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Simulating physical systems on near-term quantum computers often requires preparing states within constrained subspaces, like those with fixed particle number or spin. We use Lie algebraic techniques to prove that hardware-efficient gates are universal for state preparation in these subspaces. The key mechanism is Pauli $Z$ dressing: commutators of overlapping gates produce Pauli $Z$ operators on shared qubits, acting as spectator projectors that decompose multi-plane rotations into single-plane generators spanning the full $\mathfrak {so}(w)$ algebra, where $w$ is the dimension of the constrained subspace, thereby guaranteeing universality for real state preparation. Adding independent complex phases extends this to $\mathfrak {su}(w)$, enabling arbitrary complex state preparation. We provide a computationally efficient Jacobian criterion for verifying that a circuit can explore any direction on the target manifold from almost any parameter configuration. Our findings are applicable to many problem areas, including Fermi-Hubbard models, Bose-Hubbard models, and molecular electronic structure. We apply our framework to two physical settings: we prove the completeness of the binary encoded multi-level particles ansatz on the conserved-particle-number subspace, and we construct symmetry-preserving circuits for the fuzzy sphere regularisation of the 3D Ising conformal field theory (CFT). For the latter, we variationally prepare the ground and excited states and extract some CFT data.

Integrable quantum systems evolving from non-equilibrium initial states do not thermalize to conventional Gibbs ensembles (GE). Instead, at long times they relax to generalized Gibbs ensembles (GGEs), which incorporate the full set of local and quasi-local conserved quantities. While GGEs have been extensively studied in the literature, a quantitative analytic characterization of how different they are from ordinary GEs is still lacking. In this work, we address this question by employing the concept of athermality, which we define within quantum resource theory as the relative entropy between a given state and the closest thermal state. By means of integrability techniques we compute the athermality for several quantum quenches in paradigmatic integrable models, including the free XY spin chain, the interacting Lieb-Liniger model, the XXZ spin chain, and the harmonic chain. We find that often the athermality becomes anomalously small when the post-quench Hamiltonian is critical in its ground state, despite probing physics at a finite energy density. We also prove that it systematically develops a singularity at criticality, which is inherited from the entropy of the GGE.

We investigate multipartite entanglement via the statistical properties of pure quantum states of $n$-qubits. By analyzing the distribution of purity among balanced bipartitions, we compare Haar-typical states, uniformly distributed on the unit sphere of states, with Hadamard states, being characterized by equal weights in the computational basis. We analyze different ensembles of Hadamard states characterized by their phase distributions. Through analytical and numerical calculations, we show that Hadamard states exhibit, on average, a higher degree of entanglement than Haar-typical states. In addition, we show that a particular class of Hadamard states, characterized by real coefficients with alternating signs, known as hypergraph states, appears especially relevant in the search for maximally multipartite entangled states, both for their structural simplicity and the increased likelihood of sampling highly entangled states. These results identify Hadamard states as a tractable yet promising class for exploring multipartite entanglement structures and advancing the characterization of maximally multipartite entangled quantum states.

We compare the performance of representative cooling, adiabatic, and optimization algorithms for ground-state preparation in the presence of noise. Using an exactly solvable family of quadratic fermionic Hamiltonians subject to depolarizing noise, we derive the scaling of the achievable relative energy as a function of the noise rate and support these results with numerical simulations. The Hamiltonian exhibits two phases, separated by a quantum phase transition. As expected, the performance of the different algorithms depends on the phase: adiabatic evolution is favorable in the trivial phase, while a multi-frequency cooling algorithm, as proposed in [Phys. Rev. Res. 7 (2025) 033162 (https://doi.org/10.1103/4hx7-xnhw)], becomes competitive or superior in the topological phase, where gap-closing limits adiabatic protocols. We further present numerical results for the quantum approximate optimization algorithm [arXiv:1411.4028], showing that it performs competitively with cooling in the trivial phase but is typically outperformed in the topological regime. Finally, we show that for this model the cooling protocol exhibits enhanced robustness to parameter imperfections, highlighting its potential advantage for realistic implementations of noisy quantum state preparation. The analytical approach developed here, in conjunction with numerical validation, establishes an extendable approach to benchmarking ground-state preparation algorithms.

