# Partially Unitary Learning

arXiv: https://arxiv.org/abs/2405.10263 · HTML: https://mlsanity.com/paper/2405.10263/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 18 references that are in the map.
- First posted on arXiv 16 May 2024.

## Builds on (most cited first, 1 of 1)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)

## Built on by (most cited first, 3 of 3)

- [Superstate Quantum Mechanics](https://mlsanity.com/paper/2502.00037/) (2025)
- [Operator Learning for Schr"odinger Equation: Unitarity, Error Bounds, and Time Generalization](https://mlsanity.com/paper/2505.18288/) (2025)
- [Semidefinite Programming for Quantum Channel Learning](https://mlsanity.com/paper/2601.12502/) (2026)

## Abstract (arXiv)

The problem of an optimal mapping between Hilbert spaces IN of left|psirightrangle and OUT of left|φrightrangle based on a set of wavefunction measurements (within a phase) psil → φl, l=1dots M, is formulated as an optimization problem maximizing the total fidelity suml=1M ω(l) left|langleφl|U|psilrangleright|2 subject to probability preservation constraints on U (partial unitarity). The constructed operator U can be considered as an IN to OUT quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension dim(OUT) × dim(IN) transforming operators as AOUT=U AIN Udagger. An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.