Markovian Matrix Product Density Operators : Efficient computation of global entropy

TL;DR

This paper introduces the Markovian matrix product density operator, enabling efficient classical computation of global entropy for certain quantum states by verifying quantum Markov chain conditions, and explores the complexity of finite-temperature free energy calculations.

Contribution

It presents a new subclass of matrix product density operators that allows efficient entropy computation and analyzes the complexity of finite-temperature free energy problems for quantum spin chains.

Findings

Efficient entropy computation for Markovian matrix product states.

Verification of quantum Markov chain conditions scales polynomially.

Finite-temperature free energy problem is in NP under certain conditions.

Abstract

We introduce the Markovian matrix product density operator, which is a special subclass of the matrix product density operator. We show that the von Neumann entropy of such ansatz can be computed efficiently on a classical computer. This is possible because one can efficiently certify that the global state forms an approximate quantum Markov chain by verifying a set of inequalities. Each of these inequalities can be verified in time that scales polynomially with the bond dimension and the local Hilbert space dimension. The total number of inequalities scale linearly with the system size. We use this fact to study the complexity of computing the minimum free energy of local Hamiltonians at finite temperature. To this end, we introduce the free energy problem as a generalization of the local Hamiltonian problem, and study its complexity for a class of Hamiltonians that describe quantum spin chains. The corresponding free energy problem at finite temperature is in NP if the Gibbs state of such Hamiltonian forms an approximate quantum Markov chain with an error that decays exponentially with the width of the conditioning subsystem.