Non-Abelian adiabatic holonomies are used both to diagnose topology and to implement geometric quantum operations. Yet the standard descriptors of such transport --- Berry phases, Chern numbers, determinant phases, and Wilson-loop spectra --- do not determine whether the resulting holonomy acts as local logical operations on an extracted subsystem structure. We identify this missing information as a subgroup-reduction problem. For an isolated multiplet with pointwise product type $D=\prod _{\alpha } d_{\alpha }$, we ask whether the loop holonomy lies in the embedded local subgroup $G_{\mathbf d}=\operatorname {Im}[\prod _{\alpha }\mathrm {U}(d_{\alpha })\to \mathrm {U}(D)]$. We show that a multiplet can remain pointwise local throughout parameter space while different loops on the same control manifold realize sharply different logical actions, from almost local transport to controlled operations and entangling holonomies, even when the usual Berry or Wilson-loop data are nearly unchanged. The primary diagnostic is the distance of the microscopic holonomy to the extracted local subgroup, relative to a specified product reduction; gate-specific invariants are invoked only after reduction failure has been established. This identifies a quantum-geometric observable that is invisible to standard spectral reductions and directly motivates tomography-based detection of hidden subsystem-nonlocality. Explicit localized topological benchmarks are provided by a BHZ ribbon, a spinful SSH chain, and a BBH corner quartet.

Simulating the real-time dynamics of particle collisions is a promising application of quantum simulators, because classical methods such as tensor networks struggle to capture the highly entangled states generated in high-energy scattering. Realizing such simulations requires both the preparation of incoming wave packets and the detection of the outgoing scattering products. In this work, we propose protocols that address both challenges on programmable analog and digital quantum simulation platforms. Our state-preparation scheme uses a weakly coupled auxiliary qubit --- or, more generally, a customized local quench --- to inject a single quasiparticle with well-defined momentum. Because it relies only on conservation of energy, this scheme requires no fine-tuning or prior knowledge about particle eigenstates, making it robust against errors in calibration and implementation. The momenta of scattering products are then extracted, using only local measurements, from the interference pattern that arises when particles are reflected at the system's boundary. We validate our protocols through numerical simulations, first in a simple single-particle model and subsequently in two interacting many-body systems: a Rydberg atom chain and an Ising chain in a mixed field. We demonstrate how high-energy regimes, necessary to access inelastic scattering processes, can be reached through an adiabatic ramp, and how the wave packet shape can be optimized by spatially modulating the Hamiltonian. Finally, we show how the protocol can be generalized to systems with more than one spatial dimension. Our proposal provides a versatile approach to the quantum simulation of scattering phenomena, and is compatible with several quantum simulation platforms that are already experimentally available.

Fermionic Gaussian states are a fundamental tool in many-body physics, faithfully representing non-interacting quantum systems and allowing for efficient numerical simulations. Given a many-body wave function, it is therefore interesting to ask how much it differs from that of a Gaussian state, as quantified by the notion of non-Gaussianity. In this work, we relate measures of non-Gaussianity with the Shannon entropy of the particle-number distribution, coinciding with the particle-number asymmetry for pure states. We derive a lower bound on the relative entropy of non-Gaussianity in terms of the exponential of the Shannon entropy, and study numerically its tightness for large system sizes. Our bound is non-trivial for large values of the asymmetry and relies on the concentration of the particle-number distribution of (mixed) fermionic Gaussian states. Since the Shannon entropy of the particle-number distribution is often efficient to compute or experimentally measure, our results can be viewed as a practical way to lower bound non-Gaussianity, highlighting a non-trivial interplay with particle-number asymmetry.