Search

Quantum Fisher information (QFI) is a central concept in quantum sciences used to quantify the ultimate precision limit of parameter estimation, detect quantum phase transitions, witness genuine multipartite entanglement, or probe nonlocality. Despite this widespread range of applications, computing the QFI value of quantum many-body systems is, in general, a very demanding task. Here we combine ideas from functional theories and quantum information to develop a novel functional framework for the QFI of fermionic and bosonic ground states. By relying upon the constrained-search approach, we demonstrate that the QFI matricial values can universally be determined by the one-body reduced density matrix (1-RDM), avoiding thus the use of exponentially large wave functions. Furthermore, we show that QFI functionals can be determined from the universal 1-RDM functional by calculating its derivatives with respect to the coupling strengths, becoming thus the generating functional of the QFI. We showcase our approach with the Bose-Hubbard model and present exact analytical and numerical QFI functionals. Our results provide the first connection between the one-body reduced density matrix functional theory and the quantum Fisher information., Comment: 11 pages, 3 Figures

The weighted subspace-search variational quantum eigensolver (SSVQE) is a prominent algorithm for calculating excited-state properties of molecular quantum systems. In this work, we elaborate on some of its fundamental features with the aim of improving its practical realization. First, we demonstrate that the initial ans\"atze for various excited states could be prepared into a single pure state through a minimal number of ancilla qubits, followed by the optimization of a subsequent global unitary rotation in the targeted subspace. Since the ancillas' sole purpose is to purify an underlying ensemble $\rho_{\boldsymbol{w}}$ state with spectral weights $\boldsymbol{w}$, their measurement would just collapse $\rho_{\boldsymbol{w}}$ with probabilities $w_j$ to one of its eigenstates $|\Psi_j \rangle$. We thus observe that our realization of SSVQE is equivalent to the original SSVQE improved by importance sampling. Then, we elaborate by numerical means on the potential influence of the auxiliary weights $\boldsymbol{w}$ on the accuracy of the sought-after eigenstates and eigenenergies. Clear trends are discovered which are contrasted with some recent mathematical results concerning the ensemble variational principle that underlies SSVQE., Comment: 21 pages, 8 figures. The title has been changed

Wigner functions are broadly used to probe non-classical effects in the macroscopic world. Here we develop an orbital-free functional framework to compute the 1-body Wigner quasi-probability for both fermionic and bosonic systems. Since the key variable is a quasi-density, this theory is particularly well suited to circumvent the problem of finding the Pauli potential or approximating the kinetic energy in orbital-free density functional theory. As proof of principle, we find that the universal functional for the building block of optical lattices results from a translation, a contraction, and a rotation of the corresponding functional of the 1-body reduced density matrix, indicating a strong connection between these functional theories. Furthermore, we relate the concepts of Wigner negativity and $v$-representability, and find a manifold of ground states with negative Wigner functions., Comment: 10 pages, 6 figures

The ground states of the homogeneous electron gas and the homogeneous electron liquid are cornerstones in quantum physics and chemistry. They are archetypal systems in the regime of slowly varying densities in which the exchange-correlation energy can be estimated with a myriad of methods. For high densities, the behavior of the energy is well-known for 1, 2, and 3 dimensions. Here, we extend this model to arbitrary integer dimensions, and compute its correlation energy beyond the random phase approximation (RPA), using the celebrated approach developed by Singwi, Tosi, Land, and Sj\"olander (STLS), which is known to be remarkably accurate in the description of the full electronic density response for $2D$ and $3D$, both in the paramagnetic and ferromagnetic ground states. For higher dimensions, we compare the results obtained for the correlation energy using the STLS method with the values previously obtained using RPA. We found that at high dimensions STLS tends to be more physical in the sense that the infamous sum rules are better satisfied by the theory. We furthermore illustrate the importance of the plasmon contribution to STLS theory., Comment: 14 pages, 9 figures

State-average calculations based on mixture of states are increasingly being exploited across chemistry and physics as versatile procedures for addressing excitations of quantum many-body systems. If not too many states should need to be addressed, calculations performed on individual states is also a common option. Here we show how the two approaches can be merged into one method, dealing with a generalized yet single pure state. Implications in electronic structure calculations are discussed and for quantum computations are pointed out., Comment: 9 pages, additional calculation

The homogeneous electron gas is one of the most studied model systems in condensed matter physics. It is also at the basis of the large majority of approximations to the functionals of density functional theory. As such, its exchange-correlation energy has been extensively studied, and is well-known for systems of 1, 2, and 3 dimensions. Here, we extend this model and compute the exchange and correlation energy, as a function of the Wigner-Seitz radius $r_s$, for arbitrary dimension $D$. We find a very different behavior for reduced dimensional spaces ($D=1$ and 2), our three dimensional space, and for higher dimensions. In fact, for $D > 3$, the leading term of the correlation energy does not depend on the logarithm of $r_s$ (as for $D=3$), but instead scales polynomialy: $ -c_D /r_s^{\gamma_D}$, with the exponent $\gamma_D=(D-3)/(D-1)$. In the large-$D$ limit, the value of $c_D$ is found to depend linearly with the dimension. In this limit, we also find that the concepts of exchange and correlation merge, sharing a common $1/r_s$ dependence., Comment: 9 pages, 2 figure, minor changes

