Extraction of a One-Particle Reduced Density Matrix from a Quantum Monte Carlo Electronic Density: A New Tool for Studying Nondynamic Correlation

In this work, we present a method to build a first order reduced density matrix (1-RDM) of a molecule from variational Quantum Monte Carlo (VMC) computations by means of a given correlated mapping wave function. Such a wave function is modeled on a Generalized Valence Bond plus Complete Active Space Self Configuration Interaction form and fits at best the density resulting from the Slater-Jastrow wave function of VMC. The accuracy of the method proposed has been proved by comparing the resulting kinetic energy with the corresponding VMC value. This 1-RDM is used to analyze the amount of correlation eventually captured in Kohn-Sham calculations performed in an unrestricted approach (UKS-DFT) and with different energy functionals. We performed test calculations on a selected set of molecules that show a significant multireference character. In this analysis, we compared both local and global indicators of nondynamic and dynamic correlation. Moreover, following the natural orbital decomposition of the 1-RDM, we also compared the effective temperatures of the corresponding Fermi-like distributions. Although there is a general agreement between UKS-DFT and VMC, we found the best match with the functional LC-BLYP.

Can Wigner distribution functions with collisions satisfy complete positivity and energy conservation?

To avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modeling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner–Weyl transformation, the modeling of collisions may violate complete positivity (precluding the typical probabilistic interpretation). In this paper, we show that such quantum transport models can also strongly violate the energy conservation in the electron–photon (electron–phonon) interactions. After comparing collisions models to exact results for an electron interacting with a photon, we conclude that there is no fundamental restriction that prevents a collision model developed within the (reduced) density matrix or Wigner formalisms to satisfy simultaneously complete positivity and energy conservation. However, at the practical level, the development of such satisfactory collision model seems very complicated. Collision models with an explicit knowledge of the microscopic state ascribed to each electron seems recommendable (Bohmian conditional wavefunction), since they allow to model collisions of each electron individually in a controlled way satisfying both complete positivity and energy conservation.

Quantum simulation of molecules without fermionic encoding of the wave function

Molecular simulations generally require fermionic encoding in which fermion statistics are encoded into the qubit representation of the wave function. Recent calculations suggest that fermionic encoding of the wave function can be bypassed, leading to more efficient quantum computations. Here we show that the two-electron reduced density matrix (2-RDM) can be expressed as a unique functional of the unencoded N-qubit-particle wave function without approximation, and hence, the energy can be expressed as a functional of the 2-RDM without fermionic encoding of the wave function. In contrast to current hardware-efficient methods, the derived functional has a unique, one-to-one (and onto) mapping between the qubit-particle wave functions and 2-RDMs, which avoids the over-parametrization that can lead to optimization difficulties such as barren plateaus. An application to computing the ground-state energy and 2-RDM of H4 is presented.

Topological entanglement and hyperbolic volume

The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the Rényi entropy of index m, which captures the higher moments of the reduced density matrix. In this work, we study pure bipartite states associated with S3 complements of a two-component link which is a connected sum of a knot $$ \mathcal{K} $$ K and the Hopf link. For this class of links, the Chern-Simons theory provides the necessary setting to visualise the m-moment of the reduced density matrix as a three-manifold invariant Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ), which is the partition function of $$ {M}_{{\mathcal{K}}_m} $$ M K m . Here $$ {M}_{{\mathcal{K}}_m} $$ M K m is a closed 3-manifold associated with the knot $$ \mathcal{K} $$ K m, where $$ \mathcal{K} $$ K m is a connected sum of m-copies of $$ \mathcal{K} $$ K (i.e., $$ \mathcal{K} $$ K #$$ \mathcal{K} $$ K . . . #$$ \mathcal{K} $$ K ) which mimics the well-known replica method. We analayse the partition functions Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons coupling k. For SU(2) group, we show that Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) can grow at most polynomially in k. On the contrary, we conjecture that Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) for SO(3) group shows an exponential growth in k, where the leading term of ln Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) is the hyperbolic volume of the knot complement S3\$$ \mathcal{K} $$ K m. We further propose that the Rényi entropies associated with SO(3) group converge to a finite value in the large k limit. We present some examples to validate our conjecture and proposal.

Complexity from the reduced density matrix: a new diagnostic for chaos

We investigate circuit complexity to characterize chaos in multiparticle quantum systems. In the process, we take a stride to analyze open quantum systems by using complexity. We propose a new diagnostic of quantum chaos from complexity based on the reduced density matrix by exploring different types of quantum circuits. Through explicit calculations on a toy model of two coupled harmonic oscillators, where one or both of the oscillators are inverted, we demonstrate that the evolution of complexity is a possible diagnostic of chaos.