Determining the ensemble N-representability of Reduced Density Matrices

The N-representability problem for reduced density matrices remains a fundamental challenge in electronic structure theory. Following our previous work that employs a unitary-evolution algorithm based on an adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to probe pure-state N-representability of reduced density matrices [J. Chem. Theory Comput. 2024, 20, 9968], in this work we propose a practical framework for determining the ensemble N-representability of a p-body matrix. This is accomplished using a purification strategy consisting of embedding an ensemble state into a pure state defined on an extended Hilbert space, such that the reduced density matrices of the purified state reproduce those of the original ensemble. By iteratively applying variational unitaries to an initial purified state, the proposed algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target p-body matrix, which serves as a measure of the N-representability of the target. This methodology facilitates both error correction of defective ensemble reduced density matrices, and quantum-state reconstruction on a quantum computer, offering a route for density-matrix refinement. We validate the algorithm with numerical simulations on systems of two, three, and four electrons in both, simple models as well as molecular systems at finite temperature, demonstrating its robustness.

💡 Research Summary

The paper tackles the long‑standing N‑representability problem for reduced density matrices (RDMs) by introducing a quantum‑algorithmic framework that can assess and correct ensemble (mixed‑state) p‑body RDMs. Building on the authors’ earlier work on pure‑state N‑representability using an Adaptive Derivative‑Assembled Pseudo‑Trotter Variational Quantum Algorithm (ADAPT‑VQA), they extend the method to ensembles through a purification strategy. Any mixed state ρ = Σ_i p_i |ϕ_i⟩⟨ϕ_i| can be embedded in an enlarged Hilbert space H_s ⊗ H_b as a pure state |Ψ_sb⟩ = Σ_i √p_i |ϕ_i⟩⊗|b_i⟩, where {|b_i⟩} is an orthonormal basis of the auxiliary system. Because the p‑RDM of the purified state exactly matches the original ensemble p‑RDM, the same variational unitary‑evolution technique used for pure states can be applied.

The algorithm proceeds iteratively. Starting from an initial purified state |Ψ₀⟩ (often a simple product of a reference pure state with itself), a pool of anti‑Hermitian fermionic excitation operators {Ô} is constructed. These include one‑particle (Ô_ik = a_i† a_k – a_k† a_i) and two‑particle (Ô_ijkl = a_i† a_j† a_k a_l – a_k† a_l† a_i a_j) generators, mapped to Pauli strings via the Jordan‑Wigner transformation. In each iteration k, for every operator Ô the algorithm optimizes a rotation angle θ to minimize the Hilbert‑Schmidt distance D_k = ‖pρ_k – pρ_target‖₂² between the p‑RDM of the evolved state |Ψ_k⟩ = e^{θÔ}|Ψ_{k‑1}⟩ and the target p‑RDM. The operator and angle that achieve the greatest reduction are selected, the state is updated, and the process repeats until the change in D falls below a predefined threshold δ or a maximum number of steps is reached. The final distance D_min serves as a quantitative measure of ensemble N‑representability: D_min ≈ 0 indicates that the target matrix is physically realizable as an ensemble p‑RDM; a non‑zero D_min yields the closest physically admissible RDM, thereby providing a systematic error‑correction tool.

Implementation details: the authors coded the procedure in Python, employing OpenFermion for fermion‑to‑qubit mapping and PySCF for integral generation. Simulations were performed on a noiseless quantum‑device emulator. Test systems included two model Hamiltonians with four electrons in three (4e,3o) and four (4e,4o) spatial orbitals, as well as H₂ and H₃ molecules in the STO‑3G basis at finite temperature. For each case, both one‑ and two‑body RDMs were examined. The authors benchmarked their results against known analytical N‑representability conditions: Klyachko’s generalized Pauli constraints for pure‑state 1‑RDMs and Coleman’s ensemble constraints (eigenvalues between 0 and 1, correct trace).

Key findings: (i) In the (4e,3o) model, both pure‑state and ensemble ADAPT‑VQA converged to D_min ≈ 0 for pure (w = 0) and mixed (w = 0.5) target 1‑RDMs, reflecting that Klyachko’s constraints are satisfied in both scenarios. (ii) In the (4e,4o) model, pure‑state ADAPT‑VQA yielded a sizable residual distance (≈1.25×10⁻¹) for the mixed target (w = 0.5), indicating violation of pure‑state conditions, whereas the ensemble version achieved D_min ≈ 2.45×10⁻⁹, confirming ensemble N‑representability. (iii) Similar behavior was observed for 2‑RDMs: only the pure‑state algorithm succeeded for w = 0 (pure states), while the ensemble algorithm succeeded for both w = 0 and w = 0.5, demonstrating its ability to detect and correct ensemble‑type violations.

The study demonstrates that (a) ensemble N‑representability can be decided on a quantum computer using a modest extension of existing pure‑state variational circuits, (b) the method provides a practical means to “purify” defective RDMs by projecting them onto the nearest physical ensemble, and (c) the approach integrates naturally with thermo‑field theories for finite‑temperature electronic structure, suggesting applicability to a broad class of quantum‑chemical problems. By delivering a modular, gradient‑based algorithm that operates directly on qubits, the work paves the way for embedding N‑representability checks and RDM refinement into larger quantum‑simulation workflows, potentially improving the reliability of quantum chemistry calculations on near‑term quantum hardware.