Constrained Shadow Tomography for Molecular Simulation on Quantum Devices

TL;DR

This work tackles scalable quantum-state reconstruction by targeting the $2$-RDM rather than the full state. It introduces constrained shadow tomography as a bi-objective semidefinite program that enforces $N$-representability via $2$-positivity constraints and uses nuclear-norm regularization to suppress measurement noise from shadow data. The method yields physically valid $2$-RDMs, improves energy predictions and spectrum fidelity under realistic noise, and scales better than unconstrained approaches. Validation on simulated benchmarks and IBM hardware shows robust performance with shallower circuits, enabling robust quantum-classical workflows for molecular simulations.

Abstract

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 $N$-representability constraints and nuclear-norm regularization into the optimization, the method builds an $N$-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.

Figures (5)

• Figure 1: (a) Absolute energy error as a function of system size under a fixed total shot budget. (b) Absolute Frobenius norm of the 2-RDM error (relative to 2-RDM from FCI) plotted on a logarithmic scale. Shot budgets compared are: 16,000 unitaries × 1 shot vs. 16 unitaries × 1,000 shots for H$_4$; 36,000 unitaries × 1 shot vs. 36 unitaries × 1,000 shots for H$_6$; 160,000 unitaries × 1 shot vs. 160 unitaries × 1,000 shots for H$8$; and 300,000 unitaries × 1 shot vs. 300 unitaries × 1,000 shots for H$_{10}$. Each data point for the Fermionic Classical Shadows (FCS) and shadow variational 2-RDM (sv2RDM) methods represents the mean of 20 independent runs (10 for H$_{10}$). Error bars denote 95% confidence intervals (approximately 2$\sigma$) computed from these measurements. All calculations are performed using the minimal basis set.
• Figure 2: Lowest 2-RDM eigenvalue as a function of total shot budget for the shadow variational 2-RDM (sv2RDM) method with full 2-positivity (DQG) constraints and for the Fermionic Classical Shadows (FCS) approach applied to H$_4$ in the minimal basis set.
• Figure 3: Comparison of absolute error matrices for the H$_4$ system in the minimal basis set obtained using the shadow variational 2-RDM (sv2RDM) (a) and Fermionic Classical Shadows (FCS) (b) methods. Each matrix element represents the absolute deviation of the reconstructed 2-RDM from the FCI reference. The color intensity denotes the error magnitude, with brighter regions corresponding to larger deviations from the reference values.
• Figure 4: Potential energy curve of N$_2$ computed using the cc-pVDZ basis set and a [10,8] active space with complete active space configuration interaction (CASCI), shadow variational 2-RDM (sv2RDM), and Fermionic Classical Shadows (FCS) methods. For sv2RDM and FCS, the total shot budgets used are 100 unitaries × 1,000 shots and 100,000 unitaries × 1 shot, respectively.
• Figure 5: Potential energy curve for the H$_4$ rectangle–square–rectangle transition in the minimal basis set, computed using Density Functional Theory (DFT) with the B3LYP functional as a classical baseline, the classically computed Local Unitary Coupled Cluster with Jastrow (LUCJ) ansatz, and the shadow variational 2-RDM (sv2RDM) method computed using measurements collected on the ibm_fez quantum computer.