Quantum information theory on sparse wavefunctions and applications for Quantum Chemistry

TL;DR

This work addresses the challenge of performing quantum-information analyses on large, post-Hartree-Fock wavefunctions by introducing SparQ, a tool that maps fermionic states to qubits (e.g., via Jordan–Wigner) and exploits sparsity for efficient partial traces and information metrics. The method provides linear-time partial-trace and observable computations, enabling quantum mutual information and entropy analyses for systems far beyond traditional tensor-network limits. Demonstrations on water and benzene show meaningful insights into correlations and active-space choices, illustrating SparQ's potential to extend quantum-information analyses to large chemical systems. Overall, SparQ advances practical quantum-classical hybrid approaches for studying entanglement and correlations in complex molecules.

Abstract

In recent years Quantum Computing prominently entered in the field of Computational Chemistry, importing and transforming computational methods and ideas originally developed within other disciplines, such as Physics, Mathematics and Computer Science into algorithms able to estimate quantum properties of atoms and molecules on present and future quantum devices. An important role in this contamination process is attributed to Quantum Information techniques, having the twofold role of contributing to the analysis of electron correlation and entanglements and guiding the construction of wavefunction variational ansatzes for the Variational Quantum Eigensolver technique. This paper introduces the tool SparQ (Sparse Quantum state analysis), designed to efficiently compute fundamental quantum information theory observables on post-Hartree-Fock wavefunctions sparse in their definition space. The core methodology involves mapping fermionic wavefunctions to qubit space using fermionic-to-qubits transformations and leveraging the sparse nature of these wavefunctions to evaluate observables and properties of the wavefunction. The effectiveness of SparQ is validated by analyzing the mutual information matrices of wavefunctions for the water molecule and the total entropy of $\sim 10^2$ qubits describing the benzene molecule. This highlights its capability to handle large-scale quantum systems, limited mainly by the capabilities of quantum chemical methods used to retrieve the wavefunctions. The results indicate that quantum information theoretical analysis, so far limited to traditional tensor network methods and study of transition operators, can be applied to all post-Hartree-Fock wavefunctions, extending their applications to larger and more complex chemical systems.

Figures (3)

• Figure 1: Time requirements for the partial trace from $N$ qubits to one qubits over different values of number of components of the wavefunction a) Quadratic tracing method of Eq. \ref{['eq:undirecttrace']}, b) Tracing method of pseudocode \ref{['tracecode']} with linear scaling. As shown in \ref{['tab: systems']} each chain $H_x$ is expressed with 4$y$ qubits.
• Figure 2: Mutual information for the water molecule at CISD level. Jordan Wigner mapping is reported on the left panel, parity mapping with the two-qubit reduction on the right panel. The starting wavefunction for the mutual information was the simulation for the water molecule(H$_2$O) as detailed in Tab. \ref{['tab: systems']}.
• Figure 3: Total entropies as a function of the active space dimension and selection methods for a benzene molecule.The wavefunction analyzed is the cisd wavefunction of benzene with cc-pvdz basis. The Natural Orbitals for the natural orbitals occupation number (NOON) method were retrieved by an iterative procedure (Iterative Natural Orbitals) also based on the cisd wavefunction. Auto-cas selects the orbitals maximizing single-orbital entropy, the Fermi method selects orbitals around the Fermi level of the hf orbitals. The continuous line is the von Neumann entropy of the whole space, while the dashed line is the Shannon entropy of the whole space of the state measured in the computational basis. Further details can be found in the main text.

Theorems & Definitions (1)

• Definition 1