Variational quantum state diagonalization

TLDR

Variational hybrid quantum‑classical algorithms, which use a quantum computer to evaluate a cost function and a classical computer to optimize gate parameters, are promising for near‑term quantum computing and enable state diagonalization, a technique useful in condensed matter physics and machine learning. The authors propose a variational algorithm for diagonalizing quantum states. The algorithm defines a cost function measuring the deviation of UρU† from diagonal, evaluates it with short‑depth circuits, and optimizes U to minimize this cost, yielding an approximate diagonalization of ρ. The method allows extraction of approximate largest eigenvalues and eigenvectors of ρ, and was demonstrated on Rigetti hardware for one‑qubit states and on a simulator for the entanglement spectrum of the Heisenberg model ground state.

Abstract

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 ρ and gate sequence U , our cost function quantifies how far $$U\rho U^\dagger$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mi>ρ</mml:mi> <mml:msup> <mml:mrow> <mml:mi>U</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>†</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> 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.