Generalized Pauli Constraints Research Articles

N-representability violations in truncated equation-of-motion coupled-cluster methods.

One-electron reduced density matrices (1RDMs) from equation-of-motion (EOM) coupled-cluster with single and double excitations (CCSD) calculations are analyzed to assess their N-representability (i.e., whether they are derivable from a physical N-electron state). We identify EOM-CCSD stationary states whose 1RDMs violate either ensemble-state N-representability conditions or pure-state conditions known as generalized Pauli constraints. As such, these 1RDMs do not correspond to any physical N-electron state. Unphysical states are also encountered in the course of time-dependent EOM-CC simulations; when an external field drives transitions between a pair of stationary states with pure-state N-representable 1RDMs, the 1RDM of the time-dependent state can violate ensemble-state conditions. These observations point to potential challenges in interpreting the results of time-dependent EOM-CCSD simulations.

Open quantum system violates generalized Pauli constraints on quantum device

The Pauli exclusion principle governs the fundamental structure and function of fermionic systems from molecules to materials. Nonetheless, when such a fermionic system is in a pure state, it is subject to additional restrictions known as the generalized Pauli constraints (GPCs). Here we verify experimentally the violation of the GPCs for an open quantum system using data from a superconducting-qubit quantum computer. We prepare states of systems with three-to-seven qubits directly on the quantum device and measure the one-fermion reduced density matrix (1-RDM) from which we can test the GPCs. We find that the GPCs of the 1-RDM are sufficiently sensitive to detect the openness of the 3-to-7 qubit systems in the presence of a single-qubit environment. Results confirm experimentally that the openness of a many-fermion quantum system can be decoded from only a knowledge of the 1-RDM with potential applications from quantum computing and sensing to noise-assisted energy transfer.

Publisher’s Note: “Generalized Pauli constraints in large systems: The Pauli principle dominates” [J. Math. Phys. 62, 032204 (2021)

Publisher’s Note: “Generalized Pauli constraints in large systems: The Pauli principle dominates” [J. Math. Phys. 62, 032204 (2021)

Generalized Pauli constraints in large systems: The Pauli principle dominates

Lately, there has been a renewed interest in fermionic one-body reduced density matrices and their restrictions beyond the Pauli principle. These restrictions are usually quantified using the polytope of allowed, ordered eigenvalues of such matrices. Here, we prove that this polytope’s volume rapidly approaches the volume predicted by the Pauli principle as the dimension of the one-body space grows and that additional corrections, caused by generalized Pauli constraints, are of much lower order unless the number of fermions is small. Indeed, we argue that the generalized constraints are most restrictive in (effective) few-fermion settings with low Hilbert space dimension.

Implications of pinned occupation numbers for natural orbital expansions. II: rigorous derivation and extension to non-fermionic systems

We have explained and comprehensively illustrated in Part I (Schilling et al 2019 arXiv:1908.10938) that the generalized Pauli constraints suggest a natural extension of the concept of active spaces. In the present Part I (Schilling et al 2019 arXiv:1908.10938)I, we provide rigorous derivations of the theorems involved therein. This will offer in particular deeper insights into the underlying mathematical structure and will explain why the saturation of generalized Pauli constraints implies a specific simplified structure of the corresponding many-fermion quantum state. Moreover, we extend the results of Part I (Schilling et al 2019 arXiv:1908.10938) to non-fermionic multipartite quantum systems, revealing that extremal single-body information has always strong implications for the multipartite quantum state. In that sense, our work also confirms that pinned quantum systems define new physical entities and the presence of pinnings reflect the existence of (possibly hidden) ground state symmetries.

Implications of pinned occupation numbers for natural orbital expansions: I. Generalizing the concept of active spaces

The concept of active spaces simplifies the description of interacting quantum many-body systems by restricting to a neighborhood of active orbitals around the Fermi level. The respective wavefunction ansatzes which involve all possible electron configurations of active orbitals can be characterized by the saturation of a certain number of Pauli constraints , identifying the occupied core orbitals (ni = 1) and the inactive virtual orbitals (nj = 0). In Part I, we generalize this crucial concept of active spaces by referring to the generalized Pauli constraints. To be more specific, we explain and illustrate that the saturation of any such constraint on fermionic occupation numbers characterizes a distinctive set of active electron configurations. A converse form of this selection rule establishes the basis for corresponding multiconfigurational wavefunction ansatzes. In Part II, we provide rigorous derivations of those findings. Moroever, we extend our results to non-fermionic multipartite quantum systems, revealing that extremal single-body information has always strong implications for the multipartite quantum state. In that sense, our work also confirms that pinned quantum systems define new physical entities and the presence of pinnings reflect the existence of (possibly hidden) ground state symmetries.

