Variational hybrid quantum-classical algorithms are promising candidates for near-term implementation on quantum computers. In these algorithms, a quantum computer evaluates the cost of a gate sequence (with speedup over classical cost evaluation), and a classical computer uses this information to adjust the parameters of the gate sequence. Here we present such an algorithm for quantum state diagonalization. State diagonalization has applications in condensed matter physics (e.g., entanglement spectroscopy) as well as in machine learning (e.g., principal component analysis). For a quantum state ρ and gate sequence U, our cost function quantifies how far \(U\rho U^\dagger\) is from being diagonal. We introduce short-depth quantum circuits to quantify our cost. Minimizing this cost returns a gate sequence that approximately diagonalizes ρ. One can then read out approximations of the largest eigenvalues, and the associated eigenvectors, of ρ. As a proof-of-principle, we implement our algorithm on Rigetti’s quantum computer to diagonalize one-qubit states and on a simulator to find the entanglement spectrum of the Heisenberg model ground state.

变分混合量子经典算法是在量子计算机上近期实现的有希望的候选者。在这些算法中，量子计算机评估门控序列的成本（相对于经典成本评估而言速度有所提高），而经典计算机则使用此信息来调整门控序列的参数。在这里，我们提出了一种量子态对角化的算法。状态对角化在凝聚态物理（例如纠缠光谱）和机器学习（例如主成分分析）中都有应用。对于量子状态ρ和门序列U，我们的成本函数量化\（U \ rho U ^ \ dagger \）来自对角线。我们引入短深度量子电路来量化我们的成本。使该成本最小化将返回一个近似对角化ρ的门序列。然后，可以读出ρ的最大特征值和相关特征向量的近似值。作为原理上的证明，我们在Rigetti的量子计算机上实现对角化一个qubit态的算法，并在模拟器上实现以找到Heisenberg模型基态的纠缠谱。