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Quantum reservoir computing in finite dimensions.

Rodrigo Martínez-Peña1, Juan-Pablo Ortega2

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This study introduces vector representations for quantum reservoir computing (QRC) systems, unifying density matrix and observable spaces. This approach clarifies fading memory and echo state properties, offering new insights into QRC theory.

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Area of Science:

  • Quantum Computing
  • Information Theory
  • Theoretical Physics

Background:

  • Quantum reservoir computing (QRC) analysis often relies on the density matrix formalism.
  • Alternative representations may offer deeper insights for QRC design and assessment.

Purpose of the Study:

  • To establish system isomorphisms unifying density matrix and observable space representations in QRC.
  • To investigate the implications of these representations for QRC properties like fading memory and echo state properties.
  • To explore fundamental questions in finite-dimensional QRC theory.

Main Methods:

  • Establishing system isomorphisms between density matrix formalism and Bloch vector representations (Gell-Mann bases).
  • Connecting these vector representations to state-affine systems from classical reservoir computing.
  • Analyzing the fading memory property (FMP) and echo state property (ESP) within these unified frameworks.

Main Results:

  • Demonstrated that vector representations yield state-affine systems, linking QRC to established classical theories.
  • Showed that FMP and ESP are representation-independent.
  • Formulated necessary and sufficient conditions for ESP and FMP.
  • Characterized contractive quantum channels with trivial semi-infinite solutions.

Conclusions:

  • Vector representations offer a powerful alternative for analyzing QRC systems, providing deeper theoretical insights.
  • The established connection to state-affine systems facilitates the application of existing classical reservoir computing results to QRC.
  • This work clarifies fundamental properties of QRC in finite dimensions and characterizes specific quantum channels.