Approximating one-matrix functionals without generalized Pauli constraints

Common features are identified in the properties of the pure (PF) and ensemble (EF) energy functionals defined within the density matrix functional theory (DMFT) as well as their free-form (FFF) approximations. These features rationalize abandoning, in the common DMFT practice, prohibitively complicated generalized Pauli constraints (GPCs). Specifically, it is shown that the exact PF and EF coincide on the set of the v-representable first-order reduced density matrices (1RDMs), while FFFs approximate the common PF and EF for such 1RDMs. It is revealed that for some types of FFFs, for example for the geminal-based approximations, the GPCs are essentially trivialized to the pairwise natural occupation numbers (NONs) degeneracy conditions, routinely imposed for closed-shell systems.

Sparse non-orthogonal wave function expansions from the extension of the generalized Pauli constraints to the two-electron reduced density matrix.

Generalized Pauli constraints (GPCs) impose constraints in the form of inequalities on the natural orbital occupation numbers of the one electron reduced density matrix (1-RDM), defining the set of pure N-representable 1-RDMs, or 1-RDMs that can be derived from an N-electron wave function. Saturation of these constraints is termed "pinning" and implies a significant simplification of the N-electron wave function as the number of Slater determinants required to fully describe the system is reduced. Recent research has shown pinning to occur for the ground states of atoms and molecules with N = 3 and r = 6, where N is the number of electrons and r is the number of spin orbitals. For N = 4 and r = 8, however, pinning occurs not to the GPCs but rather to inequalities defining the pure N-representable two-electron reduced density matrices (2-RDMs). Using these more general inequalities, we derive a wave function ansatz for a system with four electrons in eight spin orbitals. We apply the ansatz to the isoelectronic series of the carbon atom and the dissociation of linear H4 where the correlation energies are recovered to fractions of a kcal/mol. These results provide a foundation for further developments in wave function and RDM theories based on "pinned" solutions, and elucidate a fundamental physical basis for the emergence of non-orthogonal bases in electronic systems of N ≥ 4.

Communication: Relating the pure and ensemble density matrix functional.

A crucial theorem in Reduced Density Matrix Functional Theory (RDMFT) suggests that the universal pure and ensemble functionals coincide on their common domain of pure N-representable one-matrices. We refute this by a comprehensive analysis of the geometric picture underlying Levy's constrained search. Moreover, we then show that the ensemble functional follows instead as the lower convex envelop of the pure functional. It is particularly remarkable that the pure functional determines the ensemble functional even outside its own domain of pure N-representable one-matrices. From a general perspective, this demonstrates that relaxing pure RDMFT to ensemble RDMFT does not necessarily circumvent the complexity of the one-body pure N-representability conditions (generalized Pauli constraints). Instead, the complexity may simply be transferred from the underlying space of pure N-representable one-matrices to the structure of the universal one-matrix functional.

Generalized Pauli constraints in small atoms

The natural occupation numbers of fermionic systems are subject to non-trivial constraints, which include and extend the original Pauli principle. A recent mathematical breakthrough has clarified their mathematical structure and has opened up the possibility of a systematic analysis. Early investigations have found evidence that these constraints are exactly saturated in several physically relevant systems, e.g., in a certain electronic state of the Beryllium atom. It has been suggested that in such cases, the constraints, rather than the details of the Hamiltonian, dictate the system's qualitative behaviour. Here, we revisit this question with state-of-the-art numerical methods for small atoms. We find that the constraints are, in fact, not exactly saturated, but that they lie much closer to the surface defined by the constraints than the geometry of the problem would suggest. While the results seem incompatible with the statement that the generalized Pauli constraints drive the behaviour of these systems, they suggest that the qualitatively correct wave-function expansions can in some systems already be obtained on the basis of a limited number of Slater determinants, which is in line with numerical evidence from quantum chemistry.

Role of the pair potential for the saturation of generalized Pauli constraints

The dependence of the (quasi-)saturation of the generalized Pauli constraints on the pair potential is studied for ground states of few-fermion systems. For this, we consider spinless fermions in one dimension which are harmonically confined and interact by pair potentials of the form $|{x}_{i}\ensuremath{-}{x}_{j}{|}^{s}$ with $\ensuremath{-}1\ensuremath{\le}s\ensuremath{\le}5$. We use the density matrix renormalization group approach and large orbital basis to achieve the convergence on more than ten digits of both the variational energy and the natural occupation numbers. Our results confirm that the conflict between energy minimization and fermionic exchange symmetry results in a universal and nontrivial quasisaturation of the generalized Pauli constraints (quasipinning), implying tremendous structural simplifications of the fermionic ground state for all $s$. Those numerically exact results are complemented by an analytical study based on a self-consistent perturbation theory which we develop for this purpose. The respective results for the weak-coupling regime eventually elucidate the singular behavior found for the specific values $s=2,4,...,$ resulting in an extremely strong quasipinning.

