RDMFTforbosons · Extending the scope of Reduced Density Matrix Functional Theory for bosons

Extending the scope of Reduced Density Matrix Functional Theory for bosons

One of the most remarkable features of quantum mechanics is the enormous size of the Hilbert spaces associated with interacting many-body quantum systems. Indeed, their exponential scaling with the system’s degrees of freedom poses significant challenges for theoretical descriptions, rendering computations highly complex and resource-intensive, if not entirely infeasible. Much of our understanding of these systems is rooted in what physicist P. Anderson called the 'continuity principle': faced with a challenging problem, insights can often be gained by studying a simpler, related problem that offers a sense of continuity in understanding and guidance for deeper exploration. Today, experimental precision and control advances allow us to observe and manipulate a wide range of strongly and ultra-strongly correlated quantum systems. These systems exhibit behaviors that are notoriously difficult, or even impossible, to capture with more traditional methods such as mean-field theories or adiabatic interaction continuations. As a result, this experimental progress is pushing the boundaries of quantum theory and underscoring the need for new frameworks capable of capturing the complexity and richness of these interactions. In the specific case of electronic or bosonic systems with pairwise interactions, it is well known that the two-body reduced density matrix alone can capture the full physics of the quantum problem, eliminating the need for (exponentially large) electronic or bosonic wave functions. Unfortunately, the practical application of such matrices is hindered by the necessity of imposing a (quite) lengthy set of representability conditions, i.e., mathematical inequalities that govern their spectra. The one-body reduced-density-matrix functional theory for fermionic or bosonic ground states has already demonstrated that universal functionals based on the one-body reduced density matrix can precisely capture quantum correlations and explain a wide range of physical phenomena by utilizing the geometric structure of the domain of one-body reduced density matrices. This approach significantly simplifies the problem, enabling solutions to the quantum many-body problem without the need for wave functions or two-body reduced density matrices, and thereby offers substantial computational advantages. Notably, using one-body reduced density matrices entails a drastic reduction in the problem’s degrees of freedom and enables the description of quantum systems in any dimensionality. The primary goal of this project is to extend the scope and applicability of reduced density matrix theory to systems involving mixtures of particles (e.g., Fermi-Bose or electron-photon quantum mixtures), dipolar gases, and quantum many-body systems at finite temperatures. Additionally, the project aims to describe systems in which certain symmetries are broken, making the problem even more challenging. Our proposed methodological approach combines novel analytical techniques with cutting-edge machine learning and quantum computing implementations. In short, our objective is to achieve a universal description of the quantum many-body problem—whether it involves a bosonic mixture or an intricate system of particles with varying statistics (e.g., electrons, phonons, photons)—that captures the essential features with the minimum set of degrees of freedom. We achieved several key results during the execution of this project. First, we developed a universally and formally exact ansatz for quantum many-body systems, applicable to fermionic systems, bosonic mixtures, and polaritonic Hamiltonians. We also demonstrated that this ansatz can be effectively learned by a contracted quantum eigensolver and that the resulting quantum data can be used to train a convolutional neural network. Additionally, we showed that the universal functional of the one-body reduced density matrices can be employed to extract the quantum Fisher information of the system. Since this is a key quantity for measuring genuine multi-particle entanglement, our results indicate that functional theories enable the computation of quantum resources at the many-particle level.

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