We review the framework of operator Hilbert space and introduce the one- and two-particle super reduced density matrices (1-SRDMs and 2-SRDMs), as well as the super mutual information (SMI). The eigenvectors of the 1-SRDMs define what we term natural single particle operator bases, and provide a way to compress vibrational and vibronic Hamiltonians with controlled error. The SMI is defined from the operator entanglement entropy of the 1-SRDMs and 2-SRDMs, and captures the correlation between operators acting on different one-mode subspaces, which may be used to reveal and quantify both direct and indirect couplings that might otherwise be difficult to extract. Efficient numerical approaches for the calculation of SRDMs and the SMI are developed and applied to a set of prototypical vibrational and vibronic Hamiltonians, as well as approximations to the corresponding time-evolution operators. Through this, we demonstrate that: (i) commonly used vibronic Hamiltonians are amenable to extremely high levels of compression without compromising accuracy, and; (ii) SMI analysis can be used to systematically and quantitatively reveal both direct and indirect couplings that might otherwise be difficult to extract, including indirect couplings of vibrational modes via intermediary electronic-vibrational interactions.
We study unitary quantum dynamics in noisy Brownian models with global continuous symmetries, such as $U(1)$ and $SU(2)$, focusing on R\'enyi entanglement entropies and hydrodynamic and non-hydrodynamic correlators. By mapping the averaged late-time dynamics to the low-energy physics of effective replica Hamiltonians, we find that the evolution is controlled by the quantum geometry of their ground-state manifolds, which is directly related to the geometry of $k$-commutants---the symmetry algebra of $k$ replicas of the system. In interacting systems, these $k$-commutants are generically determined solely by the symmetries of the system, independent of microscopic details of the noisy evolution. This allows us to use the time-dependent variational principle (TDVP) to provide simple geometric explanations for the sub-ballistic R\'enyi entanglement growth and the anomalous decay of non-hydrodynamic correlators in interacting systems with continuous symmetries. We find this behavior to be intimately connected to singularities within the $k$-commutant manifolds, arising from frozen ``void''states in the Hilbert space that exist due to continuous on-site symmetries. This also demystifies the important role of voids in the dynamics of these observables, previously identified in $U(1)$ symmetric systems. We compare these behaviors in interacting systems with Abelian and non-Abelian continuous symmetries and in free-fermion systems, which differ in the geometry of their $k$-commutants. Ultimately, this work provides a general geometric framework for systematically studying observables in noisy systems with continuous symmetries, including Haar-random circuits.
In this work, we extend the scattering quantum walk (SQW) framework to a lattice of energy-dependent point interactions. This yields, within the coined quantum walk (CQW) formalism, a coin operator that is directly related to the scattering matrix of zero-range potentials. The model thus provides a discrete-time quantum-walk (DTQW) analog of a periodic array of point interactions of the Kronig-Penney type, where the walker's wavenumber serves as a continuous, physically transparent control parameter for the coin operation. We analyze the spectra and the dynamics of position probability and entanglement, yielding distinct results for specific energies and point interactions. We relate the spectral structure to the spatial probability distribution and explicitly characterize the long-time entanglement behavior for each point interaction. The transmission modulus determines the quasienergy gap, bandwidth, and maximum group velocity, while also controlling the long-time coin-position entanglement for the initial state considered. The four families of one-dimensional point interactions ($\delta$, $\delta'$, crossed and asymmetric) realize qualitatively distinct transmission profiles and span the full range of behavior, including enhanced or strongly suppressed spreading and oscillatory entanglement.
Alana Spak dos Santos, T. T. Tsutsui, F. M. Andrade· 0 citations
Entanglement entropy (EE) is commonly studied using real-space bipartitions. We show that, in quantum impurity models, an energy-space bipartition, equivalent to the momentum-space bipartition of the bath, can display universal behavior. Motivated by poor man's scaling, we logarithmically discretize the bath and partition it into high- and low-energy sectors. For models with Fermi-liquid fixed points, including the Anderson model and fully screened or underscreened Kondo models, the low-energy EE flows to constants independent of model parameters. These constants are integer multiples of $\ln 2$ plus corrections that depend only on the logarithmic discretization parameter $\Lambda$. We show that scale invariance of the fixed-point wavefunction in energy space maps to effective translation invariance along a one-dimensional chain, allowing the fixed points to be classified by one-dimensional topological band theory. With low-energy chiral symmetry, each $\ln 2$ contribution originates from a topological edge mode. We also study transitions between distinct Fermi-liquid fixed points using the local-singlet--Kondo-singlet transition in a two-orbital Anderson model driven by an inter-orbital antiferromagnetic coupling. The local-singlet phase has an effectively decoupled impurity and nearly vanishing EE, whereas the Kondo-singlet phase has finite EE larger than $\ln 2$ per spin and orbital. When chiral symmetry holds at low energies, this distinction corresponds to a topological transition of the effective bath chain. At the non-Fermi-liquid critical point, the EE develops an unstable plateau. Its $\Lambda$ dependence resembles that of the overscreened two-channel Kondo model, supporting universality within the same non-Fermi-liquid universality class.