We train a neural network as the universal exchange-correlation functional of density-functional theory that simultaneously reproduces both the exact exchange-correlation energy and potential. This functional is extremely non-local, but retains the computational scaling of traditional local or semi-local approximations. It therefore holds the promise of solving some of the delocalization problems that plague density-functional theory, while maintaining the computational efficiency that characterizes the Kohn-Sham equations. Furthermore, by using automatic differentiation, a capability present in modern machine-learning frameworks, we impose the exact mathematical relation between the exchange-correlation energy and the potential, leading to a fully consistent method. We demonstrate the feasibility of our approach by looking at one-dimensional systems with two strongly-correlated electrons, where density-functional methods are known to fail, and investigate the behavior and performance of our functional by varying the degree of non-locality., Comment: 8 pages, 5 figures; accepted for publication in J. Phys. Chem. Lett

We present an \textit{ab initio} theory for superconductors, based on a unique mapping between the statistical density operator at equilibrium, on the one hand, and the corresponding one-body reduced density matrix $\gamma$ and the anomalous density $\chi$, on the other. This new formalism for superconductivity yields the existence of a universal functional $\mathfrak{F}_\beta[\gamma,\chi]$ for the superconductor ground state, whose unique properties we derive. We then prove the existence of a Kohn-Sham system at finite temperature and derive the corresponding Bogoliubov-de Gennes-like single particle equations. By adapting the decoupling approximation from density functional theory for superconductors we bring these equations into a computationally feasible form. Finally, we use the existence of the Kohn-Sham system to extend the Sham-Schl\"uter connection and derive a first exchange-correlation functional for our theory. This reduced density matrix functional theory for superconductors has the potential of overcoming some of the shortcomings and fundamental limitations of density functional theory of superconductivity., Comment: 18 pages

Based on recent progress on fermionic exchange symmetry we propose a way to develop new functionals for reduced density matrix functional theory. For some settings with an odd number of electrons, by assuming saturation of the inequalities stemming from the generalized Pauli principle, the many-body wave-function can be written explicitly in terms of the natural occupation numbers and natural orbitals. This leads to an expression for the two-particle density matrix and therefore for the correlation energy functional. This functional was then tested for a three-electron Hubbard model where it showed excellent performance both in the weak and strong correlation regimes., Comment: 9 pages, 5 figures

Aiming at a unified treatment of correlation and inhomogeneity effects in superconductors, Oliveira, Gross and Kohn proposed in 1988 a density functional theory for the superconducting state. This theory relies on the existence of a Kohn-Sham scheme, i.e., an auxiliary noninteracting system with the same electron and anomalous densities of the original superconducting system. However, the question of noninteracting $v$-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist in at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only., Comment: 9 pages

Fermionic natural occupation numbers do not only obey Pauli's exclusion principle but are even stronger restricted by so-called generalized Pauli constraints. Whenever given natural occupation numbers lie on the boundary of the allowed region the corresponding $N$-fermion quantum state has a significantly simpler structure. We recall the recently proposed natural extension of the Hartree-Fock ansatz based on this structural simplification. This variational ansatz is tested for the lithium atom. Intriguingly, the underlying mathematical structure yields universal geometrical bounds on the correlation energy reconstructed by this ansatz., Comment: Contribution to the special issue dedicated to professor Michael Springborg on the occasion of his 60th birthday, 7 pages

The concept of correlation is central to all approaches that attempt the description of many-body effects in electronic systems. Multipartite correlation is a quantum information theoretical property that is attributed to quantum states independent of the underlying physics. In quantum chemistry, however, the correlation energy (the energy not seized by the Hartree-Fock ansatz) plays a more prominent role. We show that these two different viewpoints on electron correlation are closely related. The key ingredient turns out to be the energy gap within the symmetry-adapted subspace. We then use a few-site Hubbard model and the stretched H$_2$ to illustrate this connection and to show how the corresponding measures of correlation compare., Comment: 6 pages, 3 figures

Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers $\{n_i\}$. A recent analysis of the pure $N$-representability problem provides a wide set of inequalities for the $\{n_i\}$, leading to constraints on these numbers. This has a strong potential impact on reduced density matrix functional theory as we know it. In this work we continue our study the nature of these inequalities for some atomic and molecular systems. The results indicate that (quasi)saturation of some of them leads to selection rules for the dominant configurations in configuration interaction expansions, in favorable cases providing means for significantly reducing their computational requirements., Comment: 12 pages, 15 figures, new references, some typos corrected and a new section added

Computational evidence shows that, when using natural orbitals to study (dynamical and non-dynamical) electron correlation, determinants with an odd number of excitations play a negligible role. Instead, doubly excited determinants rule the rostrum in this kind of configuration interaction calculations. We explain mathematically why it must be so., Comment: 4 pages, 1 figure, reference added

The Pauli exclusion principle gives an upper bound of 1 on the natural occupation numbers. Recently there has been an intriguing amount of theoretical evidence that there is a plethora of additional generalized Pauli restrictions or (in)equalities, of kinematic nature, satisfied by these numbers. Here for the first time a numerical analysis of the nature of such constraints is effected in real atoms. The inequalities are nearly saturated, or quasi-pinned. For rank-six and rank-seven approximations for lithium, the deviation from saturation is smaller than the lowest occupancy number. For a rank-eight approximation we find well-defined families of saturation conditions., Comment: 22 pages, 6 figures, minor changes, references added