Obtaining Accurate Ground-State Properties on Near-term Quantum Devices

TL;DR

This work tackles the central challenge of obtaining accurate ground-state properties on noisy, near-term quantum devices by combining a quantum variational approach with classical postprocessing that enforces $N$-representability via variational 2-RDM (v2RDM) constraints. A norm-based distance to experimental 2-RDM data is used as a physically meaningful, hardware-informed trust region, whose size $\\Delta$ is determined through a Clifford-based calibration that accounts for both hardware noise and ansatz expressivity, with $\\Delta = k\\Delta_{ref}$ and $k=2$. The method yields near-FCI energies for small molecules (e.g., $H_2$, $LiH$, $H_4$) and accurate ultrafast electron diffraction intensities for $C_6H_8$ on noisy devices, demonstrating improved robustness against noise and enhanced representational power beyond the original ansatz. These results indicate a scalable route toward quantum advantage in chemistry and materials science on NISQ hardware, and the framework is adaptable to additional constraints and observables beyond energetics.

Abstract

Accurate ground-state calculations on noisy quantum computers are fundamentally limited by restricted ansatz expressivity and unavoidable hardware errors. We introduce a hybrid-quantum classical framework that simultaneously addresses these challenges. Our method systematically purifies noisy two electron reduced density matrices from quantum devices by enforcing N-representability conditions through efficient semidefinite programming, guided by a norm-based distance constraint to the experimental data. To implement this constraint, we develop a hardware efficient calibration protocol based on Clifford circuits. We demonstrate near full configuration interaction accuracy for ground-state energies of H2, LiH, and H4, and compute precise scattering intensities for C6H8 on noisy hardware. This approach surpasses conventional methods by simultaneously overcoming both ansatz limitations and hardware noise, establishing a scalable route to quantum advantage and marking a critical step toward reliable simulations of complex molecularnsystems on noisy devices.

Figures (4)

• Figure 1: Schematic of the Noise-Aware RDM correction Framework. (a) A conceptual diagram of RDMs in matrix space. A noisy VQE experiment yields an unphysical RDM (hollow circle), which lies at a theoretical (but unknown) distance ${d}$ from the ideal ground state RDM (red star). Our method (purple square) corrects this by finding an RDM that is DQG-feasible (within the orange space) and is constrained to a trust radius $R$ around the noisy result. As shown by the optimization path (paw prints), this process yields a purified RDM that successfully lies within the physical N-representable space (yellow). This approach is contrasted with other points like the nearest DQG-feasible RDM (green triangle) and the unconstrained v2RDM global minimum (blue hexagon). The concentric circles with the hollow circle as the center represent the search results at different search radius $R$. The correction is most effective when the radius $R$ is comparable to the error distance ${d}$. (b) The procedure to estimate the unknown error distance ${d}$ and thus determine an appropriate trust radius $R$. A VQE circuit is first transformed into a "Nearly Clifford Circuit", which is then executed on both the noisy quantum hardware ($D_{\text{quantum}}$) and an ideal classical simulator ($D_{\text{classical}}$). The resulting discrepancy, ${\Delta_{\text{ref}}} = \|D_{\text{quantum}} - D_{\text{classical}}\|$, provides a measurable proxy for the true RDM error. This practical estimate of ${d}$ then guides the selection of the trust radius $R$ for the correction process on the left, ensuring an optimal correction.
• Figure 2: Ground state energy of the LiH molecule at an interatomic distance of $d = 2.8$ Å. The energy calculated with the our VQE+v2RDM method shown in Eq. (\ref{['eq:SDP']}) (solid blue line with markers) is plotted as a function of the penalty coefficient $\Delta$ defined in Eq. (\ref{['eq:SDP']}). For comparison, horizontal lines indicate the benchmark FCI energy (solid black), the noisy result of energy from a VQE-UCCSD circuit (dashed orange) with error rates of $p_1=0.001$ for single-qubit gates and $p_2=0.01$ for two-qubit gates, and the result from the v2RDM method with DQG conditions(dashed green). The energy of VQE+v2RDM theory converges to the FCI value for large $\Delta$. Zoom in view: A magnified view of the converged energy region.
• Figure 3: Potential energy curves and error with FCI for the dissociation of (a) the H$_2$ (b) the LiH and (c) a linear H$_4$ molecule for UCCSD ansatz. In all plots, results from noisy VQE calculations (orange pentagon) are compared against the exact FCI benchmark (solid black line), ideal noiseless VQE simulations (yellow triangle), and our VQE+v2RDM method (blue rhombus). The noisy simulations assume a depolarizing channel with single- and two-qubit gate error probabilities of $p_1=0.001$ and $p_2=0.01$, respectively. The dashed line at $1.6 \times 10^{-3}$ Ha marks the chemical accuracy threshold.
• Figure 4: Absolute error of the simulated UED intensity for the C$_6$H$_8$ molecule (a) eauilibrium position (b) non-eauilibrium position. The left panel shows the results at the equilibrium position , and the right panel shows the results at a non-equilibrium position. The orange dashed line with square markers ('Noisy VQE') indicates the error of the spectrum calculated on a noisy quantum computer relative to the exact, noiseless solution. The blue solid line with circle markers ('Noisy VQE+vRDM') represents the error after applying our proposed VQE+vRDM correction method.

Theorems & Definitions (6)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• proof
• proof