Constrained Shadow Tomography Reconstructs 2-RDM from Noisy Data for Quantum Molecular Simulation

Determining the complete quantum state of a molecular system remains a significant challenge, hindering progress in fields like materials science and drug discovery. Irma Avdic, Yuchen Wang, and Michael Rose, alongside colleagues at the University of Chicago and IBM Quantum, now present a new method for reconstructing these complex states using a technique called constrained shadow tomography. Their approach tackles the problem of limited and noisy data, common in quantum simulations, by combining randomised measurements with mathematical constraints that ensure the reconstructed state remains physically realistic. The team demonstrates that this bi-objective optimisation strategy significantly improves both the accuracy and scalability of molecular simulations, offering a robust pathway towards simulating complex fermionic systems on current and future quantum devices.

Efficient Quantum State Reconstruction with Constraints

Quantum state tomography aims to fully characterise an unknown quantum state, but the number of measurements required typically increases exponentially with the number of qubits. This work introduces constrained shadow tomography, a novel approach to efficiently reconstruct quantum states for molecular simulation. The method leverages constraints derived from the antisymmetry principle of fermionic wavefunctions, significantly reducing the number of measurements needed to achieve a target accuracy. Specifically, the team develops a measurement scheme tailored to fermionic systems, exploiting the known structure of molecular wavefunctions.

This allows them to discard measurement outcomes that violate the antisymmetry constraint, effectively focusing the reconstruction process on physically plausible states. The researchers demonstrate that, for systems with up to 16 spin orbitals, constrained shadow tomography requires only a modest number of measurements, scaling polynomially with system size. The resulting reconstructed states accurately predict molecular properties, such as energies and dipole moments, with high fidelity, and prove robust to experimental noise. The team validates the approach using both simulated data and experiments on a superconducting quantum processor, demonstrating its feasibility for near-term quantum devices. This advancement represents a significant step towards enabling efficient quantum state preparation and characterisation for complex molecular simulations, paving the way for more accurate and reliable quantum chemistry calculations.

Constrained Shadow Tomography for Density Matrix Reconstruction

Quantum state tomography is essential for detailed characterisation of quantum systems, but its exponential measurement and computational demands limit scalability. This work introduces 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 optimisation, the method builds an N-representable density matrix that minimises both the error with respect to the shadow measurements and the nuclear-norm of the 2-RDM. This approach effectively balances data fidelity with the desire for a physically plausible and compact representation of the quantum state, offering a pathway towards scalable quantum state reconstruction.

Reduced Density Matrices for Quantum Chemistry

Solving the Schrödinger equation for many-electron systems is computationally challenging, with traditional methods scaling exponentially with system size. Reduced density matrices (RDMs), specifically the 2-RDM, offer a simplification by containing only the information about two-particle correlations, allowing for the calculation of properties without needing the full many-body wavefunction. A key challenge is ensuring that the RDM corresponds to a physically valid many-body wavefunction, a condition known as N-representability. This research explores variational methods to find the lowest-energy RDM that satisfies these N-representability conditions, employing semidefinite programming (SDP) and combining randomised measurements with constraints to reconstruct RDMs.

The core novel method is constrained shadow tomography, a quantum state tomography technique that enforces N-representability during the reconstruction process. The problem is formulated as a multi-objective optimisation, balancing energy minimisation and constraint satisfaction. Techniques from machine learning and data reconstruction are used to improve the efficiency and accuracy of the RDM calculations.

Constrained Shadow Tomography Reconstructs Quantum States

This research introduces a new approach to reconstructing quantum states, specifically the two-particle reduced density matrix, using constrained shadow tomography. The method combines randomised measurements with mathematical optimisation, integrating physical constraints and a regularisation process to build accurate representations of quantum systems even with limited or noisy data. By directly embedding principles of physical consistency into the reconstruction process, the team achieves improved accuracy and robustness compared to standard methods. The results demonstrate that this constrained shadow tomography significantly enhances the reliability of quantum state reconstruction for systems like hydrogen chains and the nitrogen molecule, effectively suppressing errors arising from measurement limitations and noise. This advancement provides a unified framework connecting statistical inference, optimisation, and quantum chemistry, establishing a practical foundation for characterising many-body quantum simulations. Future work will likely focus on refining the algorithm by incorporating additional system symmetries or adaptive weighting strategies to enhance reconstruction fidelity, positioning constrained shadow tomography as a powerful tool for robustly characterising complex quantum systems and validating quantum simulations of correlated fermionic systems.