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arXiv:2408.11025

Enhanced Shadow Tomography of Molecular Excited States from Enforcing $N$-representability Conditions by Semidefinite Programming

Irma Avdic, David A. Mazziotti

Published Aug 20, 2024• 0 citations
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Abstract

Excited-state properties of highly correlated systems are key to understanding photosynthesis, luminescence, and the development of novel optical materials, but accurately capturing their interactions is computationally costly. We present an algorithm that combines classical shadow tomography with physical constraints on the two-electron reduced density matrix (2-RDM) to treat excited states. The method reduces the number of measurements of the many-electron 2-RDM on quantum computers by (i) approximating the quantum state through a random sampling technique called shadow tomography and (ii) ensuring that the 2-RDM represents an $N$-electron system through imposing $N$-representability constraints by semidefinite programming. This generalizes recent work on the $N$-representability-enhanced shadow tomography of ground-state 2-RDMs. We compute excited-state energies and 2-RDMs of the H$_4$ chain and analyze the critical points along the photoexcited reaction pathway from gauche-1,3-butadiene to bicyclobutane via a conical intersection. The results show that the generalized shadow tomography retains critical multireference correlation effects while significantly reducing the number of required measurements, offering a promising avenue for the efficient treatment of electronically excited states on quantum devices.

DOI
10.48550/arxiv.2408.11025

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Enhanced Shadow Tomography of Molecular Excited States from Enforcing NNN-representability Conditions by Semidefinite Programming
Excited-state properties of highly correlated systems are key to understanding photosynthesis, luminescence, and the development of novel optical materials, but accurately capturing their interactions is computationally costly. We present an algorithm that combines classical shadow tomography with physical constraints on the two-electron reduced density matrix (2-RDM) to treat excited states. The method reduces the number of measurements of the many-electron 2-RDM on quantum computers by (i) approximating the quantum state through a random sampling technique called shadow tomography and (ii) ensuring that the 2-RDM represents an NNN-electron system through imposing NNN-representability constraints by semidefinite programming. This generalizes recent work on the NNN-representability-enhanced shadow tomography of ground-state 2-RDMs. We compute excited-state energies and 2-RDMs of the H4_44​ chain and analyze the critical points along the photoexcited reaction pathway from gauche-1,3-butadiene to bicyclobutane via a conical intersection. The results show that the generalized shadow tomography retains critical multireference correlation effects while significantly reducing the number of required measurements, offering a promising avenue for the efficient treatment of electronically excited states on quantum devices.
Excited-state properties of highly correlated systems are key to understanding photosynthesis, luminescence, and the development of novel optical materials, but accurately capturing their interactions is computationally costly. We present an algorithm that combines classical shadow tomography with physical constraints on the two-electron reduced density matrix (2RDM) to treat excited states. The method reduces the number of measurements of the many-electron 2RDM on quantum computers by (i) approximating the quantum state through a random sampling technique called shadow tomography and (ii) ensuring that the 2RDM represents an NNN-electron system through imposing NNN-representability constraints by semidefinite programming. This generalizes recent work on the NNN-representability-enhanced shadow tomography of ground-state 2RDMs. We compute excited-state energies and 2RDMs of the H4{\mathrm{H}}_{4}H4​ chain and analyze the critical points along the photoexcited reaction pathway from gauche-1,3-butadiene to bicyclobutane via a conical intersection. The results show that the generalized shadow tomography retains critical multireference correlation effects while significantly reducing the number of required measurements, offering a promising avenue for the efficient treatment of electronically excited states on quantum devices.
Quantum state tomography is a fundamental task in quantum information science, enabling detailed characterization of correlations, entanglement, and electronic structure in quantum systems. However, its exponential measurement and computational demands limit scalability, motivating efficient alternatives such as classical shadows, which enable accurate prediction of many observables from randomized measurements. In this work, we introduce a bi-objective semidefinite programming approach for constrained shadow tomography, designed to reconstruct the two-particle reduced density matrix (2-RDM) from noisy or incomplete shadow data. By integrating NNN-representability constraints and nuclear-norm regularization into the optimization, the method builds an NNN-representable 2-RDM that balances fidelity to the shadow measurements with energy minimization. This unified framework mitigates noise and sampling errors while enforcing physical consistency in the reconstructed states. Numerical and hardware results demonstrate that the approach significantly improves accuracy, noise resilience, and scalability, providing a robust foundation for physically consistent fermionic state reconstruction in realistic quantum simulations.
Classical shadow tomography provides a randomized scheme for approximating the quantum state and its properties at reduced computational cost with applications in quantum computing. In this Letter we present an algorithm for realizing fewer measurements in the shadow tomography of many-body systems. Accelerated tomography of the two-body reduced density matrix (2-RDM) is achieved by combining classical shadows with necessary constraints for the 2-RDM to represent an N-body system, known as N-representability conditions. We compute the ground-state energies and 2-RDMs of hydrogen chains and the N_{2} dissociation curve. The results demonstrate a significant reduction in the number of measurements with important applications to quantum many-body simulations on near-term quantum devices.
Correlated Purification for Restoring NNN-Representability in Quantum Simulation
Classical shadow tomography offers a scalable route to estimating properties of quantum states, but the resulting reduced density matrices (RDMs) often violate constraints that ensure they represent NNN-electron states -- known as NNN-representability conditions -- because of statistical and hardware noise. We present a correlated purification framework based on semidefinite programming to restore accuracy to these noisy, unphysical two-electron RDMs. The method performs a bi-objective optimization that minimizes both the many-electron energy and the nuclear norm of the change in the measured 2-RDM. The nuclear norm, often employed in matrix completion, promotes low-rank, physically meaningful corrections to the 2-RDM, while the energy term acts as a regularization term that can improve the purity of the ground state. While the method is particularly effective for the ground state, it can also be applied to excited and non-stationary states by decreasing the weight of the energy relative to the error norm. In an application to fermionic shadow tomography of large hydrogen chains, correlated purification yields substantial reductions in both energy and 2-RDM error, achieving chemical accuracy across dissociation curves. This framework provides a robust strategy for tomography in many-body quantum simulations.
The last several decades have seen significant advances in the theoretical modeling of materials within the fields of solid-state physics and materials science, but many methods commonly applied to this problem struggle to capture strong electron correlation accurately. Recent widespread interest in quantum materials---where strong correlation plays a crucial role in the quantum effects governing their behavior---further highlights the need for theoretical methods capable of rigorously treating such correlation. Here, we present a periodic generalization of variational two-electron reduced density matrix (2-RDM) theory, a bootstrapping-type method that minimizes the ground-state energy as a functional of the 2-RDM without relying on the wave function. The 2-RDM is computed directly by semidefinite programming with NNN-representability conditions, ensuring accurate treatment of strongly correlated electronic systems. By exploiting translational symmetry, we significantly reduce computational scaling, enabling applications to realistic materials-scale systems. Additionally, we introduce an alternative to conventional energy band structures: natural-orbital occupation-number bands, which, being independent of mean-field assumptions, offer deeper insights into electron correlation effects. We demonstrate the effectiveness of this approach by applying the theory to hydrogen chains, molybdenum disulfide, and nickel oxide, showing that natural-orbital occupation bands correctly capture electronic character in regimes where density functional theory fails. This work represents a major step toward accurately describing the electronic structure of quantum materials using reduced density matrices rather than wave functions.
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.