Structure of the first order reduced density matrix in three electron systems: A generalized Pauli constraints assisted study.

We investigate the structure of the one-body reduced density matrix of three electron systems, i.e., doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use configuration interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e., the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in reduced density matrix functional theory minimization schemes, when generalized Pauli constraints (GPCs) are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost "pinned."

Applied Quantum Physics for Novel Quantum Computation Approaches: an Update

Quantum information processing is a critical part of the development of future computers, quantum computers, and quantum algorithms, where elementary particles such as photons and electrons can be applied in optomagnetic or optoelectronic devices. The computational physics behind these emerging approaches is also experiencing dramatic developments. In this paper I report on the most recent mathematical basics for quantum algorithms and quantum computing approaches. Some of these described approaches show intriguing methods for determining the states and wavefunction properties for anyons, bosons, and fermions in quantum wells that have been developed in the last years. The study also shows approaches based on N-quantum states and the reduced 1- and 2-fermion picture, which can be used for developing models for anyons and multi-fermionic states in quantum algorithms. Also, antisymmetry and generalized Pauli constraints have been given particular emphasis and include establishing a basis for pinning and quasipinning for exploring the symmetric states of many-fermionic subsets, as a foundation for quantum information processing, and are here briefly revisited. This study summarizes the developments in recent years of an advanced and important field for future computational techniques.

Sparsity of the wavefunction from the generalized Pauli exclusion principle.

Electron occupations that arise from pure quantum states are restricted by a stringent set of conditions that are said to generalize the Pauli exclusion principle. These generalized Pauli constraints (GPCs) define the boundary of the set of one-electron reduced density matrices (1-RDMs) that are derivable from at least one N-electron wavefunction. In this paper, we investigate the sparsity of the Slater-determinant representation of the wavefunction that is a necessary, albeit not sufficient, condition for its 1-RDM to lie on the boundary of the set of pure N-representable 1-RDMs or in other words saturate one of the GPCs. The sparse wavefunction, we show, is exact not only for 3 electrons in 6 orbitals but also for 3 electrons in 8 orbitals. For larger numbers of electrons and/or orbitals in the lowest spin state, the exact wavefunction does not generally saturate one of the GPCs, and hence, the sparse representation is typically an approximation. Because the sparsity of the wavefunction is a necessary but not sufficient condition for saturation of one of the GPCs, optimization of the sparse wavefunction Ansatz to minimize the ground-state energy does not necessarily produce a wavefunction whose 1-RDM exactly saturates one of the GPCs. While the sparse Ansatz can be employed with arbitrary orbitals or optimized orbitals, in this paper, we explore the Ansatz with the natural orbitals from full configuration interaction, which yields an upper bound to the ground-state energy that equals the exact energy for a given basis set if the full-configuration-interaction wavefunction saturates the Ansatz's GPC. With calculations on the boron isoelectronic sequence, the dinitrogen cation N2+, hydrogen chains, and cyclic conjugated π systems, we examine the quality of the sparse wavefunction Ansatz from the amount of correlation energy recovered.

Noise-assisted energy transfer from the dilation of the set of one-electron reduced density matrices

Noise-assisted energy transfer can be explained geometrically in terms of the set of one-electron reduced density matrices (1-RDMs) [R. Chakraborty and D. A. Mazziotti, Phys. Rev. A 91, 010101(R) (2015)]. In this paper, we examine the geometric picture of quantum noise for the seven-chromophore Fenna-Matthews-Olson (FMO) complex. Noise expands the feasible set of orbital occupation trajectories to the target state through the violation of the pure-state N-representability conditions on the 1-RDM, known as the generalized Pauli constraints. While the generalized Pauli constraints are not explicitly known for seven-electron systems, we are able to treat a seven-exciton model of the FMO complex through the use of generalized Pauli constraints for p qubits which are known for arbitrary p. In the model, we find that while dephasing noise alone produces a trajectory of ensemble states that neither expands the set of 1-RDMs nor reaches the reaction center, the inclusion of both dephasing and dissipation expands the set of 1-RDMs and exhibits an efficient energy transfer to the reaction center. The degree to which the noise expands the set of 1-RDMs, violating the generalized Pauli constraints, is quantified by the distance of the 1-RDM outside its pure set to the distance of the 1-RDM inside its ensemble set. The geometric picture of energy transfer has applications to general quantum systems in chemistry and physics.

