# Guaranteed recovery of quantum processes from few measurements

## TL;DR

This paper introduces compressed sensing methods for efficiently reconstructing low-rank quantum channels using simple measurement models without ancillary systems, providing theoretical guarantees and numerical validation.

## Contribution

It offers the first analysis of direct process tomography measurement models with recovery guarantees for low-rank channels, avoiding ancillary systems and complex operations.

## Key findings

- Recovery guarantees for three reconstruction algorithms.
- Uniform guarantees for all channels with a single measurement setup.
- Robustness to noise and low-rank violations.

## Abstract

Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum channels of low Kraus rank. Our main contribution is the analysis of a natural measurement model for this task: We assume that data is obtained by sending pure states into the channel and measuring expectation values on the output. Neither ancillary systems nor coherent operations across multiple channel uses are required. Most previous results on compressed process reconstruction reduce the problem to quantum state tomography on the channel's Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of measurements, physical implementations of such schemes would typically involve ancillary systems. A priori, it is unclear whether a measurement model tailored directly to quantum process tomography might require more measurements. We establish that this is not the case. Technically, we prove recovery guarantees for three different reconstruction algorithms. The reconstructions are based on a trace, diamond, and $\ell_2$-norm minimization, respectively. Our recovery guarantees are uniform in the sense that with one random choice of measurement settings all quantum channels can be recovered equally well. Moreover, stability against arbitrary measurement noise and robustness against violations of the low-rank assumption is guaranteed. Numerical studies demonstrate the feasibility of the approach.