Classical optimisation of reduced density matrix estimations with classical shadows using N-representability conditions under shot noise considerations

TL;DR

This work integrates classical shadow tomography with variational 2-RDM optimization under N-representability constraints to improve the readout of 2-RDM observables from quantum hardware. By introducing an improved classical-shadow estimator and two shadow-derived SDP constraints, the authors identify regimes where shot-noise-robust readout yields substantial gains (up to ~15× fewer shots) for small systems, though benefits diminish for larger systems and under high-precision requirements due to bias from the chosen cost function. The approach showcases how hybrid classical-quantum techniques can enhance chemical observables beyond energies, while highlighting remaining challenges in bias, scalability, and optimal cost-function selection. Overall, the method offers a practical path to more accurate RDM-based properties on near-term devices, with potential refinements in constraint design and cost formulations to maximize shot-efficiency. These findings inform the design of efficient readout protocols for quantum-assisted chemistry workflows, particularly when aiming to extract molecular forces or other beyond-energy observables from noisy quantum data.

Abstract

Classical shadow tomography has become a powerful tool in learning about quantum states prepared on a quantum computer. Recent works have used classical shadows to variationally enforce N-representability conditions on the 2-particle reduced density matrix. In this paper, we build upon previous research by choice of an improved estimator within classical shadow tomography and rephrasing the optimisation constraints, resulting in an overall enhancement in performance under comparable measurement shot budgets. We further explore the specific regimes where these methods outperform the unbiased estimator of the standalone classical shadow protocol and quantify the potential savings in numerical studies.

Figures (6)

• Figure 1: Top: Energy error of the estimate of the ground state energy for N2 in dependence on the number of shots spent on the quantum shadow constraints included in the optimization. Bottom: 2-RDM error in the Frobenius norm towards the true 2-RDM stemming from an FCI calculation of N2 in dependence on the number of shots spent on the quantum shadow constraints included in the optimization. Inset quantifies the factor of improvement over the estimate coming just from the classical shadow at the same level of error (red). The dot in the inset corresponds to the dashed line in the outer figure as a guide to the eye and clarification.
• Figure 2: 2-RDM error in the Frobenius norm towards the true 2-RDM stemming from an FCI calculation of benzene in a (16,16) active space in dependence of the number of shots spent on the quantum shadow constraints included in the optimization. Insert quantifies the factor of improvement over the estimate coming just from the classical shadow at the same level of error (red). The dot in the inset corresponds to the dashed line in the outer figure as a guide to the eye and clarification.
• Figure 3: Energy convergence for $\text{H}_4$ for the optimization under shadow constraint (2) in dependence of number of basis $m$ used for additional constraints during optimissation.
• Figure 4: Top: Absolute value of the 2-body coefficients of the Hamiltonian / two electron integrals for $\text{H}_4$ in STO-3G sorted descending. Bottom: Difference between the shadow estimate at a given epsilon used to build the shadow constraints and the 2-RDM estimate after the semidefinite optimization averaged over 5 runs sorted by the ordering of the upper plot.
• Figure 5: Comparison of different choices of cost functions described in Equations \ref{['eq:nrg_cost']}, \ref{['eq:nrg_nuc_cost']}, \ref{['eq:nuc_cost']}, \ref{['eq:fro_cost']} against the plain error on the shadow estimator entering the constraints. The constraints otherwise are kept the same, points are averaged over multiple runs with different errors on the shadow constraints.