Influence of the fermionic exchange symmetry beyond Pauli's exclusion principle

Pauli's exclusion principle has a strong impact on the properties of most fermionic quantum systems. Remarkably, the fermionic exchange symmetry implies further constraints on the one-particle picture. By exploiting those generalized Pauli constraints we derive a measure which quantifies the influence of the exchange symmetry beyond Pauli's exclusion principle. It is based on a geometric hierarchy induced by the exclusion principle constraints. We provide a proof of principle by applying our measure to a simple model. In that way, we conclusively confirm the physical relevance of the generalized Pauli constraints and show that the fermionic exchange symmetry can have an influence on the one-particle picture beyond Pauli's exclusion principle. Our findings provide a new perspective on fermionic multipartite correlation since our measure allows one to distinguish between static and dynamic correlations.

Spectral properties of reduced fermionic density operators and parity superselection rule

We consider pure fermionic states with a varying number of quasiparticles and analyze two types of reduced density operators: one is obtained via tracing out modes, the other is obtained via tracing out particles. We demonstrate that spectra of mode-reduced states are not identical in general and fully characterize pure states with equispectral mode-reduced states. Such states are related via local unitary operations with states satisfying the parity superselection rule. Thus, valid purifications for fermionic density operators are found. To get particle-reduced operators for a general system, we introduce the operation $$\varPhi (\varrho ) = \sum _i a_i \varrho a_i^{\dag }$$ . We conjecture that spectra of $$\varPhi ^p(\varrho )$$ and conventional p-particle reduced density matrix $$\varrho _p$$ coincide. Non-trivial generalized Pauli constraints are derived for states satisfying the parity superselection rule.

Variational optimization of the two-electron reduced-density matrix under pure-state N-representability conditions.

The direct variational optimization of the ground-state two-electron reduced-density matrix (2-RDM) is typically performed under ensemble N-representability conditions. Accordingly, variationally obtained 2-RDMs for degenerate ground states may not represent a pure state. When considering only ground-state energetics, the ensemble nature of the 2-RDM is of little consequence. However, the use of ensemble densities within an extended random phase approximation (ERPA) yields astonishingly poor estimates of excitation energies, even for simple atomic systems [H. van Aggelen et al., Comput. Theor. Chem. 1003, 50-54 (2013)]. Here, we outline an approach for the direct variational optimization of ground-state 2-RDMs that satisfy pure-state N-representability known as generalized Pauli constraints. Within the ERPA, 2-RDMs that satisfy both ensemble conditions and the generalized Pauli constraints yield much more reliable estimates of excitation energies than those that satisfy only ensemble conditions.

Pinning of fermionic occupation numbers: Higher spatial dimensions and spin

The role of the generalized Pauli constraints (GPCs) in higher spatial dimensions and by incorporating spin degrees of freedom is systematically explored for a system of interacting fermions confined by a harmonic trap. Physical relevance of the GPCs is confirmed by analytical means for the ground state in the regime of weak couplings by finding its vector of natural occupation numbers close to the boundary of the allowed region. Such quasipinning is found to become weaker in the intermediate and strong coupling regime. The study of crossovers between different spatial dimensions by detuning the harmonic trap frequencies suggests that quasipinning is essentially an effect for systems with reduced spatial dimensionality. In addition, we find that quasipinning becomes stronger by increasing the degree of spin-polarization. Consequently, the number of states available around the Fermi level plays a key role for the occurrence of quasipinning. This suggests that quasipinning emerges from the conflict between energy minimization and fermionic exchange symmetry.

Pinning of fermionic occupation numbers: General concepts and one spatial dimension

Analytical evidence for the physical relevance of generalized Pauli constraints (GPCs) has recently been provided in [PRL 110, 040404]: Natural occupation numbers $\vec{\lambda}\equiv (\lambda_i)$ of the ground state of a model system in the regime of weak couplings $\kappa$ of three spinless fermions in one spatial dimension were found extremely close, in a distance $D_{min}\sim \kappa^8$ to the boundary of the allowed region. We provide a self-contained and complete study of this quasipinning phenomenon. In particular, we develop tools for its systematic exploration and quantification. We confirm that quasipinning in one dimension occurs also for larger particle numbers and extends to intermediate coupling strengths, but vanishes for very strong couplings. We further explore the non-triviality of our findings by comparing quasipinning by GPCs to potential quasipinning by the less restrictive Pauli exclusion principle constraints. This allows us to eventually confirm the significance of GPCs beyond Pauli's exclusion principle.