Probes of chaos over the Clifford group and approach to Haar values

Abstract: Chaotic behavior of quantum systems can be characterized by the adherence of the expectation values of given probes to moments of the Haar distribution. In this work, we analyze the behavior of several probes of chaos using a technique known as Isospectral Twirling [1]. This consists in fixing the spectrum of the Hamiltonian and picking its eigenvectors at random. Here, we study the transition from stabilizer bases to random bases according to the Haar measure by T-doped random quantum circuits. We then compute the average value of the probes over ensembles of random spectra from Random Matrix Theory, the Gaussian Diagonal Ensemble and the Gaussian Unitary Ensemble, associated with non-chaotic and chaotic behavior respectively. We also study the behavior of such probes over the Toric Code Hamiltonian.

• The paper introduces a framework using isospectral twirling to compare Clifford and Haar chaos probes, highlighting key differences in scrambling metrics.
• It demonstrates that T-doped Clifford circuits converge to higher k-designs with resource estimates scaling as O(log d) for Haar-like randomness.
• The study reveals that entanglement and coherence measures merge over long times, underscoring limitations of Clifford-based randomizations.

Probes of Quantum Chaos Over the Clifford Group and Their Convergence to Haar Values

Introduction

The characterization of quantum chaos increasingly leverages statistical and dynamical probes, often benchmarked against random matrix theory (RMT) ensembles. While the Haar-random unitary ensemble is the accepted paradigmatic model for fully chaotic quantum dynamics, physically relevant systems—including those realizable in quantum computation—are frequently restricted to subgroups such as the Clifford group. This work investigates the behavior of canonical chaos probes under isospectral twirling with respect to the Clifford group, its T-doped extensions, and their approach to Haar values. Analytical results clarify the consequences of stabilizerness, non-stabilizerness, and $k$-design structure for both chaotic and integrable regimes.

Theoretical Framework and Methodology

The study exploits isospectral twirling, keeping the spectrum of the Hamiltonian fixed and randomizing only the eigenbasis according to the group under consideration. This formulation admits a direct comparison between averages with respect to the unitary (Haar), Clifford, and T-doped Clifford ensembles—enabling controlled tracking of the transition between non-chaotic, integrable, and fully chaotic regimes.

RMT provides the foundation for statistical spectral analysis, with primary focus on the Gaussian Diagonal Ensemble (GDE) and the Gaussian Unitary Ensemble (GUE). The GDE models integrable dynamics, while the GUE typifies chaotic quantum systems. A central technical component is the concept of unitary $k$-designs—ensembles of unitaries that replicate the first $k$ moments of the Haar measure. The Clifford group forms a 3-design but not a 4-design, and addition of non-Clifford elements (notably via T-doping) enables convergence to higher designs.

The primary chaos probes assessed include:

• Loschmidt Echo (second kind),
• Out-of-Time-Order Correlators (OTOCs),
• Entropic quantities such as R\'enyi entropy and tripartite mutual information,
• Quantum coherence and Wigner-Yanase-Dyson skew information.

The computations exploit the explicit structural decomposition of moments over commutants of the relevant groups and their representation-theoretic features.

Analytical Results for Chaos Probes

Distinctions Between Clifford and Haar Averages

The paper provides closed-form expressions for averaged chaos probes, sharply distinguishing between those sensitive and insensitive to stabilizerness. Specifically, scrambling metrics such as the Loschmidt Echo and OTOCs yield Clifford averages that differ asymptotically by a factor of $d$ (Hilbert space dimension) from Haar averages. In contrast, entropic and coherence-based measures coalesce for both Clifford and Haar randomized eigenbases in the long-time limit, reflecting the Clifford group's capacity to generate maximal entanglement.

At short times, all probes—regardless of group average—display identical behaviors, highlighting that rapid scrambling can emerge from stabilizer dynamics. This observation has implications for fast quantum information spreading in systems with classically simulable dynamics.

Role of Non-Stabilizerness and T-Doping

Introducing non-Clifford resources via T-doped Clifford circuits produces an interpolating family of ensembles. As T-layer depth increases, the Clifford average converges to the Haar average for all probes, with the number of layers required scaling as $\mathcal{O}(\log d)$. This result provides explicit resource estimates for achieving Haar-like randomness from Clifford-origin circuits.

Spectrally, when probing GUE (chaotic) spectra, Clifford-averaged probes exhibit suppression of the characteristic Haar-induced oscillations and approach different equilibrium values than their Haar counterparts. For GDE (integrable) spectra and the Toric Code, which serves as a prototypical non-chaotic stabilizer Hamiltonian, the long-time and oscillatory behaviors are likewise group-sensitive.

Clifford Spectral Form Factors and Structural Implications

The analysis identifies and characterizes "Clifford spectral form factors" $\tilde{g}_k(t)$, which nuance the connection between group average and spectral statistics. Notably, due to the Clifford group's 3-design property, the fourth-order Clifford average does not depend on the standard four-point form factor $g_4(t)$, but rather on a modified three-point function $\tilde{g}_3(t)$. The hierarchy of design sensitivity determines the structural origin of quantum memory retention and equilibration bounds in the system.

The technical requirement that averaging over the Clifford group only yields stabilizer states with stabilizer Hamiltonians is elucidated. This constraint ensures controlled resource accounting and sharp discrimination between stabilizer and general Haar randomness in the eigenbasis.

Group Averaging and $k$-Design Transitions

T-doped Clifford circuits enable continuous interpolation between Clifford and Haar behavior via a detailed analysis of the relevant commutant structure and explicit diagonalization of the doping operator's action. Projectors onto the symmetric and antisymmetric irreducible representations of the permutation group $S_4$ play a central role in resolving the transition to higher $k$0-designs. The matrix framework developed permits precise quantification of convergence rates as a function of doping.

Implications and Prospects

The results demonstrate that:

• For probes sensitive to eigenvector complexity, Clifford-group averaging leads to partial scrambling and memory retention, whereas Haar-randomization maximally erases initial-state information.
• Entanglement and coherence properties cannot distinguish between Clifford and Haar averages for large system size due to the 3-design property and the ability of Clifford circuits to saturate entanglement limits.
• Group-based averages provide a principled, resource-aware method to interpolate between efficient (classically simulable) and maximally random quantum dynamics.

Theoretically, this delineates the boundaries of Clifford-based simulation efficacy and the necessity of non-stabilizer resources for physical processes tied to higher design conditions. The results empower future work on quantum simulation, benchmarking, and complexity certification in near-term quantum devices, as well as extending diagnostic tools to higher moments (designs), with cryptographic and tomography implications.

Conclusion

This work presents a rigorous framework for analyzing the approach to Haar randomness of chaos probes under Clifford and T-doped Clifford group averages. It establishes the kinematic and dynamical boundaries between classically tractable (Clifford) and genuinely quantum-random (Haar) regimes, quantifies convergence rates via T-doping, and elucidates the differentiability of chaos indicators contingent on their design sensitivity. These advances set the stage for systematic exploration of design properties in chaotic dynamics, foundational studies of complex quantum systems, and practical resource assessments for randomization protocols in quantum technologies.

---

Reference:

"Probes of chaos over the Clifford group and approach to Haar values" (2603.29695).