# Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry

**Authors:** Faris M. Sbahi, Antonio J. Martinez, Sahil Patel, Dmitri Saberi, Jae Hyeon Yoo, Geoffrey Roeder, Guillaume Verdon  
**arXiv:** [2206.04663](https://arxiv.org/abs/2206.04663)  
**Date:** 2022-06-09  
**Summary page:** https://www.summarizepaper.com/en/paper/2206.04663v1/

## Key points

- Challenges in quantum Hamiltonian learning and quantum Gibbs sampling in low-temperature regime
- Proposal of a generalization of quantum natural gradient descent to parameterized mixed states
- Introduction of Quantum-Probabilistic Mirror Descent as a robust first-order approximating algorithm
- Demonstration of data sample efficiency using information geometry and quantum metrology
- Extension of classical Fisher efficiency to a variational quantum algorithm for the first time
- Flexibility in model choice, including spectrally-decomposed models like Quantum Hamiltonian-Based Models
- Derivation of the first-order algorithm using a novel quantum generalization of classical mirror descent duality
- Use of Bogoliubov-Kubo-Mori metric for both proposed approaches
- Comparison with existing baselines on the task of quantum Gibbs sampling for the transverse field Ising model
- Proposal of an initialization strategy leveraging geometric locality for modeling sequences of states from quantum-stochastic processes
- Empirical demonstrations showing effectiveness for real and imaginary time evolution and potential broader applications

## Summary

The paper titled "Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry" addresses the challenges faced by algorithms for quantum Hamiltonian learning and quantum Gibbs sampling in the low-temperature regime. These tasks often suffer from intractabilities, such as poor sample or time complexity. To overcome these issues, the authors propose a generalization of quantum natural gradient descent to parameterized mixed states and introduce a robust first-order approximating algorithm called Quantum-Probabilistic Mirror Descent. The authors demonstrate data sample efficiency for these dual tasks using tools from information geometry and quantum metrology. This result extends the seminal classical Fisher efficiency to a variational quantum algorithm for the first time. The proposed approaches also allow for flexibility in model choice, including spectrally-decomposed models like Quantum Hamiltonian-Based Models, which can potentially circumvent intractable time complexities. The first-order algorithm is derived using a novel quantum generalization of the classical mirror descent duality. Both results rely on a special choice of metric known as the Bogoliubov-Kubo-Mori metric. To evaluate the performance of their proposed algorithms, the authors compare them to existing baselines on the task of quantum Gibbs sampling for the transverse field Ising model. Additionally, they propose an initialization strategy that leverages geometric locality for modeling sequences of states arising from quantum-stochastic processes. Empirical demonstrations show its effectiveness for both real and imaginary time evolution while defining a broader class of potential applications. Overall, this paper presents novel techniques that address intractabilities in quantum Hamiltonian learning and quantum Gibbs sampling tasks through generalizations of existing methods and efficient initialization strategies. The results provide valuable insights into variational generative modeling of quantum many-body systems and have implications for physics and chemistry research.