Reduced density matrices are central to describing observables in many-body quantum systems. In electronic structure theory, the two-particle reduced density matrix (2-RDM) suffices to determine the energy and other key properties. Recent work has used matrix completion, leveraging the low-rank structure of RDMs and approximate theoretical models, to reconstruct the 2-RDM from partial data and thus reduce computational cost. However, matrix completion is, in general, an under-determined problem. Revisiting Rosina's theorem [M. Rosina, Queen's Papers on Pure and Applied Mathematics No. 11, 369 (1968)], we here show that the matrix completion is unique under certain conditions, identifying the subset of 2-RDM elements that enables its exact reconstruction from incomplete information. Building on this, we introduce a hybrid quantum-stochastic algorithm that achieves exact matrix completion, demonstrated through applications to the Fermi-Hubbard model.
Fermionic reduced density matrices summarize the key observables in Fermionic systems. In electronic systems, the two-particle reduced density matrix (2-RDM) is sufficient to determine the energy and most physical observables of interest. Here, we consider the possibility of using matrix completion to reconstruct the two-particle reduced density matrix to chemical accuracy from partial information. We consider the case of noiseless matrix completion, where the partial information corresponds to a subset of the 2-RDM elements, as well as noisy completion, where the partial information corresponds to both a subset of elements and statistical noise in their values. Through experiments on a set of 24 molecular systems, we find that 2-RDM can be efficiently reconstructed from a reduced amount of information. In the case of noisy completion, this results in a multiple orders of magnitude reduction in the number of measurements needed to determine the 2-RDM with chemical accuracy. These techniques can be readily applied to both classical and quantum algorithms for quantum simulations.
We present a novel machine learning algorithm for the many-electron problem, predicting the convex combination of two-electron reduced density matrices (2-RDMs)─obtained from upper- and lower-bound energy calculations─that closely approximates the exact energy. In contrast to other recently developed approaches based on the wave function or one-electron density, our 2-RDM machine-learning approach predicts energies and properties without steep scaling or functional approximation. As conjectured by Preskill and co-workers, a small amount of data in a physics-based machine learning algorithm─in this case, information about the RDMs and their violation of selected higher-order N -representability conditions─yields highly accurate electronic energies that capture both dynamic and static correlation. We demonstrate the method by predicting the potential energy curves for BH and N 2 within a few millihartrees of results from exact diagonalization. This machine learning algorithm provides a general framework for improving electronic structure calculations, with the potential for wide-reaching applications to both moderately and strongly correlated molecular systems.
We derive necessary conditions for any two-fermion reduced density matrix (2-RDM) to be representable by a pure
N-fermion density matrix $\mathrm{\ensuremath{\Psi}}{\mathrm{\ensuremath{\Psi}}}^{*}$ where $\mathrm{\ensuremath{\Psi}}$ is the wave function. These pure
N-representability conditions of the 2-RDM are important because they provide stringent constraints beyond those from the Pauli and the generalized Pauli constraints on the structure of many-fermion 2-RDMs and their wave functions. The pure 2-RDM conditions are derived as generalized Pauli constraints on effective one-fermion reduced density matrices (1-RDMs) generated by the removal or addition of a fermion from the wave function. Computationally, we show for four-electron molecules that the derived pure
N-representability conditions are nontrivially active for exact ground-state 2-RDMs and that they provide significant restrictions beyond the D, Q, and G ensemble
N-representability conditions. Constraints on higher-order
p-RDMs where
p>2 are derived in a similar fashion. The constraints have potentially significant applications to computing strongly correlated many-fermion states with enhanced accuracy and decreased computational complexity.
Variational reduced-density-matrix theory is applied to calculating the ground-state energy and two-electron reduced density matrices (2-RDMs) of the one-dimensional Hubbard model for a range of interaction strengths. The 2-RDM is constrained to represent an
N-particle wave function by two sets of
N-representability conditions, known as the 2- and partial 3-positivity conditions. Variational optimization of the energy with the 2-RDM constrained by
N-representability conditions is performed using a first-order semidefinite-programming algorithm that was developed for treating atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Accurate energies for a broad range of interaction strengths indicate that the variational 2-RDM method is a valuable tool for studying strongly correlated electrons.
Variationally calculating the ground state of a many-electron quantum system using only the two-electron reduced-density-matrix (2-RDM) requires
N-representability conditions that constrain the 2-RDM to correspond to an
N-electron wave function. A systematic hierarchy of
N-representability conditions, known as
p-positivity conditions, has been developed [D. A. Mazziotti and R. M. Erdahl, Phys. Rev. A 63, 042113 (2001)], and many-electron atoms and molecules in nonminimal basis sets have been solved with useful accuracy by a variational 2-RDM method with 2-positivity conditions [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. This paper considers two forms of partial 3-positivity conditions, the lifting conditions and the
T1∕T2 conditions, to further enhance the accuracy of the 2-RDM methods without the computational cost of full 3-positivity conditions. Variational 2-RDM methods with different
N-representability constraints including 2-positivity conditions, the two types of partial 3-positivity conditions, as well as the complete 3-positivity conditions are applied to compute the ground state of the Lipkin spin model. The energies and 2-RDMs are compared to the results from full and truncated configuration interaction, many-body perturbation theory, and couple cluster theory with single and double excitations. Implications of using partial 3-positivity for variational 2-RDM calculations of many-electron atoms and molecules will be discussed.
We recently presented a constructive solution to the
N-representability problem of the two-electron reduced density matrix (2-RDM)---a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic
N-electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. (to be published)]. In this paper we provide additional details and derive further
N-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the
(2,q)-positivity conditions, where the
q indicates their derivation from the non-negativity of
q-body operators. In addition to the known T1 and T2 conditions, we derive another class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is, conditions which arise from lifting lower
(2,q)-positivity conditions to higher
(2,q+1)-positivity conditions, and (ii) pure conditions, that is, conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure
(2,q)-positivity conditions for
q>3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the derived
N-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation.