N-representability Problem Research Articles

Determining the N-Representability of a Reduced Density Matrix via Unitary Evolution and Stochastic Sampling.

The N-representability problem consists in determining whether, for a given p-body matrix, there exists at least one N-body density matrix from which the p-body matrix can be obtained by contraction, that is, if the given matrix is a p-body reduced density matrix (p-RDM). The knowledge of all necessary and sufficient conditions for a p-body matrix to be N-representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the p-body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the N-representability conditions grows exponentially with system size, and hence, the procedure quickly becomes intractable for practical applications. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N-representability conditions. The algorithm consists of applying to an initial N-body density matrix a sequence of unitary evolution operators constructed from a stochastic process that successively approaches the reduced state of the density matrix on a p-body subsystem, represented by a p-RDM, to a target p-body matrix, potentially a p-RDM. The generators of the evolution operators follow the well-known adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented by using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide whether a given p-body matrix is N-representable, establishing a criterion to determine its quality and correcting it. We apply the proposed hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, from the reduced Bardeen-Cooper-Schrieffer model with constant pairing, and from the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices toward different targets.

An exact one-particle theory of bosonic excitations: from a generalized Hohenberg–Kohn theorem to convexified N-representability

Motivated by the Penrose–Onsager criterion for Bose–Einstein condensation we propose a functional theory for targeting low-lying excitation energies of bosonic quantum systems through the one-particle picture. For this, we employ an extension of the Rayleigh–Ritz variational principle to ensemble states with spectrum w and prove a corresponding generalization of the Hohenberg–Kohn theorem: the underlying one-particle reduced density matrix determines all properties of systems of N identical particles in their w -ensemble states. Then, to circumvent the v-representability problem common to functional theories, and to deal with energetic degeneracies, we resort to the Levy–Lieb constrained search formalism in combination with an exact convex relaxation. The corresponding bosonic one-body w -ensemble N-representability problem is solved comprehensively. Remarkably, this reveals a complete hierarchy of bosonic exclusion principle constraints in conceptual analogy to Pauli’s exclusion principle for fermions and recently discovered generalizations thereof.

Special entangled fermionic systems and exceptional symmetries

Special fermionic systems entered the realm of quantum chemistry in the seventies in the work of Borland and Dennis in the form of a toy model. This work was leading to a detailed study of the N-representability problem by Klyachko. The topic then has been reconsidered in the light of entanglement theory boiling down to the notion of entanglement polytopes. Recently building on certain properties of such special fermionic systems, a connection between the coupled cluster method and entanglement has been established. In this paper we show that precisely such a special class of systems also provides an interesting physical realization for structures related to the Lie algebras of exceptional groups. This result draws such exotic symmetry structures under the umbrella of entangled systems of physical relevance.

Characterizing quantum states via sector lengths

Correlations in multiparticle systems are constrained by restrictions from quantum mechanics. A prominent example for these restrictions are monogamy relations, limiting the amount of entanglement between pairs of particles in a three-particle system. A powerful tool to study correlation constraints is the notion of sector lengths. These quantify, for different k, the amount of k-partite correlations in a quantum state in a basis-independent manner. We derive tight bounds on the sector lengths in multi-qubit states and highlight applications of these bounds to entanglement detection, monogamy relations and the n-representability problem. For the case of two- and three qubits we characterize the possible sector lengths completely and prove a symmetrized version of strong subadditivity for the linear entropy.

Two-point weighted density approximations for the kinetic energy density functional

We construct a model for the one-electron reduced density matrix that is symmetric and which satisfies the diagonal of the idempotency constraint and then use this model to evaluate the kinetic energy. This strategy for designing density functionals directly addresses the N-representability problem for kinetic energy density functionals. Results for atoms and molecules are encouraging, especially considering the simplicity of the model. However, like all of the other kinetic energy functionals in the literature, quantitative accuracy is not achieved.

Towards a formal definition of static and dynamic electronic correlations.

Some of the most spectacular failures of density-functional and Hartree-Fock theories are related to an incorrect description of the so-called static electron correlation. Motivated by recent progress in the N-representability problem of the one-body density matrix for pure states, we propose a method to quantify the static contribution to the electronic correlation. By studying several molecular systems we show that our proposal correlates well with our intuition of static and dynamic electron correlation. Our results bring out the paramount importance of the occupancy of the highest occupied natural spin-orbital in such quantification.

