Learning algebraic models of quantum entanglement
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Learning algebraic models of quantum entanglement Hamza Jaffali1
· Luke Oeding2
Received: 5 October 2019 / Accepted: 22 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We review supervised learning and deep neural network design for learning membership on algebraic varieties. We demonstrate that these trained artificial neural networks can predict the entanglement type for quantum states. We give examples for detecting degenerate states, as well as border rank classification for up to 5 binary qubits and 3 qutrits (ternary qubits). Keywords Quantum entanglement · Classification · Algebraic varieties · Machine learning · Neural networks
1 Introduction Recent efforts to unite Quantum Information, Quantum Computing, and Machine Learning have largely been centered on integrating quantum algorithms and quantum information processing into machine learning architectures [9,55,65,83–85,87,99,100, 103]. Our approach is quite the opposite–we leverage Machine Learning techniques to build classifiers to distinguish different types of quantum entanglement. Machine Learning has been used to address problems in quantum physics and quantum information, such as quantum-state tomography [81], quantum error correction code [75] and wave-function reconstruction [5]. Here, we focus on quantum entanglement. While we were inspired by the approach of learning algebraic varieties in [14], our methods differ in that we are not trying to find intrinsic defining equations of the algebraic models for entanglement types, but rather building a classifier that directly determines the entanglement class. Distinguishing entanglement types may be useful for quantum information processing and in improving and increasing the efficiency of quantum
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Hamza Jaffali [email protected] Luke Oeding [email protected]
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Femto-ST/UTBM, Université de Bourgogne Franche-Comté, 90010 Belfort, France
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Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA 0123456789().: V,-vol
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algorithms, quantum communication protocols, quantum cryptographic schemes, and quantum games. Our methods generalize to cases where a classification of all entanglement types is not known, and cases where the number of different classes is not finite (see for instance [59, Ch.10], or [94]). We only focus on pure states for representing quantum systems, which is sufficient for studying quantum computations and quantum algorithms. This is opposed to the noisy approach with density matrices and mixed states, which is used when one needs to account for the noise and the interaction with the environment [76]. 1.1 Basic notions A basic reference for tensors is [59]. The quantum state of a particle can be represented by a unit vector |ψ in a Hilbert space H (typically H = Cd or Rd ), with basis {|x | x ∈ 0, d − 1} in decimal notation. The state of an n-qudit quantum system is represented by a unit vector |ψ in a tensor product H⊗n of the state spaces for each particle, wh
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