# Faster Stochastic First-Order Method for Maximum-Likelihood Quantum State Tomography

arXiv: https://arxiv.org/abs/2211.12880 · HTML: https://mlsanity.com/paper/2211.12880/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 34 references that are in the map.
- First posted on arXiv 23 Nov 2022.

## Builds on (most cited first, 11 of 11)

- [A Refined Laser Method and Faster Matrix Multiplication](https://mlsanity.com/paper/2010.05846/) (2020)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Online Learning of Quantum States](https://mlsanity.com/paper/1802.09025/) (2018)
- [Pushing the Efficiency-Regret Pareto Frontier for Online Learning of Portfolios and Quantum States](https://mlsanity.com/paper/2202.02765/) (2022)
- [Analysis of the Frank-Wolfe Method for Convex Composite Optimization involving a Logarithmically-Homogeneous Barrier](https://mlsanity.com/paper/2010.08999/) (2020)
- [Composite Self-Concordant Minimization](https://mlsanity.com/paper/1308.2867/) (2013)
- [Bandit Principal Component Analysis](https://mlsanity.com/paper/1902.03035/) (2019)
- [Self-Concordant Analysis of Frank-Wolfe Algorithms](https://mlsanity.com/paper/2002.04320/) (2020)
- [Convergence of the Exponentiated Gradient Method with Armijo Line Search](https://mlsanity.com/paper/1712.08480/) (2017)
- [Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States](https://mlsanity.com/paper/2210.00997/) (2022)
- [Minimizing Quantum Renyi Divergences via Mirror Descent with Polyak Step Size](https://mlsanity.com/paper/2109.06054/) (2021)

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

- [Fast Minimization of Expected Logarithmic Loss via Stochastic Dual Averaging](https://mlsanity.com/paper/2311.02557/) (2023)

## Abstract (arXiv)

In maximum-likelihood quantum state tomography, both the sample size and dimension grow exponentially with the number of qubits. It is therefore desirable to develop a stochastic first-order method, just like stochastic gradient descent for modern machine learning, to compute the maximum-likelihood estimate. To this end, we propose an algorithm called stochastic mirror descent with the Burg entropy. Its expected optimization error vanishes at a O ( √ ( 1 / t ) d log t  ) rate, where d and t denote the dimension and number of iterations, respectively. Its per-iteration time complexity is O ( d3 ), independent of the sample size. To the best of our knowledge, this is currently the computationally fastest stochastic first-order method for maximum-likelihood quantum state tomography.