Storage capacity and learning capability of quantum neural networks

ORAL

Abstract

We study the storage capacity of quantum neural networks (QNNs), described by completely positive trace preserving (CPTP) maps acting on a $N$-dimensional Hilbert space. We demonstrate that attractor QNNs can store in a non-trivial manner up to $N$ linearly independent pure states. For $n$ qubits, QNNs can reach an exponential storage capacity, $\mathcal O(2^{n})$, clearly outperforming standard classical neural networks whose storage capacity scales linearly with the number of neurons $n$. We estimate, employing the Gardner program, the relative volume of CPTP maps with $M\leq N$ stationary states and show that this volume decreases exponentially with $M$ and shrinks to zero for $M\geq N+1$. We generalize our results to QNNs storing mixed states as well as input-output relations for feed-forward QNNs. Our approach opens the path to relate storage properties of QNNs to the quantum features of the input-output states. This work is dedicated to the memory of Peter Wittek.

*We acknowledge financial support from:ERC-AdG NOQIA,Spanish MINECO:FIS2016-79508-P,FIS2016-80681-P(AEI/FEDER,UE),"Severo Ochoa" program for Centers of Excellence in R\&D(CEX2019-000910-S),Spanish Agencia Estatal de Investigacion:(PID2019-107609GB100,PID2019-106901GB-I00 /10.13039/501100011033,FPI),European Social Fund,Generalitat de Catalunya:(CIRIT 2017-SGR-1341,2017-SGR-1127,AGAUR FI-2018-B01134,CERCA Program and QuantumCAT/001-P-001644,QuantumCAT\_U16-011424 co-funded by ERDF Operational Program of Catalonia 2014-2020),Fundacio Privada Cellex,Fundacio Mir-Puig,MINCIN-EU QuantERA MAQS funded by the State Research Agency (AEI):(PCI2019-111828-2,10.13039/501100011033).This project has received funding from the European Union Horizon 2020:PROBIST 754510,Marie Sklodowska-Curie grant agreement No.754510 and No.847648,FET-OPEN OPTOLogic No.899794,the National Science Centre Poland-Symfonia Grant No.2016/20/W/ST4/00314,the ”la Caixa” Foundation(ID 100010434) with code(LCF/BQ/PI20/11760031).

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Publication: Storage capacity and learning capability of quantum neural networks,
M Lewenstein, A Gratsea, A Riera-Campeny, A Aloy, V Kasper, A Sanpera
Quantum Science and Technology, 2021

Presenters

  • Aikaterini Gratsea

    • ICFO-The Institute of Photonic Sciences

Authors

  • Aikaterini Gratsea

    • ICFO-The Institute of Photonic Sciences

  • Valentin Kasper

    • Harvard University

  • Maciej A Lewenstein

    • ICFO-The Institute of Photonic Sciences

  • Anna Sanpera

    • UAB

  • Albert Alloy

    • ICFO

  • Andreu Riera-Campeny

    • UAB