Extracting eigenvalues and eigenvectors of exponentially large matrices will be an important application of near-term quantum computers. The variational quantum eigensolver (VQE) treats the case when the matrix is a Hamiltonian. Here, we address the case when the matrix is a density matrix ρ. We introduce the variational quantum state eigensolver (VQSE), which is analogous to VQE in that it variationally learns the largest eigenvalues of ρ as well as a gate sequence V that prepares the corresponding eigenvectors. VQSE exploits the connection between diagonalization and majorization to define a cost function \(C={{{\rm{Tr}}}}(\tilde{\rho }H)\) where H is a non-degenerate Hamiltonian. Due to Schur-concavity, C is minimized when \(\tilde{\rho }=V\rho {V}^{{\dagger} }\) is diagonal in the eigenbasis of H. VQSE only requires a single copy of ρ (only n qubits) per iteration of the VQSE algorithm, making it amenable for near-term implementation. We heuristically demonstrate two applications of VQSE: (1) Principal component analysis, and (2) Error mitigation.

提取指数大矩阵的特征值和特征向量将是近期量子计算机的重要应用。变分量子特征求解器 (VQE) 处理矩阵是哈密顿量的情况。在这里，我们解决矩阵是密度矩阵ρ的情况。我们引入了变分量子态特征求解器 (VQSE)，它类似于 VQE，因为它变分地学习ρ的最大特征值以及准备相应特征向量的门序列V。VQSE 利用对角化和主要化之间的联系来定义成本函数\(C={{{\rm{Tr}}}}(\tilde{\rho }H)\)，其中H是非退化哈密顿量。由于 Schur 凹度，当\(\tilde{\rho }=V\rho {V}^{{\dagger} }\)在H的特征基中是对角线时， C被最小化。VQSE 在 VQSE 算法的每次迭代中只需要一个ρ副本（仅n 个量子位），使其适合近期实施。我们启发式地展示了 VQSE 的两个应用：（1）主成分分析，和（2）错误缓解。