# Gradient-descent quantum process tomography by learning Kraus operators

arXiv: https://arxiv.org/abs/2208.00812 · HTML: https://mlsanity.com/paper/2208.00812/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 108 references that are in the map.
- First posted on arXiv 1 Aug 2022.

## Builds on (most cited first, 13 of 13)

- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [Efficient Riemannian Optimization on the Stiefel Manifold via the Cayley Transform](https://mlsanity.com/paper/2002.01113/) (2020)
- [Quantum State Tomography with Conditional Generative Adversarial Networks](https://mlsanity.com/paper/2008.03240/) (2020)
- [Experimental neural network enhanced quantum tomography](https://mlsanity.com/paper/1904.05902/) (2019)
- [Classification and reconstruction of optical quantum states with deep neural networks](https://mlsanity.com/paper/2012.02185/) (2020)
- [A Framework of Constraint Preserving Update Schemes for Optimization on Stiefel Manifold](https://mlsanity.com/paper/1301.0172/) (2013)
- [Machine learning assisted quantum state estimation](https://mlsanity.com/paper/2003.03441/) (2020)
- [Efficient Quantum State Sample Tomography with Basis-dependent Neural-networks](https://mlsanity.com/paper/2009.07601/) (2020)
- [Expressiveness and Learning of Hidden Quantum Markov Models](https://mlsanity.com/paper/1912.02098/) (2019)
- [Partial Trace Regression and Low-Rank Kraus Decomposition](https://mlsanity.com/paper/2007.00935/) (2020)

## Built on by (most cited first, 6 of 6)

- [Artificial intelligence for representing and characterizing quantum systems](https://mlsanity.com/paper/2509.04923/) (2025)
- [Quantum computing and artificial intelligence: status and perspectives](https://mlsanity.com/paper/2505.23860/) (2025)
- [Fast Quantum Process Tomography via Riemannian Gradient Descent](https://mlsanity.com/paper/2404.18840/) (2024)
- [Superstate Quantum Mechanics](https://mlsanity.com/paper/2502.00037/) (2025)
- [Active Learning with Variational Quantum Circuits for Quantum Process Tomography](https://mlsanity.com/paper/2412.20925/) (2024)
- [Kraus Constrained Sequence Learning For Quantum Trajectories from Continuous Measurement](https://mlsanity.com/paper/2603.05468/) (2026)

## Abstract (arXiv)

We perform quantum process tomography (QPT) for both discrete- and continuous-variable quantum systems by learning a process representation using Kraus operators. The Kraus form ensures that the reconstructed process is completely positive. To make the process trace-preserving, we use a constrained gradient-descent (GD) approach on the so-called Stiefel manifold during optimization to obtain the Kraus operators. Our ansatz uses a few Kraus operators to avoid direct estimation of large process matrices, e.g., the Choi matrix, for low-rank quantum processes. The GD-QPT matches the performance of both compressed-sensing (CS) and projected least-squares (PLS) QPT in benchmarks with two-qubit random processes, but shines by combining the best features of these two methods. Similar to CS (but unlike PLS), GD-QPT can reconstruct a process from just a small number of random measurements, and similar to PLS (but unlike CS) it also works for larger system sizes, up to at least five qubits. We envisage that the data-driven approach of GD-QPT can become a practical tool that greatly reduces the cost and computational effort for QPT in intermediate-scale quantum systems.