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.
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.
Enhanced Shadow Tomography of Molecular Excited States from Enforcing N-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 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 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.
Correlated Purification for Restoring N-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 N-electron states -- known as N-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.
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 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 2RDMs. We compute excited-state energies and 2RDMs of the 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 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.
Many applications of quantum simulation require one to prepare and then characterize quantum states by efficiently estimating k-body reduced density matrices (k-RDMs), from which observables of interest may be obtained. For instance, the fermionic 2-RDM contains the energy, charge density, and energy gradients of an electronic system, while the qubit 2-RDM contains the spatial correlation functions of magnetic systems. Naive estimation of such RDMs requires repeated state preparations for each matrix element, which makes for prohibitively large computation times. However, commuting matrix elements may be measured simultaneously, allowing for a significant cost reduction. In this work, we design schemes for such a parallelization with near-optimal complexity in the system size N. We first describe a scheme to sample all elements of a qubit k-RDM using only O(3klogk-1N) unique measurement circuits, an exponential improvement over prior art. We then describe a scheme for sampling all elements of the fermionic 2-RDM using only O(N2) unique measurement circuits, each of which requires only a local O(N)-depth measurement circuit. We prove a lower bound of ω(ϵ-2Nk) on the number of state preparations, Clifford circuits, and measurement in the computational basis required to estimate all elements of a fermionic k-RDM, making our scheme for sampling the fermionic 2-RDM asymptotically optimal. We finally construct circuits to sample the expectation value of a linear combination of ω anticommuting two-body fermionic operators with only O(ω) gates on a linear array. These circuits allows for sampling any linear combination of fermionic 2-RDM elements in O(N4/ω) time, with a significantly lower measurement circuit complexity than prior art. Our results improve the viability of near-term quantum simulation of molecules and strongly correlated material systems.
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 two-particle reduced density matrix. In this paper, we build upon previous research by choosing an improved estimator within classical shadow tomography and rephrasing the optimization 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 stand-alone classical shadow protocol and quantify the potential savings in numerical studies.