Variational Quantum State Diagonalization

TL;DR

This paper introduces a variational hybrid quantum-classical algorithm for quantum state diagonalization, enabling extraction of eigenvalues and eigenvectors, with applications in physics and machine learning, demonstrated on real quantum hardware and simulators.

Contribution

It presents a novel variational algorithm with short-depth circuits for quantum state diagonalization, suitable for near-term quantum computers.

Findings

Successfully implemented on Rigetti's quantum computer for one-qubit states.

Demonstrated on a simulator to find the entanglement spectrum of the Heisenberg model.

Introduced a cost function to quantify how close a state is to being diagonalized.

Abstract

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.