The primary goal of this project was to extend the theory of reduced density matrices to describe strongly correlated quantum many-body systems involving more than just fermionic particles (or electrons), which is the framework in which the standard version of the theory is typically defined.
One of our main goals was to engineer universal functionals for bosonic particles that account for disorder, focusing on uncovering general properties such as convexity and gradient divergences. In our research, we discovered a remarkably general approach that allowed us to access highly correlated regimes of quantum many-body systems, applicable not only to bosons or fermions but to any type of particle or interaction permitted by our universe (as long as the corresponding Hamiltonian is written in second quantization). This is admittedly a surprising result, as we, when drafting our proposal in 2021, thought such outcomes were beyond reach. However, by using a generalized version of the contracted Schrödinger equation, rather than the more standard Schrödinger equation, we were able to construct a fully universal, exact ansatz for the quantum many-body wave function, and consequently for the functionals of the reduced-density matrices.
Additionally, we uncovered key connections between the functionals of the one-body reduced density matrices and both the quantum Fisher theory of quantum resources (see Fig. 1) and the phase-space theory of quantum physics. By employing the so-called constrained-search approach, we demonstrated that the terms of the quantum Fisher information matrix can be universally determined by the one-body reduced density matrix. Furthermore, we showed that quantum Fisher information functionals can be derived from the universal functional of the one-body reduced density matrix by calculating its derivatives with respect to the coupling strengths, thus serving as the generating functional of the quantum Fisher information. These results lead us to an unexpectedly clear understanding of the quantum many-body problem in its broadest sense.
When discussing the time evolution of quantum many-body systems, the problem is perhaps more challenging than for ground states, as the quantum state evolves within an exponentially large Hilbert space. Numerical methods for solving the corresponding equations of motion generally fall into two categories: iterative-based approaches, such as Runge-Kutta, and methods based on the time-dependent variational principle, which minimizes the residual of the equations of motion solution. However, despite varying levels of sophistication, these algorithms are often hindered by the high computational cost of iteratively propagating the dynamics. A second important result of our project was the introduction and development of the concept of neural quantum propagator, a universal neural network framework that solves driven-dissipative quantum dynamics by approximating propagators rather than wave functions or density matrices. Neural quantum propagators can handle arbitrary initial quantum states, adapt to various external fields, and simulate long-time dynamics, even when trained on much shorter time windows. Furthermore, by appropriately configuring the external fields, our trained neural propagators can be adapted to systems governed by different Hamiltonians.
A second objective focused on studying dipolar gases and mixtures of bosons in the relevant regimes of supersolidity and droplets. The dilute regime of two-component Bose-Einstein condensates (BEC) is typically described by the first beyond mean-field contribution, known as the Lee-Huang-Yang (LHY) correction, which accounts for the effects of quantum fluctuations on the ground state energy of bosonic quantum gases. Our goal, however, was to move beyond this weakly correlated regime and investigate the universal functional of (Bose-Bose or even Bose-Bose-Bose) mixtures of gases, formulated in terms of the one-particle reduced density matrix (1RDM) of each species.
The achievement of the exact ansatz demonstrated our ability to solve, at least in principle, any bosonic mixture through a universal, formally exact exponential ansatz capable of handling any type of particles or any type of interactions. Originally, derived for fermionic systems, the contracted Schrödinger equation was extended to mixed Boson-Fermion systems. The main result of that extension is an exact ansatz that can be implemented directly on quantum devices to find eigenstates of arbitrary mixed-particle Hamiltonians. Remarkably, this equation uniquely allows the ansatz to have the same degrees of freedom as the original many-body Hamiltonian (e.g. a purely two-body ansatz for the traditional electronic structure problem). Based on those previous results, we presented a quantum algorithm for molecular polaritonic chemistry and supersolids. Yet, although the final form of the new LHY correction is still under development, the theoretical foundations have already been laid.
