# Maximum-Likelihood Quantum State Tomography by Soft-Bayes

arXiv: https://arxiv.org/abs/2012.15498 · HTML: https://mlsanity.com/paper/2012.15498/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Dec 2020.

## 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)

Quantum state tomography (QST), the task of estimating an unknown quantum state given measurement outcomes, is essential to building reliable quantum computing devices. Whereas computing the maximum-likelihood (ML) estimate corresponds to solving a finite-sum convex optimization problem, the objective function is not smooth nor Lipschitz, so most existing convex optimization methods lack sample complexity guarantees; moreover, both the sample size and dimension grow exponentially with the number of qubits in a QST experiment, so a desired algorithm should be highly scalable with respect to the dimension and sample size, just like stochastic gradient descent. In this paper, we propose a stochastic first-order algorithm that computes an varepsilon-approximate ML estimate in O( ( D log D ) / varepsilon ^ 2 ) iterations with O( D3 ) per-iteration time complexity, where D denotes the dimension of the unknown quantum state and varepsilon denotes the optimization error. Our algorithm is an extension of Soft-Bayes to the quantum setup.