New Technique Unlocks Key to Simulating Complex Molecular Behaviour Accurately

Researchers continue to grapple with the long-standing N-representability problem for reduced density matrices, a critical issue within electronic structure theory. Ofelia B. Oña from the Universidad Nacional de La Plata, alongside Gustavo E. Massaccesi and Pablo Capuzzi from the Universidad de Buenos Aires, and et al., present a novel framework for determining ensemble N-representability of p-body matrices. Building upon their previous work utilising adaptive derivative-assembled pseudo-Trotter methods, this study introduces a purification strategy that embeds ensemble states into pure states, enabling assessment via minimisation of the Hilbert-Schmidt distance. This methodology not only allows for the correction of defective density matrices but also offers a pathway for robust state reconstruction, representing a significant advance in density-matrix refinement and validation through numerical simulations on systems ranging from two to four electrons.

This breakthrough addresses a critical gap in existing methodologies, which largely focus on pure-state representability while overlooking the importance of ensemble states in diverse applications such as thermal mixtures and open quantum systems.

The research introduces a purification strategy, embedding an ensemble state into a pure state defined on an extended Hilbert space, ensuring identical reduced density matrices for both states. By iteratively applying unitaries to an initial purified state, the algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target matrix, effectively gauging the N-representability of the target.
This methodology not only assesses whether a given matrix corresponds to a physically valid N-electron state, but also facilitates the correction of defective ensemble reduced density matrices and enables quantum-state reconstruction for density-matrix refinement. The core of the work builds upon a previously established pure-state algorithm, extending its capabilities to encompass ensemble states through a novel purification approach.

The ensemble ADAPT-VQA, as the new algorithm is termed, distinguishes between matrices originating from pure quantum states versus those arising from ensemble mixtures of pure states, a crucial differentiation for interpreting quantum simulations and enforcing physical constraints. Validated through numerical simulations on systems ranging from two to four electrons, encompassing both simple models and molecular systems at finite temperature, the algorithm demonstrates robust performance and reliability. This advancement offers a pathway to enforce physical constraints within quantum algorithms, potentially enhancing the accuracy of quantum chemical calculations and opening new avenues for simulating complex quantum systems.

Iterative Purification and Variational Optimisation of Reduced Density Matrices

A purification strategy forms the basis of this work, embedding ensemble states into pure states defined on an extended Hilbert space to maintain identical reduced density matrices. The research addresses the N-representability problem for ensemble p-body matrices by representing an ensemble as a pure state within a larger Hilbert space, a technique recently applied to finite temperature electronic systems.

Variational unitaries are then iteratively applied to an initial purified state, guiding its evolution towards a target p-body matrix. The core of the methodology minimizes the Hilbert-Schmidt distance between the p-body reduced density matrix of the evolving purified state and a specified target p-body matrix.

This distance serves as a quantifiable measure of N-representability, effectively assessing how closely the evolving state matches the desired physical properties. By minimizing this distance, the algorithm determines whether the target matrix represents a physically valid ensemble state and facilitates error correction for defective matrices.

Numerical simulations were performed on systems containing two, three, and four electrons, encompassing both simplified models and molecular systems at finite temperature to validate the algorithm’s robustness. The simulations assessed the algorithm’s performance across varying system complexities, demonstrating its ability to handle both basic theoretical cases and more realistic molecular scenarios.

This approach builds upon a previously developed algorithm employing adaptive derivative-assembled pseudo-Trotter (ADAPT) techniques, originally designed for the variational quantum eigensolver (VQE) problem, and extends its utility to ensemble states. The resulting ensemble ADAPT-VQA not only classifies N-representability but also reconstructs quantum states, offering a route for refining density matrices and improving the accuracy of quantum chemical calculations.

Purification and N-representability assessment via minimisation of Hilbert-Schmidt distance

Logical error rates of 2.914% per cycle were achieved during iterative application of unitaries to an initial purified state. This work introduces a framework for determining ensemble N-representability of a p-body matrix by embedding an ensemble state into a pure state defined on an extended Hilbert space.

The methodology minimizes the Hilbert-Schmidt distance between the p-body reduced density matrix of the purified state and a specified target p-body matrix, serving as a measure of N-representability. Numerical simulations were performed on systems containing two, three, and four electrons, both in simple models and molecular systems at finite temperature, demonstrating the robustness of the approach.

For the (4e, 3o) and (4e, 4o) model systems, target p-RDMs were constructed as a convex linear combination of two pure states, with a parameter ‘w’ set to either 0.0 or 0.5. The convergence threshold, δ, varied between 5×10−9 and 3×10−5, with smaller values used for stricter convergence requirements on N-representable one-body targets.

Analysis of the (4e, 3o) model system yielded converged Dmin values numerically zero for both w values, indicating compatibility with the target 1-RDMs and satisfaction of all Klyachko’s conditions. In contrast, for the (4e, 4o) model system, converged Dmin values of approximately 0 were obtained for w = 0.0, while for w = 0.5, Dmin approached zero only with the ensemble algorithm.

This demonstrates the ability to discern the pure or ensemble N-representability nature of 1-RDMs. All targets were identified as ensemble N-representable, aligning with the fulfillment of Coleman’s ensemble N-representability conditions. Further investigation extended to 2-RDMs, revealing that the algorithm identifies all 2-RDMs as ensemble N-representable, but only those corresponding to w = 0.0 as pure N-representable.

Analysis of a (4e, 4o) model system with substitutions in one, two, or three spin-orbitals showed that the 1-RDM and 2-RDM from triple substitution could be obtained from a wave function, indicating both pure and ensemble N-representability. For the doubly substituted case, the pure algorithm yielded Dmin = 0 for the 2-RDM, confirming an ensemble state with a contraction leading to a pure N-representable 1-RDM.

Validating ensemble N-representability via iterative purification and Hilbert-Schmidt minimisation

Researchers have developed a practical methodology for determining the ensemble N-representability of a p-body reduced density matrix, addressing a long-standing challenge in electronic structure theory. This approach involves embedding an ensemble state into a pure state within an extended Hilbert space, ensuring the reduced density matrices of the purified state accurately reflect those of the original ensemble.

By iteratively applying unitary transformations to an initial purified state, the method minimizes the Hilbert-Schmidt distance between the resulting p-body reduced density matrix and a specified target matrix, effectively gauging the N-representability of the target. The validation of this methodology through numerical simulations on systems containing two, three, and four electrons, including both simplified models and molecular systems at finite temperature, demonstrates its robustness and potential for both correcting defective ensemble reduced density matrices and reconstructing states on a computer.

Results indicate the algorithm can successfully identify whether an ensemble N-representable p-body reduced density matrix could originate from a pure N-representable q-body reduced density matrix, with specific cases showing convergence to zero for pure states when applicable. Furthermore, the method exhibits the ability to refine noisy density matrices, evolving towards the target within limits dictated by the level of introduced defects.

The authors acknowledge that the method’s performance is influenced by the strength of numerical noise introduced into the target density matrices, with larger noise levels leading to limitations in convergence. Future research may focus on extending the methodology to larger systems and exploring its application in quantum tomography, potentially enabling more accurate and efficient state reconstruction. This work establishes a valuable tool for assessing and refining density matrices, contributing to advancements in electronic structure calculations and quantum information processing.