Meta-learning of Gibbs states for many-body Hamiltonians with applications to Quantum Boltzmann Machines

TL;DR

This work tackles the challenge of efficiently preparing quantum Gibbs states for parametrized many-body Hamiltonians on NISQ devices. It introduces two meta-learning frameworks, Meta-VQT and NN Meta-VQT, to learn a single, generalizable circuit that outputs Gibbs states $\rho(\vec{h})$ across unseen Hamiltonian parameters by encoding $\vec{h}$ into the circuit and optimizing a finite-temperature Gibbs free energy objective. The methods demonstrate accurate Gibbs-state generation for the Transverse Field Ising Model up to 8 qubits, robust finite-temperature behavior in a Kitaev ring, and substantial speedups in Quantum Boltzmann Machine training (up to 30×) compared with VarQITE-based approaches, with NN Meta-VQT providing superior scalability in larger systems. These results indicate a scalable, practical route to memory-efficient Gibbs-state preparation and accelerated quantum-machine-learning workflows on near-term devices.

Abstract

The preparation of quantum Gibbs states is a fundamental challenge in quantum computing, essential for applications ranging from modeling open quantum systems to quantum machine learning. Building on the Meta-Variational Quantum Eigensolver framework proposed by Cervera-Lierta et al.(2021) and a problem driven ansatz design, we introduce two meta-learning algorithms: Meta-Variational Quantum Thermalizer (Meta-VQT) and Neural Network Meta-VQT (NN-Meta VQT) for efficient thermal state preparation of parametrized Hamiltonians on Noisy Intermediate-Scale Quantum (NISQ) devices. Meta-VQT utilizes a fully quantum ansatz, while NN Meta-VQT integrates a quantum classical hybrid architecture. Both leverage collective optimization over training sets to generalize Gibbs state preparation to unseen parameters. We validate our methods on upto 8-qubit Transverse Field Ising Model and the 2-qubit Heisenberg model with all field terms, demonstrating efficient thermal state generation beyond training data. For larger systems, we show that our meta-learned parameters when combined with appropriately designed ansatz serve as warm start initializations, significantly outperforming random initializations in the optimization tasks. Furthermore, a 3- qubit Kitaev ring example showcases our algorithm's effectiveness across finite-temperature crossover regimes. Finally, we apply our algorithms to train a Quantum Boltzmann Machine (QBM) on a 2-qubit Heisenberg model with all field terms, achieving enhanced training efficiency, improved Gibbs state accuracy, and a 30-fold runtime speedup over existing techniques such as variational quantum imaginary time (VarQITE)-based QBM highlighting the scalability and practicality of meta-algorithm-based QBMs.

Figures (14)

• Figure 1: Overview of meta-algorithm for Gibbs state preparation :(a) Depicts the core task addressed by our meta-algorithm: preparing thermal states for a parametrized many-body spin Hamiltonian at finite temperature. Shown are illustrative profiles of the Gibbs free energy and the purity of the corresponding Gibbs state as functions of a Hamiltonian parameter. The parameter space is split into training (green circles) and test (blue star) sets. (b) Illustrates the training process, where parameters from the training set are encoded into a variational ansatz composed of a hardware-efficient and a Hamiltonian variational ansatz, separated by a linear arrangement of CNOT gates. The global cost, evaluated across all training points, is minimized using a classical optimizer. (c) Shows inference, where test parameters are input into the trained ansatz, initialized with the optimized parameters from (b), to generate Gibbs states. These are then used to reconstruct the Gibbs free energy profile.
• Figure 2: Workflow of VQT, Meta-VQT and NN Meta-VQT : The workflow of VQT, where the PQC is trained for approximating the Gibbs state corresponding to a particular parameter value, $\vec{h}_{1}$, of the Hamiltonian is shown in (a). For a different parameter, this workflow needs to be run again. The workflow of Meta-VQT and NN Meta-VQT is shown in (b). Unlike (a) here a set of Hamiltonian parameters $\vec{h}_{train}$ (Step 1), are encoded in the PQC (Step 3, pink colored box) or classical neural network (Step 3, green colored box), and a global loss function is evaluated (Step 7) that does collective optimization over these set of parameters. After training, the PQC can take any value in $\Vec{h}_{test} \notin \Vec{h}_{train}$ as well as those that lies within the range of $\vec{h}_{train}$ as input and accurately prepare Gibbs state corresponding to these parameters without the need for further training. In both (a) and (b), the blue and orange boxes represent modules that are run on classical and quantum computers respectively.
• Figure 3: Schematic workflow of meta-learning algorithms : In the workflow presented here, different types of training data denoted by training data 1,..,training data N are associated with distinct learning algorithms and models, each defined by a separate loss function. The meta-learning approach optimizes a combined loss function based on $P_{i,j}$ and shared feature representations and configurations, $m_{j}~\text{and}~\lambda_{i}$, enabling the resulting meta-learner model to perform effectively on unseen new data that was not included in the training dataset. This schematic workflow has been adapted from Bahri et al.meta_learning_workflow.
• Figure 4: Meta-VQT ansatz : The Hamiltonian parameters, $\Vec{h}$, are encoded in the angles of the single qubit parametrized gates through a linear transformation. This along with the linear implementation of CNOTs forms the encoding layer. The output state from the encoding layer is further processed by the parametrized HVA. The gates in HVA are chosen according to the Hamiltonian whose Gibbs states needs to be prepared. An example HVA for a transverse field Ising model is shown, where $XX = e^{-i\theta\sigma_{i}^{x}\sigma_{i+1}^{x}}$. $l$ represents the number of encoding or processing layers. After the final processing layer, measurement on the system qubits yield a density matrix, $\tilde{\rho} (\vec{\omega},\vec{\phi},\vec{\theta})$, that is further optimized to approximate the true Gibbs state of the system. The number of ancillas are taken to be same as the number of system qubits.
• Figure 5: NN Meta-VQT ansatz : In NN Meta-VQT, the neural network encodes the Hamiltonian parameters, $\vec{h}$, and provides the circuit parameters, $\vec{\theta}$, as output. The quantum circuit is constructed out of both hardware-efficient and HVA as shown above. Similar to Meta-VQT, after this final processing layer, measurement on the system qubits yield a density matrix, $\tilde{\rho} (\vec{\theta})$, that is further optimized to approximate the true Gibbs state of the system.