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arXiv:2403.08045 · Faraday Discussions

What can quantum information theory offer to quantum chemistry?

Damiano Aliverti-Piuri, Kaustav Chatterjee, Lexin Ding, Ke Liao, Julia Liebert, Christian Schilling

Published Jan 1, 2024• 17 citations
arXivHTMLPDF

Abstract

It is the ultimate goal of this work to foster synergy between quantum chemistry and the flourishing field of quantum information theory. For this, we first translate quantum information concepts, such as entanglement and correlation, into the context of quantum chemical systems. In particular, we establish two conceptually distinct perspectives on 'electron correlation', leading to a notion of orbital and particle correlation. We then demonstrate that particle correlation equals total orbital correlation minimized over all orbital bases. Accordingly, particle correlation resembles the minimal, thus intrinsic, complexity of many-electron wave functions, while orbital correlation quantifies their complexity relative to a basis. We illustrate these concepts of intrinsic and extrinsic correlation complexity in molecular systems, which also manifests the crucial link between the two correlation pictures. Our results provide theoretical justification for the long-favored natural orbitals for simplifying electronic structures, and open new pathways for developing more efficient approaches towards the electron correlation problem.

DOI
10.1039/d4fd00059e

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It is the ultimate goal of this work to foster synergy between quantum chemistry and the flourishing field of quantum information theory. For this, we first translate quantum information concepts, such as entanglement and correlation, into the context of quantum chemical systems. In particular, we establish two conceptually distinct perspectives on 'electron correlation', leading to a notion of orbital and particle correlation. We then demonstrate that particle correlation equals total orbital correlation minimized over all orbital bases. Accordingly, particle correlation resembles the minimal, thus intrinsic, complexity of many-electron wave functions, while orbital correlation quantifies their complexity relative to a basis. We illustrate these concepts of intrinsic and extrinsic correlation complexity in molecular systems, which also manifests the crucial link between the two correlation pictures. Our results provide theoretical justification for the long-favored natural orbitals for simplifying electronic structures, and open new pathways for developing more efficient approaches towards the electron correlation problem.
A recent development in quantum chemistry has established the quantum mutual information between orbitals as a major descriptor of electronic structure. This has already facilitated remarkable improvements in numerical methods and may lead to a more comprehensive foundation for chemical bonding theory. Building on this promising development, our work provides a refined discussion of quantum information theoretical concepts by introducing the physical correlation and its separation into classical and quantum parts as distinctive quantifiers of electronic structure. In particular, we succeed in quantifying the entanglement. Intriguingly, our results for different molecules reveal that the total correlation between orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding. Our work also shows that implementing the fundamental particle number superselection rule, so far not accounted for in quantum chemistry, removes a major part of correlation and entanglement seen previously. In that respect, realizing quantum information processing tasks with molecular systems might be more challenging than anticipated.
Scientists often refer to matter, energy, and information as the three basic categories, though they are related by various scientific laws. Compared to the concepts of matter and energy, the scientific definition of information is relatively new: it was only in the 1940s that Claude Shannon introduced the definition of information as a reduction of uncertainty, laying the foundation for modern information theory. In molecular electronic structure theory, we routinely encounter nonnegative quantities that sum to unity, forming valid probability distributions. These include properly normalized electron densities, eigenvalues of reduced density matrices, squared wave function coefficients in orthonormal bases, and related quantities. The incorporation of information-theoretic concepts into quantum chemistry has yielded profound insights into electronic system behavior. The two dominant frameworks in electronic structure theory are quantum many-body theory and density functional theory. This thesis systematically integrates these approaches with: classical information theory, yielding the information-theoretic approach, and quantum information theory, generating distinct tools like orbital entanglement measures and quantum fidelity metrics. In this thesis, we subsequently demonstrate applications of these information-theoretic methods in quantum chemistry for analyzing electronic structure and chemical phenomena.
Abstract The second quantum revolution is all about exploiting the quantum nature of atoms and molecules to execute quantum information processing tasks. To boost this growing endeavor and by anticipating the key role of quantum chemistry therein, our work establishes a framework for systematically exploring, quantifying and dissecting correlation effects in molecules. By utilizing the geometric picture of quantum states we compare—on a unified basis and in an operationally meaningful way—total, quantum and classical correlation and entanglement in molecular ground states. To unlock and maximize the quantum informational resourcefulness of molecules an orbital optimization scheme is developed, leading to a paradigm-shifting insight: a single covalent bond equates to the entanglement 2 ln ( 2 ) . This novel and more versatile perspective on electronic structure suggests a generalization of valence bond theory, overcoming deficiencies of modern chemical bonding theories.
The intersection of quantum chemistry and quantum computing has led to significant advancements in understanding the potential of using quantum devices for the efficient calculation of molecular energies. Simultaneously, this intersection enhances the comprehension of quantum chemical properties through the use of quantum computing and quantum information tools. This paper tackles a key question in this relationship: Is the nature of the orbital-wise electron correlations in wavefunctions of realistic prototypical cases classical or quantum? We address this question with a detailed investigation of molecular wavefunctions in terms of Shannon and von Neumann entropies, common tools of classical and quantum information theory. Our analysis reveals a notable distinction between classical and quantum mutual information in molecular systems when analyzed with Hartree-Fock canonical orbitals. However, this difference decreases dramatically, by ∼100-fold, when natural orbitals are used as reference. This finding suggests that orbital correlations, when viewed through the appropriate basis, are predominantly classical. Consequently, our study underscores the importance of using natural orbitals to accurately assess molecular orbital correlations and to avoid their overestimation.
Natural orbitals, defined in electronic structure and quantum chemistry as the molecular orbitals diagonalizing the one-particle reduced density matrix of the ground state, have been conjectured for decades to be the perfect reference orbitals to describe electron correlation. In the present work we applied the Wave function-Adapted Hamiltonian Through Orbital Rotation (WAHTOR) method to study correlated empirical ansätze for quantum computing. In all representative molecules considered, we show that the converged orbitals are coinciding with natural orbitals. Interestingly, the resulting quantum mutual information matrix built on such orbitals is also maximally sparse, providing a clear picture that such orbital choice is indeed able to provide the optimal basis to describe electron correlation. The correlation is therefore encoded in a smaller number of qubit pairs contributing to the quantum mutual information matrix.
The basic concepts of orbital entanglement and its application to chemistry are briefly reviewed. The calculation of orbital entanglement measures from correlated wavefunctions is discussed in terms of reduced n ‐particle density matrices. Possible simplifications in their evaluation are highlighted in case of seniority‐zero wavefunctions. Specifically, orbital entanglement allows us to dissect electron correlation effects in its strong and weak contributions, to determine bond orders, to assess the quality and stability of active space calculations, to monitor chemical reactions, and to identify points along the reaction coordinate where electronic wavefunctions change drastically. Thus, orbital entanglement represents a useful and intuitive tool to interpret complex electronic wavefunctions and to facilitate a qualitative understanding of electronic structure and how it changes in chemical processes. © 2014 Wiley Periodicals, Inc.