We introduce a variational framework for approximating the time-evolution operator $\hat{U}(t)$ within a physically motivated operator manifold, reformulating quantum dynamics as a tractable problem in operator space using stationary action principle. For periodically driven systems, the resulting approximate evolution operator directly yields an effective Floquet Hamiltonian, offering a non-perturbative alternative to conventional expansion-based methods. The framework is systematically improvable by enlarging the operator pool and naturally incorporates symmetries and physical constraints. When the operator manifold is chosen from the terms of a truncated Magnus expansion, the variational procedure effectively resums the Magnus series within the restricted space, significantly enhancing accuracy. We benchmark the approach on the driven Rabi model, the driven Lipkin-Meshkov-Glick model, and the one-dimensional driven Ising chain, yielding effective Floquet Hamiltonians that are systematically more accurate than low-order Magnus expansions, particularly in regimes where the latter converge poorly, and illustrating applicability to systems with exponentially large Hilbert spaces. Although we focus here on Floquet systems, the formalism applies equally to generic time-dependent Hamiltonians, providing a versatile tool for non-equilibrium quantum dynamics.
We study the one-sided time evolution of thermofield double (TFD) states in random matrix theory, where the Hamiltonian is taken to be a $D\times D$ random matrix drawn from a unitarily invariant emsemble of Hermitian matrices. We argue that the Krylov basis with the maximally entangled state taken as the initial vector gives a semiclassical Hilbert space description of these TFD states in random matrix theory at any $O(1)$ temperature and time in the $D\to \infty$ limit, very analogous to the ``chord Hilbert space''construction in the double-scaled SYK (DSSYK) model. We study this semiclassical description in detail for ensembles where the spectral density in the $D\to \infty$ limit is even, compactly supported on an interval and has square root edges. With a few more conditions on the analytic structure of the spectral density, we observe that the semiclassical Hamiltonian has the same asymptotic behavior at large Krylov depth as that of DSSYK, with the corresponding parameter $\mathfrak{q}=e^{-\lambda}$ being related to the location of the nearest zero of the spectral density away from the spectral cut. Furthermore, in a large class of models corresponding to ultraviolet deformations of the DSSYK spectral density, i.e., where the spectrum in the UV is modified while leaving the near-edge behavior unchanged, we show that the semiclassical effective Hamiltonian in the Krylov basis reduces to the Liouville Hamiltonian of JT gravity in a low-energy, continuum limit. This suggests that our semiclassical Hilbert space should be interpreted as the bulk Hilbert space of a dual gravity description.
The quantum-selected configuration interaction identifies important determinantal basis functions through real-time evolution of a reference wavefunction and diagonalizing the Hamiltonian matrix in the resulting selected subspace. However, implementing the full electronic Hamiltonian on noisy quantum devices leads to rapidly increasing circuit complexity, limiting its scalability. To address this issue, we identify the dominant fermionic excitation operators and perform reference-state fidelity loss analysis to construct a compact Hamiltonian, reducing computational overhead while retaining high precision. Applied to Group IIIA monofluorides (BF, AlF, GaF, InF, and TlF), the proposed framework achieves a near-quadratic improvement in Hamiltonian-term scaling, enabling resource-efficient simulations. We employ this framework to compute the relativistic ground-state energies and permanent electric dipole moments (PDMs) of the systems under consideration. After validating the framework via simulations, we demonstrate hardware execution for AlF and TlF on the IBM Marrakesh processor using active spaces of up to 20 qubits. For a 20-qubit TlF system, the reduced Hamiltonian yields a reduction of higher than $ 98\%$ in both circuit depth and two-qubit gate counts, with the resulting PDMs from quantum hardware matching complete active space configuration interaction values within $99.99\%$. These results demonstrate the scalability of this approach on noisy intermediate-scale quantum devices.
Suprava Sahoo, Abdul Kalam, Kenji Sugisaki et al.· 0 citations
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