[Paper Review] Variational quantum Gibbs state preparation with a truncated Taylor series

Youle Wang, Guangxi Li+1 more|arXiv (Cornell University)|May 18, 2020

Quantum Computing Algorithms and Architecture76 references77 citations

TL;DR

This paper proposes a variational quantum algorithm for preparing quantum Gibbs states using a truncated Taylor series expansion of the von Neumann entropy, enabling efficient free energy minimization on near-term quantum hardware. By using shallow parameterized quantum circuits with only one additional qubit, the method achieves fidelity above 95% for Ising and XY spin chains, reaching 99% fidelity at inverse temperatures β > 2, demonstrating practical feasibility for NISQ devices.

ABSTRACT

The preparation of quantum Gibbs state is an essential part of quantum computation and has wide-ranging applications in various areas, including quantum simulation, quantum optimization, and quantum machine learning. In this paper, we propose variational hybrid quantum-classical algorithms for quantum Gibbs state preparation. We first utilize a truncated Taylor series to evaluate the free energy and choose the truncated free energy as the loss function. Our protocol then trains the parameterized quantum circuits to learn the desired quantum Gibbs state. Notably, this algorithm can be implemented on near-term quantum computers equipped with parameterized quantum circuits. By performing numerical experiments, we show that shallow parameterized circuits with only one additional qubit can be trained to prepare the Ising chain and spin chain Gibbs states with a fidelity higher than 95%. In particular, for the Ising chain model, we find that a simplified circuit ansatz with only one parameter and one additional qubit can be trained to realize a 99% fidelity in Gibbs state preparation at inverse temperatures larger than 2.

Motivation & Objective

• To develop a resource-efficient method for preparing quantum Gibbs states on near-term quantum computers.
• To overcome the challenge of estimating von Neumann entropy in variational quantum algorithms.
• To reduce quantum resource requirements such as qubit count, gate count, and circuit depth for Gibbs state preparation.
• To enable high-fidelity Gibbs state preparation using shallow parameterized quantum circuits with minimal ancilla qubits.
• To demonstrate practical feasibility of the approach on realistic many-body Hamiltonians like the Ising and XY spin chains.

Proposed method

• The method uses a truncated Taylor series expansion of the von Neumann entropy up to order K to approximate the free energy, which serves as the loss function.
• The truncated free energy loss function is a linear combination of system energy and higher-order state overlaps (tr(ρ²), tr(ρ³)), which are efficiently estimable via quantum circuits.
• Parameterized quantum circuits (PQCs) are trained via a hybrid quantum-classical optimization loop to minimize the truncated free energy.
• The algorithm employs quantum gadgets to estimate energy and state overlaps, enabling implementation on NISQ devices without requiring complex subroutines like quantum phase estimation.
• Analytical gradients of the loss function are derived using parameter shift rules, enabling efficient classical optimization.
• A simplified ansatz with only one parameter and one ancilla qubit is shown to achieve high fidelity for the Ising chain model.

Experimental results

Research questions

• RQ1 Can a truncated Taylor series of the von Neumann entropy enable effective free energy minimization for Gibbs state preparation on near-term quantum hardware?
• RQ2 What is the minimum circuit depth and ancilla resource requirement to achieve high-fidelity Gibbs state preparation?
• RQ3 How does the truncation order K of the Taylor series affect the convergence and fidelity of the prepared Gibbs state?
• RQ4 Can a simple parameterized quantum circuit with only one parameter and one additional qubit achieve high-fidelity Gibbs state preparation for the Ising model?
• RQ5 Does the method remain effective at both low and high temperatures for realistic many-body Hamiltonians?

Key findings

• The method achieves a fidelity higher than 95% for Gibbs state preparation of the Ising chain and XY spin-1/2 chain using shallow parameterized quantum circuits with only one additional qubit.
• For the Ising chain model, a simplified ansatz with one parameter and one ancilla qubit achieves a fidelity exceeding 99% at inverse temperatures β > 2.
• The fidelity between the output state and the target Gibbs state is lower bounded by 1 / sqrt(1 + (N/2 - 1) * e^(-βΔ)), where N is the system size and Δ is the spectral gap.
• Theoretical analysis confirms that fidelity increases with higher truncation order K, and order K=2 suffices for high-fidelity preparation in numerical experiments.
• The approach is effective at both low and high temperatures, demonstrating robustness across different thermal regimes.
• The method avoids costly quantum subroutines like quantum phase estimation, making it suitable for near-term quantum devices.