The extended Koopmans' theorem for orbital-optimized methods: Accurate computation of ionization potentials

The extended Koopmans' theorem (EKT) provides a straightforward way to compute ionization potentials (IPs) from any level of theory, in principle. However, for non-variational methods, such as Møller-Plesset perturbation and coupled-cluster theories, the EKT computations can only be performed as by-products of analytic gradients as the relaxed generalized Fock matrix (GFM) and one- and two-particle density matrices (OPDM and TPDM, respectively) are required [J. Cioslowski, P. Piskorz, and G. Liu, J. Chem. Phys. 107, 6804 (1997)]. However, for the orbital-optimized methods both the GFM and OPDM are readily available and symmetric, as opposed to the standard post Hartree-Fock (HF) methods. Further, the orbital optimized methods solve the N-representability problem, which may arise when the relaxed particle density matrices are employed for the standard methods, by disregarding the orbital Z-vector contributions for the OPDM. Moreover, for challenging chemical systems, where spin or spatial symmetry-breaking problems are observed, the abnormal orbital response contributions arising from the numerical instabilities in the HF molecular orbital Hessian can be avoided by the orbital-optimization. Hence, it appears that the orbital-optimized methods are the most natural choice for the study of the EKT. In this research, the EKT for the orbital-optimized methods, such as orbital-optimized second- and third-order Møller-Plesset perturbation [U. Bozkaya, J. Chem. Phys. 135, 224103 (2011)] and coupled-electron pair theories [OCEPA(0)] [U. Bozkaya and C. D. Sherrill, J. Chem. Phys. 139, 054104 (2013)], are presented. The presented methods are applied to IPs of the second- and third-row atoms, and closed- and open-shell molecules. Performances of the orbital-optimized methods are compared with those of the counterpart standard methods. Especially, results of the OCEPA(0) method (with the aug-cc-pVTZ basis set) for the lowest IPs of the considered atoms and closed-shell molecules are substantially accurate, the corresponding mean absolute errors are 0.11 and 0.15 eV, respectively.

Compatible quantum correlations: Extension problems for Werner and isotropic states

We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reduced states of a global N-partite quantum state. Intuitively, we say that the multipartite state "joins" the underlying correlations. Determining whether, for a given set of states and a given joining structure, a compatible N-partite quantum state exists is known as the quantum marginal problem. We restrict to bipartite reduced states that belong to the paradigmatic classes of Werner and isotropic states in d dimensions, and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The second is m-n sharability of a Werner state across N subsystems, which may be seen as a variant of the N-representability problem to the case where subsystems are partitioned into two groupings of m and n parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we determine necessary and sufficient conditions for three-party joinability and 1-n sharability for arbitrary d. Our results explicitly show that although entanglement is required for sharing limitations to emerge, correlations beyond entanglement generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. The relationship between joinability and quantum cloning as well as implications for the joinability of arbitrary bipartite states are discussed.

On the size-consistency of the reduced-density-matrix method and the unitary invariant diagonal N-representability conditions

A promising variational approach for determining the ground state energy and its properties is by using the second-order reduced density matrix (2-RDM). However, the leading obstacle with this approach is the N-representability problem. By employing a subset of conditions (typically the P, Q, G, T1 and T2′ conditions) results comparable to those of CCSD(T) can be achieved. However, these conditions do not guarantee size-consistency. In this work, we show that size-consistency can be satisfied if the 2-RDM satisfies the following conditions: (i) the 2-RDM is unitary invariant diagonal N-representable; (ii) the 2-RDM corresponding to each (unspecified) subsystem is the eigenstate of the number of corresponding electrons; and (iii) the 2-RDM satisfies at least one of the P, Q, G, T1 and T2′ conditions. This is the first time that a computationally feasible (though demanding) sufficient condition for the RDM method that guarantees size-consistency in all chemical systems has been published in the literature.

Structure of Fermionic Density Matrices: CompleteN-Representability Conditions

We present a constructive solution to the N-representability problem: a full characterization of the conditions for constraining the two-electron reduced density matrix to represent an N-electron density matrix. Previously known conditions, while rigorous, were incomplete. Here, we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompositions of model Hamiltonians. Existing conditions D, Q, G, T1, and T2, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems.

Significant conditions for the two-electron reduced density matrix from the constructive solution ofNrepresentability

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. 108, 263002 (2012)]. 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 nonnegativity of q-body operators. In addition to the known T1 and T2 conditions, we derive a new 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 new 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.

Rank reduction for the local consistency problem

We address the problem of how simple a solution can be for a given quantum local consistency instance. More specifically, we investigate how small the rank of the global density operator can be if the local constraints are known to be compatible. We prove that any compatible local density operators can be satisfied by a low rank global density operator. Then we study both fermionic and bosonic versions of the N-representability problem as applications. After applying the channel-state duality, we prove that any compatible local channels can be obtained through a global quantum channel with small Kraus rank.

