# Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm

arXiv: https://arxiv.org/abs/1910.02071 · HTML: https://mlsanity.com/paper/1910.02071/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2019.

## Built on by (most cited first, 27 of 27)

- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Subtleties in the trainability of quantum machine learning models](https://mlsanity.com/paper/2110.14753/) (2021)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Learning many-body Hamiltonians with Heisenberg-limited scaling](https://mlsanity.com/paper/2210.03030/) (2022)
- [Variational inference with a quantum computer](https://mlsanity.com/paper/2103.06720/) (2021)
- [Ab-initio study of interacting fermions at finite temperature with neural canonical transformation](https://mlsanity.com/paper/2105.08644/) (2021)
- [Practical Black Box Hamiltonian Learning](https://mlsanity.com/paper/2206.15464/) (2022)
- [A semi-agnostic ansatz with variable structure for quantum machine learning](https://mlsanity.com/paper/2103.06712/) (2021)
- [Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?](https://mlsanity.com/paper/2109.01051/) (2021)
- [Training-efficient density quantum machine learning](https://mlsanity.com/paper/2405.20237/) (2024)
- [Learnability of the output distributions of local quantum circuits](https://mlsanity.com/paper/2110.05517/) (2021)
- [Quantum-probabilistic Hamiltonian learning for generative modelling & anomaly detection](https://mlsanity.com/paper/2211.03803/) (2022)
- [Quantum Neural Estimation of Entropies](https://mlsanity.com/paper/2307.01171/) (2023)
- [Physics-Informed Learning Using Hamiltonian Neural Networks with Output Error Noise Models](https://mlsanity.com/paper/2305.01338/) (2023)
- [F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits](https://mlsanity.com/paper/2110.04253/) (2021)
- [Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry](https://mlsanity.com/paper/2206.04663/) (2022)
- [On the similarity of bandwidth-tuned quantum kernels and classical kernels](https://mlsanity.com/paper/2503.05602/) (2025)
- [Meta-learning of Gibbs states for many-body Hamiltonians with applications to Quantum Boltzmann Machines](https://mlsanity.com/paper/2507.16373/) (2025)
- [Covariance matrix preparation for quantum principal component analysis](https://mlsanity.com/paper/2204.03495/) (2022)
- [Improving thermal state preparation of Sachdev-Ye-Kitaev model with reinforcement learning on quantum hardware](https://mlsanity.com/paper/2501.11454/) (2025)
- [The quantum low-rank approximation problem](https://mlsanity.com/paper/2203.00811/) (2022)
- [MEMO-QCD: Quantum Density Estimation through Memetic Optimisation for Quantum Circuit Design](https://mlsanity.com/paper/2406.08591/) (2024)
- [Statistical inference for quantum singular models](https://mlsanity.com/paper/2411.16396/) (2024)
- [Trainability Beyond Linearity in Variational Quantum Objectives](https://mlsanity.com/paper/2604.18846/) (2026)
- [An End-to-End Hybrid Quantum-Classical Sampling Workflow for Discrete Markov Random Fields: A Reproducible Case Study](https://mlsanity.com/paper/2607.09893/) (2026)
- [Degeneracy Counting Quantum Algorithm using Decoherence](https://mlsanity.com/paper/2608.14941/) (2026)

## Abstract (arXiv)

We introduce a new class of generative quantum-neural-network-based models called Quantum Hamiltonian-Based Models (QHBMs). In doing so, we establish a paradigmatic approach for quantum-probabilistic hybrid variational learning, where we efficiently decompose the tasks of learning classical and quantum correlations in a way which maximizes the utility of both classical and quantum processors. In addition, we introduce the Variational Quantum Thermalizer (VQT) for generating the thermal state of a given Hamiltonian and target temperature, a task for which QHBMs are naturally well-suited. The VQT can be seen as a generalization of the Variational Quantum Eigensolver (VQE) to thermal states: we show that the VQT converges to the VQE in the zero temperature limit. We provide numerical results demonstrating the efficacy of these techniques in illustrative examples. We use QHBMs and the VQT on Heisenberg spin systems, we apply QHBMs to learn entanglement Hamiltonians and compression codes in simulated free Bosonic systems, and finally we use the VQT to prepare thermal Fermionic Gaussian states for quantum simulation.