum anti-flatness ($\mathcal F_A = p_3 - (p_2)^2$) and capacity of entanglement directly track phase transitions, providing a robust, model-independent phase portrait (Viscardi et al., 11 Mar 2025).

Deep in trivial or stabilizer-like phases, $M_2 \approx 0$ and $\mathcal F_A \approx 0$. At quantum critical points or symmetry-protected topological (SPT) boundaries, $M_2$ and $\mathcal F_A$ both peak, while in SPT and topological phases, residual extensive magic reflects persistent complexity.

4. Random Circuits, Universal Magic Saturation, and Magic Transitions

In the context of random Clifford circuits doped with $T$ (non-Clifford) gates, complexity transitions are universally characterized by a critical T-gate density $q_{c, \alpha}$ above which SREs saturate their maximum: $$q_{c,\alpha} \approx \frac{M_\alpha^{\max}}{N M_\alpha^T}$$ This threshold is $O(1)$ in the large-$N$ limit, independent of $N$ and $\alpha$ (Haug et al., 2024). At the threshold, the derivative of $M_2$ for different $N$ exhibits finite-size scaling collapse—a marker of a true phase transition in resource content.

Distinct universality classes emerge: for $\alpha<1$, SRE rapidly becomes large with very little magic, closely tracking the transition to classical intractability; for $\alpha>1$, saturation signifies the scaling of circuit fidelity estimation cost. These differences encode operationally distinct transitions in circuit complexity and state certification rates.

5. Measurement-Induced and Hybrid Phase Structure

Hybrid random circuits with injected non-Clifford resources and projective measurements demonstrate the entanglement–magic separation phenomenon: the critical measurement rate for the transition from volume-law to area-law entanglement ($p_c^{\mathrm{ent}}$) differs from that for magic ($p_c^{\mathrm{mag}}$), with $p_c^{\mathrm{mag}} > p_c^{\mathrm{ent}}$ (Fux et al., 2023). In the intermediate regime, states are area-law entangled yet possess sub-extensive (power-law) magic, marking a regime of classical intractability that goes beyond standard entanglement-based criteria. This bifurcation highlights magic as a more stringent indicator for quantum advantage.

6. Lattice Gauge Theories, Operator Resources, and Quantum Simulability

In lattice gauge theories (LGTs) with Abelian and non-Abelian symmetries, the complexity cost and entanglement content are governed not solely by group structure but also by superselection sectors and local encoding. Discrete gauge theories (e.g., $\mathbb{Z}_N$, $D_3$) exhibit strong peaking of SRE and multipartite entanglement in crossover regions, but stabilizer-like (classically tractable) limits at extremes. For continuous groups (SU(2)), magic remains extensive throughout, representing a severe complexity bottleneck for simulations (Santra et al., 8 Oct 2025).

The scaling of SRE and empirical thresholds map directly onto the classical hardness of simulating such gauge models, tightly linking resource theory to emergent quantum advantage and NISQ/fault-tolerant boundaries.

7. Synthesis: Topological, Dynamical, and Algorithmic Manifestations

Non-stabilizer complexity transitions admit both physical and topological interpretations. In $SU(2)_1$ Chern-Simons theory and related topological constructions, non-stabilizer states (e.g., $W_n$, Dicke) are associated with increased genus in the path integral manifold and higher insertion number of Wilson loops—a geometric "complexity monotone" that parallels magic resource scaling. Abrupt jumps in topological complexity coincide with shifts in entropy per subsystem, mirroring phase transitions at the level of quantum resources (Munizzi et al., 16 Oct 2025).

Algorithmically, classical simulation capacity is sharply delimited by the presence of non-stabilizer entropy, as captured by NsEE. In Clifford-augmented MPS approaches, the hardness transition aligns with the emergence of nonzero residual entanglement that resists Clifford disentangling—validating the link between resource measures and simulability thresholds (Huang et al., 2024).

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References:

• (Odavić et al., 2022): "Complexity of frustration: a new source of non-local non-stabilizerness"
• (Catalano et al., 2024): "Magic phase transition and non-local complexity in generalized W State"
• (Viscardi et al., 11 Mar 2025): "Interplay of entanglement structures and stabilizer entropy in spin models"
• (Haug et al., 2024): "Probing quantum complexity via universal saturation of stabilizer entropies"
• (Fux et al., 2023): "Entanglement-magic separation in hybrid quantum circuits"
• (Santra et al., 8 Oct 2025): "Quantum Resources in Non-Abelian Lattice Gauge Theories: Nonstabilizerness, Multipartite Entanglement, and Fermionic Non-Gaussianity"
• (Munizzi et al., 16 Oct 2025): "Topological Preparation of Non-Stabilizer States and Clifford Evolution in SU(2)_1 Chern-Simons Theory"
• (Huang et al., 2024): "Non-stabilizerness Entanglement Entropy: a measure of hardness in the classical simulation of quantum many-body systems"