The discovery of the superdiffusive regime sparked an interest in the interplay of transport with integrability and non-abelian symmetries. In this respect, the one-dimensional Fermi-Hubbard model represents an interesting playground: it is integrable and possesses a global U(1)⊗SU(2) symmetry in the charge and spin sectors, respectively, raised to SU(2)⊗SU(2) at half-filling. Hence, the possibility of selectively breaking SU(2) symmetry (in either channel) or breaking integrability establishes the Hubbard chain as a paradigmatic case to explore the connection between integrability, non-abelian symmetries, and anomalous diffusion.

It is possible to identify different transport regimes: [1,3]

1. ballistic (z=1) with Lieb-Robinson velocity in the non-interacting limit;

2. superdiffusive (z=3/2) at any finite interaction in SU(2)-symmetric sector;

3. reducing charge or spin-SU(N) to U(1) spoils superdiffusion in the corresponding channels;

4. diffusive (z=2) by breaking integrability (e.g., with a nearest-neighbor interaction). 

The full counting statistics P(Γ) characterize the fluctuation of a conserved charge in a subsystem. Its simulation is generally challenging and limited to relatively short timescales. An alternative route is to obtain the quantum generating function, which can be represented as a matrix product operator with limited entanglement growth, thus allowing to probe unprecedentedly-long timescales. 

We characterize the full counting statistics for the spin-1/2 isotropic Heisenberg (XXZ) chain in terms of the first few cumulants (mean, variance, skewness, and kurtosis). One can conclude that (i) the distribution is symmetric (i.e., all odd cumulants are identically zero) and (ii) while the variance displays a KPZ-like superdiffusive scaling, fluctuations are asymptotically Gaussian, This corroborate Google's conclusions but at the same time, it extend the analysis far beyond the typical timescales that are experimentally accessible — limited by the coherence time of the underlying superconductive qubit architecture. In short, we demonstrate that spin superdiffusion in integrable quantum spin chains is incompatible with KPZ universality [2]. 

Furthermore, it can be shown that the quantum analog of the classical roughness (proportional to the standard deviation of the spin transfer distribution) satisfies the self-similar Family-Vicsek scaling relation lᐨᵅ W(l,t/lᙆ) ~ tᵝ which describes the spin fluctuations of a subsystem of size l, the roughness exponent α and the growth exponent β are connected to the dynamical exponent by the relation: z=α/β. 

It can be shown that the Family-Vicsek universal scaling emerges in all transport regimes of SU(N) spin chains, irrespective of integrability of any underlying symmetry [3], and it likely extends to other models in 1+1 dimensions and to other transport channels (e.g., charge transport in the Hubbard model). In this respect, the QGF numerical approach plays a pivotal role in reaching the hitherto unachievable length- and time-scales required for a robust scaling analysis.  

Quantum complexity is a forefront trope in quantum computing, although a pinpoint definition of what is considered 'simple' or 'complicated' is yet elusive. A typical question is "how difficult is to prepare a quantum state?" which can be reformulated by how complex is the quantum circuit (i.e., the set of qubit operations) to prepare such a state. 

For a long time, entanglement has been identified as the paradigmatic discriminant for quantumness and proposed to quantify quantum complexity. However, with the rise of the stabilizer formalism, it became clear that stabilizer states, generated by Clifford operations on the computational basis states, do not conform to this belief. Despite stabilizer states being potentially highly entangled, the corresponding Clifford circuits can be simulated efficiently (i.e., with an algorithm running in polynomial time) on a classical computer. The lack of complexity of stabilizer states is encoded in the peculiar spectral properties of the corresponding Clifford circuits [4] — see also: Clifford orbits, below. 

In this respect, nonstabilizerness (or more colloquially referred to as magic) emerges as a valuable metric to quantify the non-Clifford resources required to prepare a quantum state, and in a sense, represents the hardness to simulate it classically. Without magic, a quantum computer cannot perform any task beyond the reach of a classical computer: magic is therefore pivotal for achieving quantum advantage. 

Both entanglement and nonstabilizerness can be framed within quantum resource theory — see this [link] for a brief introduction.  Following a classification of quantum states in terms of quantum resources, the Hilbert space can be divided into different regimes, in particular, one can identify entanglement-dominated states (e.g., stabilizer states) and magic-dominated states (e.g., matrix product states), whereas typical Haar random states are resourceful in both entanglement and magic [5]. While the two resources are not independent, and, e.g., entanglement is necessary to generate high-magic states, hitherto, little is known about their relation [5].

[4] Szombathy et al., Phys. Rev. Research 7, 043080 (2025)  
[5] Szombathy et al., Phys. Rev. Research 7, 043072 (2025)

Related reading:

• Chitambar & Gour, Rev. Mod. Phys. 91, 025001 (2019)

• Gu, Oliviero, Leone, PRX Quantum 6, 020324 (2025)

Clifford Orbits

Since Clifford operations map Pauli strings to Pauli strings, they generate periodic orbits — see more details at this [link]. It can be shown that: (i) the orbits are related to the eigenvalues of Clifford operators — which are distributed on the unit circle (and therefore are reduced to phases), and (ii) the phase correlation function and level-spacing statistics display peculiar properties, such as high degeneracies. As mentioned, such properties reflect the lack of complexity of stabilizer states, i.e., quantum states prepared solely with Clifford operations.

It can be shown that the characteristic Clifford spectral properties are rapidly destroyed by non-Clifford operations (e.g., a T gate) and the correlation function quickly converges to its counterparts from the circular unitary ensemble (CUE) from random matrix theory, suggesting that a transition to a chaotic behavior can be possibly induced already by a single injected T gate [4].

The one-qubit case is key to understanding complex circuits. We identify interesting states on the Bloch sphere: 

• stabilizer states (S): with zero magic;

• T-type states, generated as, e.g., THTH|0⟩: with magic saturating the upper theoretical bound log(3/2);

• H-type states, generated as, e.g., TH|0⟩: with magic corresponding to the non-stabilizing power of the T gate. It is also the most probable magic value, as the magic distribution exhibits a logarithmic (van Hove) singularity. 

For Clifford+T circuits on an N-qubit register, in the dilute limit (i.e., for ≪ N injected T gates) interference between T-gates is statistically irrelevant: the magic distribution is discrete, magic is generated in (approximate) quanta and increases linearly with the number of injected T gates. For O(N) injected T gates the magic distribution becomes quasi-continuous, skewed, and concentrated towards high magic values, and eventually converges to the magic distribution of Haar-random unitaries [4,5].

The average magic of the ensemble displays a universal scaling with the T-gate density. In the thermodynamic limit, it grows linearly up to a critical T-gate density, and above it saturates the theoretical upper bound log((d+1)/2)  [4].