# Local Stochastic Factored Gradient Descent for Distributed Quantum State Tomography

arXiv: https://arxiv.org/abs/2203.11579 · HTML: https://mlsanity.com/paper/2203.11579/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 33 references that are in the map.
- First posted on arXiv 22 Mar 2022.

## Builds on (most cited first, 19 of 19)

- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Local SGD Converges Fast and Communicates Little](https://mlsanity.com/paper/1805.09767/) (2018)
- [Sparsity in Deep Learning: Pruning and growth for efficient inference and training in neural networks](https://mlsanity.com/paper/2102.00554/) (2021)
- [Deep Gradient Compression: Reducing the Communication Bandwidth for Distributed Training](https://mlsanity.com/paper/1712.01887/) (2017)
- [Tighter Theory for Local SGD on Identical and Heterogeneous Data](https://mlsanity.com/paper/1909.04746/) (2019)
- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [Is Local SGD Better than Minibatch SGD?](https://mlsanity.com/paper/2002.07839/) (2020)
- [QSGD: Communication-Efficient SGD via Gradient Quantization and Encoding](https://mlsanity.com/paper/1610.02132/) (2016)
- [Local SGD with Periodic Averaging: Tighter Analysis and Adaptive Synchronization](https://mlsanity.com/paper/1910.13598/) (2019)
- [Cooperative SGD: A unified Framework for the Design and Analysis of Communication-Efficient SGD Algorithms](https://mlsanity.com/paper/1808.07576/) (2018)
- [Low-rank Solutions of Linear Matrix Equations via Procrustes Flow](https://mlsanity.com/paper/1507.03566/) (2015)
- [SGD and Hogwild! Convergence Without the Bounded Gradients Assumption](https://mlsanity.com/paper/1802.03801/) (2018)
- [On the convergence properties of a K-step averaging stochastic gradient descent algorithm for nonconvex optimization](https://mlsanity.com/paper/1708.01012/) (2017)
- [Communication-Efficient Algorithms for Decentralized and Stochastic Optimization](https://mlsanity.com/paper/1701.03961/) (2017)
- [Dropping Convexity for Faster Semi-definite Optimization](https://mlsanity.com/paper/1509.03917/) (2015)
- [Finding Low-Rank Solutions via Non-Convex Matrix Factorization, Efficiently and Provably](https://mlsanity.com/paper/1606.03168/) (2016)
- [Fast quantum state reconstruction via accelerated non-convex programming](https://mlsanity.com/paper/2104.07006/) (2021)

## Built on by (most cited first, 1 of 1)

- [Adaptive Federated Learning with Auto-Tuned Clients](https://mlsanity.com/paper/2306.11201/) (2023)

## Abstract (arXiv)

We propose a distributed Quantum State Tomography (QST) protocol, named Local Stochastic Factored Gradient Descent (Local SFGD), to learn the low-rank factor of a density matrix over a set of local machines. QST is the canonical procedure to characterize the state of a quantum system, which we formulate as a stochastic nonconvex smooth optimization problem. Physically, the estimation of a low-rank density matrix helps characterizing the amount of noise introduced by quantum computation. Theoretically, we prove the local convergence of Local SFGD for a general class of restricted strongly convex/smooth loss functions, i.e., Local SFGD converges locally to a small neighborhood of the global optimum at a linear rate with a constant step size, while it locally converges exactly at a sub-linear rate with diminishing step sizes. With a proper initialization, local convergence results imply global convergence. We validate our theoretical findings with numerical simulations of QST on the Greenberger-Horne-Zeilinger (GHZ) state.