Quantum State Tomography with Conditional Generative Adversarial Networks

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Quantum State Tomography with Conditional Generative Adversarial Networks

Shahnawaz Ahmed, Carlos Sánchez Muñoz, Franco Nori, and Anton Frisk Kockum

Phys. Rev. Lett. 127, 140502 – Published 27 September 2021

Abstract

Quantum state tomography (QST) is a challenging task in intermediate-scale quantum devices. Here, we apply conditional generative adversarial networks (CGANs) to QST. In the CGAN framework, two dueling neural networks, a generator and a discriminator, learn multimodal models from data. We augment a CGAN with custom neural-network layers that enable conversion of output from any standard neural network into a physical density matrix. To reconstruct the density matrix, the generator and discriminator networks train each other on data using standard gradient-based methods. We demonstrate that our QST-CGAN reconstructs optical quantum states with high fidelity, using orders of magnitude fewer iterative steps, and less data, than both accelerated projected-gradient-based and iterative maximum-likelihood estimation. We also show that the QST-CGAN can reconstruct a quantum state in a single evaluation of the generator network if it has been pretrained on similar quantum states.

Received 14 December 2020
Revised 21 May 2021
Accepted 10 June 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.140502

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Machine learning Quantum optics Quantum tomography

Artificial neural networks

Tomography

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Shahnawaz Ahmed ^1,*, Carlos Sánchez Muñoz ², Franco Nori ^3,4, and Anton Frisk Kockum ^1,†

¹Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96 Gothenburg, Sweden
²Departamento de Fisica Teorica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autonoma de Madrid, Madrid 28049, Spain
³Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
⁴Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*shahnawaz.ahmed95@gmail.com
^†anton.frisk.kockum@chalmers.se

Classification and reconstruction of optical quantum states with deep neural networks

Shahnawaz Ahmed, Carlos Sánchez Muñoz, Franco Nori, and Anton Frisk Kockum

Phys. Rev. Research 3, 033278 (2021)

Article Text

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Issue

Vol. 127, Iss. 14 — 1 October 2021

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Images

Figure 1
Illustration of the CGAN architecture for QST. Data $d$ sampled from measurements of a set of measurement operators ${O_{i}}$ on a quantum state is fed into both the generator $G$ and the discriminator $D$ . The other input to $D$ is the generated statistics from $G$ . The next to last layer of $G$ outputs a physical density matrix and the last layer computes measurement statistics using this density matrix. The discriminator compares the measurement data and the generated data for each measurement operator and outputs a probability that they match.
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Figure 2
Observables for an optical quantum state, the “cat state” $| α ⟩ + | - α ⟩$ (up to a normalization), with coherent amplitude $α = 2$ . (a) The Wigner function. (b) The Husimi $Q$ function. The stars mark specific $β$ used to sample the data used as input to the QST-CGAN.
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Figure 3
QST-CGAN performance. The data are the Husimi $Q$ function of the cat state in Fig. 2. (a) Reconstruction fidelity $F (ρ, ρ^{'}) = tr (\sqrt{\sqrt{ρ} ρ^{'} \sqrt{ρ}})^{2}$ as a function of iterations for the QST-CGAN (red), iMLE (blue), and APG-MLE (dashed black). We use 1024 displacements $β$ in a $32 \times 32$ grid. The weights of the QST-CGAN and the starting density matrix of the iMLE are randomly initialized. The APG-MLE runs 13 iterations of conjugate-gradient line search from the maximally mixed state before switching to APG. The solid lines show the mean $F$ for 100 runs; the shaded areas show one standard deviation from the mean. (b) Average $F$ as a function of the number of $β$ . For each number, 10 sets of displacements are randomly selected from within a disk with $| β | \leq 5$ for the state in Fig. 2. We show the average $F$ reached after 1000 iterations for QST-CGAN and iMLE and 10 000 iterations for APG-MLE.
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Figure 4
(a) Reconstruction of a Wigner-negative state by a QST-CGAN from (b) noisy experimental data. Inset: the target state. The reconstruction uses 4281 data points of the Wigner function measured for $β$ inside the dashed circle. The data outside the circle, e.g., the Wigner-negative region in the top left, are not as reliable due to measurement calibration problems at higher photon numbers. We also attempt reconstruction with a subset of the data points inside the circle and find that $\sim 600$ data points are enough to achieve a fidelity $\sim 0.9$ with the full reconstruction.
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Figure 5
Single-shot reconstructions of 200 cat states (cf. Fig. 2, $| α | \in [1, 3]$ , up to six coherent states in superposition), using a pretrained QST-CGAN. (a) Fidelity distribution of the reconstructions after training on a $32 \times 32$ grid of data points. (b) Average fidelity (solid line) within one standard deviation (shaded region) after further iterations.
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