Variational Thermal State Preparation on Digital Quantum Processors Assisted by Matrix Product States

TL;DR

The paper introduces an MPS-assisted variational framework to prepare quantum Gibbs states on digital quantum processors by classically evaluating the Helmholtz free energy $F(\rho)=E(\rho)-\beta^{-1}S(\rho)$ for a purified, variationally generated state. It benchmarks two ansatz families—the thermofield-double purification (TFDA) and a hardware-efficient ansatz (HEA)—and finds HEA is better suited for near-term devices, enabling scalable simulations of 1D and 2D lattice thermodynamics and a hardware demonstration on IBM hardware with substantial error mitigation. Large-scale noiseless simulations reach 1D systems up to 30 spins and 2D systems up to $6\times 6$ (up to 42 qubits estimated), accurately capturing energy, susceptibility, specific heat, and two-point correlations, especially at low temperatures. Hardware experiments on a 30-spin 1D TFIM with error mitigation show practical viability, reducing relative errors by more than a factor of two for key observables and illustrating the approach’s potential for studying finite-temperature quantum phases on near-term devices.

Abstract

The preparation of quantum Gibbs states at finite temperatures is a cornerstone of quantum computation, enabling applications in quantum simulation of many-body systems, machine learning via quantum Boltzmann machines, and optimization through thermal sampling techniques. In this work, we introduce a variational framework that leverages matrix product states for the efficient classical evaluation of the Helmholtz free energy, combining scalable entanglement entropy computation with a hardware efficient ansatz to accurately approximate thermal states in one- and two-dimensional systems. We conduct extensive benchmarking on key observables, including energy density, susceptibility, specific heat, and two-point correlations, comparing against exact analytical results for 1D systems and quantum Monte Carlo simulations for 2D lattices across various temperatures and ansatz configurations. Our large-scale numerical simulations demonstrate the capability to prepare high-quality Gibbs states for 1D lattice models with up to 30 sites and 2D systems with up to 6x6 sites, using up to 42 qubits. Finally, we demonstrate the framework's practical viability on a 156-qubit IBM Heron processor by preparing the approximate Gibbs state of a 30-site transverse-field Ising model. Leveraging a combination of error mitigation techniques, we reduce the relative errors in energy and susceptibility measurements by over 50% compared to unmitigated results.

Figures (7)

• Figure 1: Schematic of the MPS-assisted variational Gibbs state preparation algorithm, showcasing two ansatzes benchmarked in this work: (a) the TFDA for the 1D transverse-field Ising model, and (b) the HEA. The green- and red-shaded ovals highlight the ancilla and physical qubits, respectively. In the TFDA, the numbers of these two types of qubits are always equal, while in HEAs, the number of ancilla qubits can be adjusted independently. Moreover, the structure of the TFDA is model-dependent, while the HEA is more generic and can be applied to different models. In both circuits, the orange gates represent parameterized gates, including single-qubit rotations such as $R_X$ and $R_Y$, and two-qubit entangling gates such as $R_{ZZ}$ and $R_{XX}$. The gray dashed boxes represent the part of circuit that can be repeated for multiple layers. The parameterized quantum circuit prepares a pure state $\ket{\psi(\boldsymbol{\theta})}$ on an enlarged Hilbert space containing the physical and ancilla qubits. The MPS representation of $\ket{\psi(\boldsymbol{\theta})}$ is built classically, from which the energy and von Neumann entropy of the resulting mixed state $\rho(\boldsymbol{\theta})$ are computed to obtain the free energy $F(\boldsymbol{\theta})$. The parameters $\boldsymbol{\theta}$ are iteratively optimized using a classical optimizer in the standard VQE fashion. The optimal parameters $\boldsymbol{\theta}^*$ are finally used to prepare the approximate Gibbs state $\tilde{\rho}_\text{Gibbs} = \rho(\boldsymbol{\theta}^*)$ on the quantum device, where various observables of interest are measured to characterize the prepared state.
• Figure 2: Performance comparison of the TFDA and HEA on the 1D (a) TFIM and (b) XXZ model, each of 4 and 6 spins at various inverse temperatures $\beta$. The left panels show the infidelity of the prepared Gibbs state with respect to the exact one, where the TFDA results are shown as gray squares and the HEA results as gray circles. The right panels show the absolute difference between the estimated and exact thermal energies, where the TFDA results are represented as blue squares and the HEA results as blue circles. Different shades of the same color represent different system sizes, with lighter colors for 4 spins and darker colors for 6 spins. For both ansatzes, the number of ancilla qubits used is equal to $N$, where $N$ is the number of physical spins in the system. The number of layers is $L = N/2$. Ten optimization runs with different random initial parameters are performed for each $\beta$ and the best result is shown.
• Figure 3: Thermal energy estimates of the variationally prepared Gibbs states for 1D TFIM with (a) 20 and (b) 30 spins, and 2D TFIM with (c) $4\times 4$ and (d) $6\times 6$ spins, at various inverse temperatures $\beta = 0, 1, \cdots, 6$. Each row corresponds to a different number of ancilla qubits, $N_a = 4, 6, 8$. Curves with markers in different shades of blue represent different numbers of layers $L$ in the HEA, with $L = 3, 5, 7$. The black plus markers (+) denote the exact thermal energies of the Gibbs states for the 1D systems, while the black crosses ($\times$) with error bars are the results from Monte Carlo simulations for the 2D systems. Each data point is the best result from the 40 (20) optimization runs with different initial parameters for the 1D (2D) systems.
• Figure 4: Thermal estimates of magnetic susceptibility of the variationally prepared Gibbs states for 1D TFIM with (a) 20 and (b) 30 spins, and 2D TFIM with (c) $4\times 4$ and (d) $6\times 6$ spins, at inverse temperatures $\beta = 0, 1, \cdots, 6$. Each row corresponds to a different number of ancilla qubits, $N_a = 4, 6, 8$. Curves with markers in different shades of green represent different numbers of layers $L$ in the HEA, with $L = 3, 5, 7$. The black crosses ($\times$) with error bars denote the results from quantum Monte Carlo simulations. Each data point is the best result from the 40 (20) optimization runs with different initial parameters for the 1D (2D) systems.
• Figure 5: Thermal estimates of specific heat of the variationally prepared Gibbs states for (a) 1D TFIM with 10 spins and (b) 2D TFIM with $3\times 3$ spins, at inverse temperatures $\beta = 0, 1, \cdots, 6$. Each row corresponds to a different number of ancilla qubits, $N_a = 4, 6, 8$. Curves with markers in different shades of orange represent different numbers of layers $L$ in the HEA, with $L = 3, 5, 7$. The black plus markers (+) denote the exact results obtained based on the exact Gibbs states. Each data point is the best result from the 20 optimization runs with different initial parameters.