Periodic Reporting for period 1 - RDMFTforbosons (Extending the scope of Reduced Density Matrix Functional Theory for bosons)
Reporting period: 2022-09-01 to 2024-08-31
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.
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.
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.
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.
Our main results are:
(1) We developed a universal, formally exact ansatz for quantum many-body systems and wrote a quantum algorithm that implemented it on real quantum devices. Based on the theory of reduced density matrices, we developed an iterative algorithm that minimizes the residual of the contracted Schrödinger equation for general quantum many-body systems. While the algorithm's mathematical structure was known in the context of electronic structure theory, we made the first generalization to mixed systems of fermions and bosons.
(2) Our quantum algorithm uses the residual of the contracted Schrödinger equation to guide a sequence of trial wave functions toward an exact eigenstate by iteratively applying a sequence of exponential transformations. The scheme is agnostic to the statistics of the system and thus can be used for fermions, bosons, or spin systems.
(3) We demonstrated that when combined with the Rayleigh–Ritz variational principle for mixed quantum states, the ground-state contracted quantum eigensolver can be generalized to compute any number of quantum eigenstates simultaneously. We introduced two excited-state (anti-Hermitian) eigensolvers that perform the excited-state calculation while inheriting many of the remarkable features of the original ground-state version of the algorithm, such as its scalability.
(4) We initiated and developed a functional-theoretical framework for the quantum Fisher information. We showed that for ground states of identical particles, the one-body reduced density matrix can determine the quantum Fisher information matrix, thus avoiding the pre-computation of wave functions that expand into exponentially large Hilbert spaces. We unveiled two surprising links between functional theories and quantum information: (i) the energy functional of the one-body reduced-density-matrix can be fully reconstructed from the functionals of the quantum Fisher information and (ii) the quantum Fisher information functionals correspond to the derivatives of the functionals of the one-body reduced density matrix with respect to the coupling strengths.
(5) We implemented neural networks for quantum propagators which is a universal neural network framework that solves driven-dissipative quantum dynamics by approximating propagators rather than wavefunctions or density matrices. Neural propagators can handle arbitrary initial quantum states, adapt to various external fields, and simulate long-time dynamics, even when trained on far shorter time windows. Furthermore, by appropriately configuring the external fields, our trained NQP can be transferred to systems governed by different Hamiltonians.
(6) We developed an orbital-free functional framework to compute one-body quasiprobabilities for both fermionic and bosonic systems. Since the key variable is a quasidensity, this theory circumvents the problems of finding the Pauli potential or approximating the kinetic energy that are known to limit the applicability of standard orbital-free density functional theory. We presented a set of strategies to (a) compute the one-body Wigner quasiprobability in an orbital-free manner from the knowledge of the universal functional and (b) obtain those functionals from the functionals of the one-body reduced density matrix.
(1) We developed a universal, formally exact ansatz for quantum many-body systems and wrote a quantum algorithm that implemented it on real quantum devices. Based on the theory of reduced density matrices, we developed an iterative algorithm that minimizes the residual of the contracted Schrödinger equation for general quantum many-body systems. While the algorithm's mathematical structure was known in the context of electronic structure theory, we made the first generalization to mixed systems of fermions and bosons.
(2) Our quantum algorithm uses the residual of the contracted Schrödinger equation to guide a sequence of trial wave functions toward an exact eigenstate by iteratively applying a sequence of exponential transformations. The scheme is agnostic to the statistics of the system and thus can be used for fermions, bosons, or spin systems.
(3) We demonstrated that when combined with the Rayleigh–Ritz variational principle for mixed quantum states, the ground-state contracted quantum eigensolver can be generalized to compute any number of quantum eigenstates simultaneously. We introduced two excited-state (anti-Hermitian) eigensolvers that perform the excited-state calculation while inheriting many of the remarkable features of the original ground-state version of the algorithm, such as its scalability.
(4) We initiated and developed a functional-theoretical framework for the quantum Fisher information. We showed that for ground states of identical particles, the one-body reduced density matrix can determine the quantum Fisher information matrix, thus avoiding the pre-computation of wave functions that expand into exponentially large Hilbert spaces. We unveiled two surprising links between functional theories and quantum information: (i) the energy functional of the one-body reduced-density-matrix can be fully reconstructed from the functionals of the quantum Fisher information and (ii) the quantum Fisher information functionals correspond to the derivatives of the functionals of the one-body reduced density matrix with respect to the coupling strengths.
(5) We implemented neural networks for quantum propagators which is a universal neural network framework that solves driven-dissipative quantum dynamics by approximating propagators rather than wavefunctions or density matrices. Neural propagators can handle arbitrary initial quantum states, adapt to various external fields, and simulate long-time dynamics, even when trained on far shorter time windows. Furthermore, by appropriately configuring the external fields, our trained NQP can be transferred to systems governed by different Hamiltonians.
(6) We developed an orbital-free functional framework to compute one-body quasiprobabilities for both fermionic and bosonic systems. Since the key variable is a quasidensity, this theory circumvents the problems of finding the Pauli potential or approximating the kinetic energy that are known to limit the applicability of standard orbital-free density functional theory. We presented a set of strategies to (a) compute the one-body Wigner quasiprobability in an orbital-free manner from the knowledge of the universal functional and (b) obtain those functionals from the functionals of the one-body reduced density matrix.