The N-representability problem consists in determining whether, for a given p-body matrix, there exists at least one N-body density matrix from which the p-body matrix can be obtained by contraction, that is, if the given matrix is a p-body reduced density matrix (p-RDM). The knowledge of all necessary and sufficient conditions for a p-body matrix to be N-representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the p-body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the N-representability conditions grows exponentially with system size, and hence, the procedure quickly becomes intractable for practical applications. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N-representability conditions. The algorithm consists of applying to an initial N-body density matrix a sequence of unitary evolution operators constructed from a stochastic process that successively approaches the reduced state of the density matrix on a p-body subsystem, represented by a p-RDM, to a target p-body matrix, potentially a p-RDM. The generators of the evolution operators follow the well-known adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented by using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide whether a given p-body matrix is N-representable, establishing a criterion to determine its quality and correcting it. We apply the proposed hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, from the reduced Bardeen-Cooper-Schrieffer model with constant pairing, and from the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices toward different targets.

中文翻译：

---

通过幺正进化和随机采样确定降密度矩阵的 n 表示性。


N 可表示性问题在于确定对于给定的 p 体矩阵，是否至少存在一个 N 体密度矩阵，p体矩阵可以通过收缩获得，也就是说，如果给定的矩阵是 p 体约化密度矩阵 （p-RDM）。了解 p 体矩阵可 N 表示的所有必要和充分条件后，可以对 p 体密度矩阵的多体哈密顿期望值进行约束最小化，从而确定其确切的基态。但是，完成 N 表示条件的约束数量随着系统大小呈指数级增长，因此，该过程很快就会变得难以处理。这项工作引入了一种混合量子随机算法来有效地替换 N 表示性条件。该算法包括将一系列幺正进化算子应用于初始 N 体密度矩阵，该算子由随机过程构建，该过程连续接近 p 体子系统上密度矩阵的约化状态，由 p-RDM 表示，到目标 p 体矩阵，可能是 p-RDM。进化算子的生成器遵循众所周知的自适应导数组装伪 Trotter 方法 （ADAPT），而随机分量则通过使用模拟退火过程来实现。生成的算法独立于任何基础哈密顿量，可用于确定给定的 p 体矩阵是否为 N 表示，从而建立确定其质量的标准并对其进行校正。 我们将提出的混合 ADAPT 算法应用于量子化学电子哈密顿量、常数配对的简化 Bardeen-Cooper-Schrieffer 模型以及 Heisenberg XXZ 自旋模型中的所谓约化密度矩阵。在所有情况下，所提出的方法都与 1-RDM 和 2-RDM 的预期行为一样，将初始矩阵演向不同的目标。