Non-Stabilizerness in Qubit-Regularised QFT

• The paper demonstrates that stabilizer Rényi entropies, particularly the second-Rényi measure, quantitatively diagnose magic in qubit-regularised QFT by linking it to OTOC fluctuations.
• It details methodologies that assess magic in many-body systems, contrasting nonlocal W-states with GHZ-type states to highlight non-stabilizer characteristics.
• The work connects non-stabilizerness to quantum chaos and computational intractability, introducing protocols for real-time magic tracking and NNQS-based evaluations.

Non-stabilizerness, also termed "magic," quantifies the extent to which quantum states in qubit-regularised quantum field theories (QFTs) deviate from the stabilizer manifold—the set of states and operations accessible to Clifford circuits and therefore efficiently simulable classically. As a pivotal resource for quantum computational advantage, understanding, quantifying, and controlling non-stabilizerness is essential in assessing complexity, chaos, and the simulatability of qubit-based field-theoretic models.

1. Quantifying Non-Stabilizerness: Magic Measures and Diagnostic Protocols

In qubit-regularised QFTs, the field degrees of freedom are discretized onto a lattice of qubits, with the computational resource content of the state measured by its distance from the stabilizer set. The key family of monotones used are the stabilizer Rényi entropies (SRE), particularly the second-Rényi variant: $$M_2(|\Psi_N\rangle) = -\log_2 \left( \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle\Psi_N|P|\Psi_N\rangle^4 \right)$$ where the sum is over all $N$-qubit Pauli operators. $M_2 = 0$ for any stabilizer state, while $M_2 > 0$ diagnoses non-stabilizerness. For systems of prime odd dimension, related measures such as mana, defined in terms of the negativity of the discrete Wigner function,

$$\mathcal{M}(\rho) = \log \left(\sum_u |W_\rho(u)| \right)$$

and its Rényi generalizations, are appropriate (Tarabunga, 2023).

Operationally, evaluating these measures exactly is exponentially expensive for large systems. However, recent work has established a direct relationship between $M_2$ (and mana) and the fluctuations of out-of-time-order correlators (OTOCs), providing an efficient protocol for magic estimation. The OTOC

$$\mathrm{OTOC}(t) = \mathrm{Re} \langle A^\dagger(t)B^\dagger A(t) B \rangle, \quad A(t) = U^\dagger(t)A(0)U(t)$$

serves as a scrambling witness, and the standard deviation of OTOC samples across Clifford circuit realizations, $\delta_{\mathrm{OTOC}}$, is tightly linked to $M_2$ via an approximate linearity: $$M_2 \approx -\alpha \log_2(\delta_{\mathrm{OTOC}}) + \beta$$ with $\alpha,\beta$ set by ensemble details (Ahmadi et al., 2022). This approach enables real-time tracking of magic growth during lattice QFT dynamics, as illustrated by sandwiching Hamiltonian evolution between Clifford layers.

2. Structure and Dynamics of Non-Stabilizerness in Many-Body and Field Systems

The manifestation of magic in qubit-regularised quantum field theories is nuanced by underlying many-body structure. In models with ground states built as symmetric superpositions over classically degenerate configurations—such as W-states in topologically frustrated spin chains—the SRE exhibits a logarithmic increase with system size,

$M_2(|W\rangle) = 3\log_2 L - \log_2(7L-6)$

signaling delocalized, nonlocal quantum complexity without classical analog (Odavić et al., 2022). This contribution is uniquely nonlocal, distinguishing it sharply from GHZ-type (Clifford-orbit) states which, while highly entangled, have zero non-stabilizerness.

Critical lattice models derived from conformal field theory, such as the three-state Potts chain, display universal scaling of "mutual mana"—a magic-based analogue of entanglement entropy—at criticality: $$I_{\mathcal{M}}(\ell, L) \sim \log \left(\frac{L}{\pi} \sin\left( \frac{\pi \ell}{L} \right) \right)$$ This logarithmic scaling, which mirrors that of the entanglement entropy, provides both a diagnostic for conformal invariance and a sharp indicator of classical intractability (Tarabunga, 2023). In integrable/gapped phases, mutual mana saturates, reflecting reduced complexity.

Magic generation in explicitly dynamical settings, such as multi-particle quantum walks in the XXZ chain, follows light-cone propagation: single-particle dynamics in the easy-plane regime ($\Delta < 1$) yield $M_2(t) \sim \log_2 v_F t$, while strong interactions ($\Delta > 1$) cause the formation of slowly propagating doublons and much slower magic growth (Moca et al., 28 Apr 2025). The statistical structure of the Pauli spectrum asymptotically acquires universal (Poisson) statistics, robustly independent of interaction strength or particle number.

3. Magic, Quantum Chaos, and Simulatability

Quantifying non-stabilizerness also provides a sharp tool for diagnosing quantum chaos and complexity. In the Sachdev-Ye-Kitaev (SYK) model, the expansion coefficients of a many-fermion ground state in the Majorana basis (the "Majorana spectrum") are found to be Gaussian-distributed for the maximally chaotic SYK case, but Laplace-distributed in non-chaotic variants (SYK$_2$) (Bera et al., 3 Feb 2025). The stabilizer Rényi entropy again scales extensively: $M_\alpha \sim D_\alpha N + c_\alpha$, with a larger slope—hence higher magic—for chaotic models, establishing a resource-theoretic basis for chaos identification.

