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 $\rho$ and gate
sequence $U$, our cost function quantifies how far $ U\rho U^{\dagger}$ is from
being diagonal. We introduce novel short-depth quantum circuits to quantify our
cost. Minimizing this cost returns a gate sequence that approximately
diagonalizes $\rho$. One can then read out approximations of the largest
eigenvalues, and the associated eigenvectors, of $\rho$. 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.

本研究提出了一种基于变分混合量子 - 经典算法的量子状态对角化算法，该算法可以通过量子计算评估门序列的成本（比经典成本评估更快），并且经典计算机使用此信息来调整门序列的参数，主要适用于凝聚态物理和机器学习领域中的状态对角化问题，它可以通过最小化成本函数来返回一个近似对角化的门序列，从而获得量子态的最大特征值和相关的特征向量的近似值。