# Fast quantum state reconstruction via accelerated non-convex programming

arXiv: https://arxiv.org/abs/2104.07006 · HTML: https://mlsanity.com/paper/2104.07006/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 107 references that are in the map.
- First posted on arXiv 14 Apr 2021.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Low-rank Solutions of Linear Matrix Equations via Procrustes Flow](https://mlsanity.com/paper/1507.03566/) (2015)
- [Guaranteed Matrix Completion via Non-convex Factorization](https://mlsanity.com/paper/1411.8003/) (2014)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees](https://mlsanity.com/paper/1509.03025/) (2015)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Quantum State Tomography with Conditional Generative Adversarial Networks](https://mlsanity.com/paper/2008.03240/) (2020)
- [Dropping Convexity for Faster Semi-definite Optimization](https://mlsanity.com/paper/1509.03917/) (2015)
- [Non-square matrix sensing without spurious local minima via the Burer-Monteiro approach](https://mlsanity.com/paper/1609.03240/) (2016)
- [A Convergent Gradient Descent Algorithm for Rank Minimization and Semidefinite Programming from Random Linear Measurements](https://mlsanity.com/paper/1506.06081/) (2015)
- [Provable Inductive Matrix Completion](https://mlsanity.com/paper/1306.0626/) (2013)
- [A geometric alternative to Nesterov's accelerated gradient descent](https://mlsanity.com/paper/1506.08187/) (2015)
- [Accelerated Stochastic Power Iteration](https://mlsanity.com/paper/1707.02670/) (2017)
- [Operator splitting for a homogeneous embedding of the linear complementarity problem](https://mlsanity.com/paper/2004.02177/) (2020)
- [Accelerated Methods for Non-Convex Optimization](https://mlsanity.com/paper/1611.00756/) (2016)
- [Classification and reconstruction of optical quantum states with deep neural networks](https://mlsanity.com/paper/2012.02185/) (2020)
- [Nonconvex Matrix Factorization from Rank-One Measurements](https://mlsanity.com/paper/1802.06286/) (2018)
- [IHT dies hard: Provable accelerated Iterative Hard Thresholding](https://mlsanity.com/paper/1712.09379/) (2017)
- [Finding Approximate Local Minima Faster than Gradient Descent](https://mlsanity.com/paper/1611.01146/) (2016)
- [Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems](https://mlsanity.com/paper/1606.01316/) (2016)
- [Provable quantum state tomography via non-convex methods](https://mlsanity.com/paper/1711.02524/) (2017)
- [Estimating expectation values using approximate quantum states](https://mlsanity.com/paper/2011.04754/) (2020)

## Built on by (most cited first, 5 of 5)

- [Tensor Train Quantum State Tomography using Compressed Sensing](https://mlsanity.com/paper/2506.23560/) (2025)
- [One Rank at a Time: Cascading Error Dynamics in Sequential Learning](https://mlsanity.com/paper/2505.22602/) (2025)
- [Local Stochastic Factored Gradient Descent for Distributed Quantum State Tomography](https://mlsanity.com/paper/2203.11579/) (2022)
- [AdaPaD: Adaptive Parallel Deflation for PEFT with Self-Correcting Rank Discovery](https://mlsanity.com/paper/2605.10741/) (2026)
- [A Block Tensor Train Burer-Monteiro Framework for Low-Rank Quantum State Tomography](https://mlsanity.com/paper/2609.09457/) (2026)

## Abstract (arXiv)

We propose a new quantum state reconstruction method that combines ideas from compressed sensing, non-convex optimization, and acceleration methods. The algorithm, called Momentum-Inspired Factored Gradient Descent (textttMiFGD), extends the applicability of quantum tomography for larger systems. Despite being a non-convex method, textttMiFGD converges provably close to the true density matrix at an accelerated linear rate, in the absence of experimental and statistical noise, and under common assumptions. With this manuscript, we present the method, prove its convergence property and provide Frobenius norm bound guarantees with respect to the true density matrix. From a practical point of view, we benchmark the algorithm performance with respect to other existing methods, in both synthetic and real experiments performed on an IBM's quantum processing unit. We find that the proposed algorithm performs orders of magnitude faster than state of the art approaches, with the same or better accuracy. In both synthetic and real experiments, we observed accurate and robust reconstruction, despite experimental and statistical noise in the tomographic data. Finally, we provide a ready-to-use code for state tomography of multi-qubit systems.