This resource perspective is reinforced in circuit models: interleaving non-Clifford operations with randomized Clifford gates induces a "thermalization" of non-stabilizerness toward its Haar-typical maximum,

$$\langle m_p(U^{(t)}) \rangle = \bar{m}_p [1 - (1 - m_p(U)/\bar{m}_p)^t]$$

where $m_p$ is the non-stabilizing power of $U$, and $\bar{m}_p$ is its Haar average. Therefore, even sparse non-Clifford "seeding" completely "thermalizes" magic content over time, with Clifford operations acting as thermalization agents for quantum computational complexity (Varikuti et al., 20 May 2025).

4. Measurement, Resource Theories, and Magic Dynamics

Non-stabilizerness is critically affected by measurements and noise. Recent resource theory developments define "magic-breaking channels" as those mapping any quantum state to a stabilizer state (rendering it classically simulable). For qubits, this occurs if and only if the output Bloch ellipsoid of the channel is contained in the stabilizer polytope $\{|m_1| + |m_2| + |m_3| \leq 1\}$. Importantly, tensor products of magic-breaking channels may fail to break magic in correlated (multiqubit) inputs due to surviving joint magic (Patra et al., 6 Sep 2024), identifying a subtlety in dynamical resource theories of magic preservability relevant for noisy field-theoretic simulators.

Moreover, monitored quantum circuits—systems subject to repeated projective measurements—display robust protection of magic in the presence of Clifford scrambling: in the computational basis, erasing magic (as measured by the stabilizer nullity) requires exponentially many measurements. Allowing measurements in non-Clifford (magic) bases can both destroy and—in contrast—generate nontrivial steady-state magic, with the stabilizer Rényi entropy providing a fine-grained probe (scaling as $\theta_m^2$ with rotation angle) (Scocco et al., 15 Jul 2025).

The temporal dynamics of magic also underlie quantum optimization and phase transitions. In the Quantum Approximate Optimization Algorithm and adiabatic quantum annealing, passage from an initial to a solution stabilizer state traverses a "magic barrier": a transient regime of high non-stabilizerness, the crossing of which is necessary for algorithmic success (Capecci et al., 22 May 2025). The extent of demagication—i.e., the drop in magic from maximum to final state—has a quantitative relationship with the solution fidelity. This central feature imposes resource costs on shallow/noisy quantum optimization and may have analogues in complexity transitions in field-theoretic models.

5. Practical Protocols and Neural Quantum States

Sampling the stabilizer Rényi entropy or mana directly is feasible for small or structured systems, but for large or strongly correlated QFTs, variational neural network quantum states (NNQS) such as Restricted Boltzmann Machines offer scalable representations. Monte Carlo methods based on replicated estimator schemes (e.g., the fourfold replica estimator for $M_2$)

$$\exp(-M_2(|\psi\rangle)) = \langle\Phi|\hat{U}|\Phi\rangle / \langle\Phi|\Phi\rangle$$

with $|\Phi\rangle$ the replicated state and $\hat{U}$ the appropriate unitary, enable direct evaluation of SRE even in high-entanglement/high-magic regimes (Sinibaldi et al., 13 Feb 2025, Crew et al., 13 Aug 2025).

This approach has been implemented for the Schwinger model with a topological term; the observed SRE exhibits strong dependence on the separation between external probe charges. Magic content increases as charges are brought into proximity—a signature that the variationally optimized ground state becomes less classically simulable in the confined regime, a direct diagnostic of classical intractability for gauge theories with non-trivial infrared structure (Crew et al., 13 Aug 2025).

6. Long-Range Non-Stabilizerness and Classification of Phases

A further refinement distinguishes "long-range non-stabilizerness": the portion of magic robust against the action of any shallow local quantum circuit. This is operationally defined for matrix product state (MPS) ground states as the noninteger value of the Shannon entropy of block probabilities in renormalization-group fixed points,

$H(\{|\alpha_k^{(N)}|^2\}) \not\in \mathbb{N}$

If the mutual information between distant regions, $I_{A,B} = H(\{p_k\})$, is non-integer in the thermodynamic limit, the state possesses long-range magic that is robust to polylogarithmic-depth circuits (Korbany et al., 26 Feb 2025). This robustness serves as a fingerprint of nontrivial quantum phases, akin to the role of topological entanglement entropy, and may assist in classifying phases in both many-body and field-theoretic models.

7. Outlook and Applications

Non-stabilizerness is now recognized as a central computational resource in qubit-regularised quantum field theory, directly informing quantum advantage, complexity, chaos, and the boundaries of classical simulatability. Its measurement—via stabilizer entropies, mana, OTOC fluctuations, entanglement spectrum flatness, or neural quantum state sampling—enables both theoretical insight and algorithmic benchmarking.

Key conceptual advances include:

• Universal scaling of magic measures at criticality (diagnostic for conformal invariance).
• Protocols for experimental and numerical measurement of magic in large systems, including measurement-induced transitions.
• Robustness and preservation of magic under Clifford scrambling, and limitations imposed by magic-breaking processes.
• The non-amplifiability of magic through cloning or broadcasting, a fundamental no-go in resource theory (Gupta et al., 27 Jan 2025).
• The direct connection between non-stabilizerness, quantum information scrambling, and the emergence of many-body quantum chaos.

Future directions involve optimizing resource-efficient simulation strategies, developing hybrid error-mitigation protocols sensitive to magic preservability, and characterizing long-range non-stabilizerness as a novel invariant for quantum phases, both in QFTs and in system architectures motivated by quantum computation.