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Expressivity of Quantum Reservoir Computers

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arxiv 2501.15528 v3 pith:7FIW3KLE submitted 2025-01-26 quant-ph

Expressivity of Quantum Reservoir Computers

classification quant-ph
keywords quantumexpressivitysizefourieroutputreservoirseriessystem
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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Using Hamiltonian encoding to inject an input into parameterized quantum circuits (PQCs), the output of the PQC can be written as truncated Fourier series. In recent years, the expressivity of PQCs was established as the number of frequencies contained in this Fourier series. While this concept has also been applied to other quantum machine learning (QML) paradigms, a clear notion of expressivity for temporal information processing with quantum systems is still lacking. Here, we introduce such a notion to the field of quantum reservoir computing (QRC). We analytically derive an expression for the readouts showing that the output of a QRC can be interpreted as a multi-dimensional Fourier series. We give a formula for the growth of expressivity induced by the sequential information injection, which we corroborate with numerical simulations, calculating explicitly the number of multi-dimensional output functions which can be generated from the readouts. Our results show that the specific interplay between system size, input encoding, and memory time gives rise to a boundary on the system size beyond which it is obstructive to further increase the reservoir size in extreme scrambling systems. We propose a recipe for determining this maximal system size for a given QRC setup.

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Cited by 8 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Fisher-Orthogonal Memory in Quantum Reservoir Computing

    quant-ph 2026-07 conditional novelty 7.0

    Clifford-routed quantum reservoirs that store each past input along a distinct Pauli direction have diagonal Fisher memory matrices and outperform optimized random Ising reservoirs in simulated finite-shot delay tasks.

  2. Controllable Quantum Memory Capacity in Quantum Reservoir Networks with Tunable partial-SWAPs

    quant-ph 2026-05 unverdicted novelty 7.0

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    A hardware-realizable tunable partial-SWAP is introduced to control the rate of memory dissipation in recurrent quantum reservoir computing architectures, validated via simulation and IBM QPUs.

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    quant-ph 2026-04 unverdicted novelty 7.0

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  5. Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach

    quant-ph 2026-02 unverdicted novelty 7.0

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  7. Controllable Quantum Memory Capacity in Quantum Reservoir Networks with Tunable partial-SWAPs

    quant-ph 2026-05 unverdicted novelty 6.0

    Introduces tunable partial-SWAP for controllable memory capacity in quantum reservoir networks, modeled as controlled amplitude-damping and validated via STMC and NARMA-5 benchmarks on simulators and IBM QPUs.

  8. Quantum Reservoir Computing: Recent Advances and Future Directions

    quant-ph 2026-07 accept novelty 4.0

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