Generalized P, Q and G conditions

The N-Representability Problem entails characterizing the set of second order reduced states that are contractions of N-electron states of the Fermion Fock algebra. This problem is formulated in the form of finding the conditions that a positive linear functional defined on a subspace of this algebra must satisfy in order to be extended to the whole algebra. As this algebra is a w*-algebra one can utilize a theorem by Kadison that shows it is sufficient to consider the values of linear functionals on projectors contained in the subspace in order to determine whether they have positive extensions. Thus we find the form of projectors belonging to the subspace of one and two particle operators and subsequently show that the extension conditions needed in the N-Representability Problem correspond to generalized P, Q and G conditions plus the additional constraints that the functionals be dispersion free on the number operator and their values on one particle operators determined by their values on two particle operators.

Interacting Boson Problems Can Be QMA Hard

Computing the ground-state energy of interacting electron problems has recently been shown to be hard for quantum Merlin Arthur (QMA), a quantum analogue of the complexity class NP. Fermionic problems are usually hard, a phenomenon widely attributed to the so-called sign problem. The corresponding bosonic problems are, according to conventional wisdom, tractable. Here, we demonstrate that the complexity of interacting boson problems is also QMA hard. Moreover, the bosonic version of N-representability problem is QMA complete. Consequently, these problems are unlikely to have efficient quantum algorithms.

Obtaining the two-body density matrix in the density matrix renormalization group method

We present an approach that allows to produce the two-body density matrix during the density matrix renormalization group (DMRG) run without an additional increase in the current disk and memory requirements. The computational cost of producing the two-body density matrix is proportional to O(M3k2+M2k4). The method is based on the assumption that different elements of the two-body density matrix can be calculated during different steps of a sweep. Hence, it is desirable that the wave function at the convergence does not change during a sweep. We discuss the theoretical structure of the wave function ansatz used in DMRG, concluding that during the one-site DMRG procedure, the energy and the wave function are converging monotonically at every step of the sweep. Thus, the one-site algorithm provides an opportunity to obtain the two-body density matrix free from the N-representability problem. We explain the problem of local minima that may be encountered in the DMRG calculations. We discuss theoretically why and when the one- and two-site DMRG procedures may get stuck in a metastable solution, and we list practical solutions helping the minimization to avoid the local minima.

Investigations of random pair densities and the application to the N-representability problem

We show that random pair densities that can arise from a many-body wavefunction have a distinct distribution of their eigenvalues, when interpreted as a kernel of an integral operator. This gives a probabilistic condition for pair densities as well as two-particle density matrices to be obtainable from a wavefunction. The condition depends on the number of particles in the system.

Quantum Computational Complexity of theN-Representability Problem: QMA Complete

We study the computational complexity of the N-representability problem in quantum chemistry. We show that this problem is quantum Merlin-Arthur complete, which is the quantum generalization of nondeterministic polynomial time complete. Our proof uses a simple mapping from spin systems to fermionic systems, as well as a convex optimization technique that reduces the problem of finding ground states to N representability.

Quantum marginal problem and N-representability

A fermionic version of the quantum marginal problem was known from the early sixties as N-representability problem. In 1995 it was mentioned by the National Research Council of the USA as one of ten most prominent research challenges in quantum chemistry. In spite of this recognition the progress was very slow, until a couple of years ago the problem came into focus again, now in the framework of quantum information theory. In the paper I give a survey of the recent development.

Generalizations of the Hohenberg-Kohn theorem: I. Legendre Transform Constructions of Variational Principles for Density Matrices and Electron Distribution Functions

Given a general, N-particle Hamiltonian operator, analogs of the Hohenberg-Kohn theorem are derived for functions that are more general than the particle density, including density matrices and the diagonal elements thereof. The generalization of Lieb's Legendre transform ansatz to the generalized Hohenberg-Kohn functional not only solves the upsilon-representability problem for these entities, but, more importantly, also solves the N-representability problem. Restricting the range of operators explored by the Legendre transform leads to a lower bound on the true functional. If all the operators of interest are incorporated in the restricted maximization, however, the variational principle dictates that exact results are obtained for the systems of interest. This might have important implications for practical work not only for density matrices but also for density functionals. A follow-up paper will present a useful alternative approach to the upsilon- and N-representability problems based on the constrained search formalism.

Using the Kohn–Sham formalism in pair density-functional theories

There has been some recent interest in ‘generalized’ density functional theories, in which the fundamental variable is the density of electron pairs (pair density), instead of the electron density. Much of the work along these lines has focused on Weisacker-type functionals of the density and has not explicitly considered the N-representability problem. As is discussed here, such theories cannot be accurate. Some new functionals for the kinetic energy, which may address the problem, are proposed, and the correlation component is discussed as a relatively small correction to the Kohn–Sham kinetic energy.