[Paper Review] A Variational Quantum Algorithm for Preparing Quantum Gibbs States

Anirban Chowdhury, Guang Hao Low+1 more|arXiv (Cornell University)|Jan 31, 2020

Quantum Computing Algorithms and ArchitectureComputer Science24 references53 citations

TL;DR

The paper introduces a variational algorithm to prepare quantum Gibbs states by minimizing free energy, using a Fourier-series-based entropy estimate and Hamiltonian energy evaluation on near-term quantum devices.

ABSTRACT

Preparation of Gibbs distributions is an important task for quantum computation. It is a necessary first step in some types of quantum simulations and further is essential for quantum algorithms such as quantum Boltzmann training. Despite this, most methods for preparing thermal states are impractical to implement on near-term quantum computers because of the memory overheads required. Here we present a variational approach to preparing Gibbs states that is based on minimizing the free energy of a quantum system. The key insight that makes this practical is the use of Fourier series approximations to the logarithm that allows the entropy component of the free-energy to be estimated through a sequence of simpler measurements that can be combined together using classical post processing. We further show that this approach is efficient for generating high-temperature Gibbs states, within constant error, if the initial guess for the variational parameters for the programmable quantum circuit are sufficiently close to a global optima. Finally, we examine the procedure numerically and show the viability of our approach for five-qubit Hamiltonians using Trotterized adiabatic state preparation as an ansatz.

Motivation & Objective

• Motivate the need to prepare Gibbs (thermal) states for quantum simulation and machine learning tasks.
• Develop a practical variational approach that reduces memory overhead for near-term quantum hardware.
• Provide a method to estimate von Neumann entropy efficiently via a Fourier-series representation and density-matrix exponential techniques.

Proposed method

• Formulate Gibbs state preparation as free-energy minimization over parameterized quantum circuits.
• Estimate von Neumann entropy with a Fourier-series approximation using traces of ρ cos(ρ t) and ρ sin(ρ t) terms implemented via density-matrix exponentiation.
• Evaluate energy by decomposing the Hamiltonian into a sum of unitaries and using a Hadamard-test-like circuit with amplitude estimation.
• Analyze gradient-based optimization of the free energy and provide complexity bounds for gradient estimation.
• Demonstrate feasibility with a five-qubit Hamiltonian using Trotterized adiabatic state preparation as an Ansatz.

Experimental results

Research questions

• RQ1 Can a variational approach efficiently prepare Gibbs states suitable for quantum simulations and Boltzmann-machine-like learning on near-term devices?
• RQ2 How can the von Neumann entropy term of the free energy be estimated efficiently without full state tomography?
• RQ3 What are the resource implications (queries, gates) of entropy and energy estimation in this framework?
• RQ4 Is the proposed method viable for high-temperature Gibbs states and what are the convergence guarantees under reasonable assumptions?
• RQ5 How does the method perform with realistic ansatzes such as Trotterized adiabatic state preparation on small quantum systems?

Key findings

• Entropy can be approximated as a finite Fourier series of traces of unitary evolutions with respect to ρ.
• The entropy estimation cost scales as Õ(1/(ε p_min^2)) in terms of U_ρ queries, making it feasible for near-term devices under favorable p_min.
• Energy can be estimated efficiently when H is a linear combination of unitaries, with query complexity Õ(∥α∥_1/ε).
• The overall framework yields a hybrid quantum-classical variational algorithm that targets a purification of the thermal state for a fixed Hamiltonian.
• Numerical examination shows viability for five-qubit Hamiltonians using a Trotterized adiabatic-state-preparation Ansatz.
• Complexity bounds for gradient evaluation indicate feasible optimization under smoothness and (pseudo) strong convexity assumptions.