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Quantum reservoir processing
Sanjib Ghosh, Andrzej Opala, Michał Matuszewski, Tomasz Paterek, Timothy C. H. Liew
TL;DR
The paper addresses how quantum tasks can be processed with reservoir computing rather than conventional recurrent-network training. It introduces a fermionic quantum reservoir that recognizes entanglement and estimates nonlinear state properties, with experiments indicating recognition beyond the training class. The platform can simplify some experiments by replacing multiple-observable measurements with one trained-reservoir measurement, while its scope includes limitations for certain training procedures and state classes.
Problem
Recurrent neural-network training is typically inefficient and computationally expensive, motivating a simpler reservoir-computing approach for quantum information tasks.
Method
The paper uses a randomly coupled 2D fermionic-lattice quantum reservoir with trained readout weights to process optical quantum inputs.
Results
The processor recognizes entanglement beyond the training class and estimates nonlinear functions including logarithmic negativity, entropy, purity, and traces of powers.
Takeaways & Limitations
A suitably trained QRP can replace measurements of multiple quantum observables with one reservoir measurement followed by post-processing.
Takeaways & Limitations
An entangled cat-state example is not recognized when the processor is trained using only squeezed-thermal states.
Abstract
from arXiv · showhide
The concurrent rise of artificial intelligence and quantum information poses opportunity for creating interdisciplinary technologies like quantum neural networks. Quantum reservoir processing, introduced here, is a platform for quantum information processing developed on the principle of reservoir computing that is a form of artificial neural network. A quantum reservoir processor can perform qualitative tasks like recognizing quantum states that are entangled as well as quantitative tasks like estimating a non-linear function of an input quantum state (e.g. entropy, purity or logarithmic negativity). In this way experimental schemes that require measurements of multiple observables can be simplified to measurement of one observable on a trained quantum reservoir processor.
INTRODUCTION
The paper introduces quantum reservoir processing, applying reservoir computing to quantum inputs with a fermionic lattice. The platform supports entanglement recognition and quantitative estimation of nonlinear state properties, potentially replacing multiple-observable measurements with one trained-reservoir measurement.
- Recurrent neural networks handle temporal tasks through feedback-driven internal dynamics, but their training is typically inefficient and computationally expensive.
- Reservoir computing simplifies training by optimizing only readout weights on a randomly connected dynamical reservoir.This approach is also suitable for hardware implementation.
- A trained processor can combine occupation-number readouts to identify a quantum state and estimate several of its properties.
- Quantum reservoir processing applies reservoir computing to quantum inputs using a 2D fermionic lattice with random intersite coupling.The input is an optical quantum field incident on the reservoir.
- A quantum reservoir processor recognizes entangled states, including states beyond the training class such as bipartite bound entangled states.
- The platform estimates logarithmic negativity, von Neumann entropy, purity, and traces of powers of an input quantum state.
The model
The quantum reservoir is a randomly coupled 2D fermionic lattice driven by sequential input modes. Site occupations are measured after interaction and linearly combined through trained output weights for the target task.
- The model: The reservoir consists of fermions arranged in a 2D lattice with random nearest-neighbour hopping and is defined by a Fermi-Hubbard Hamiltonian.
- The model: Random hopping amplitudes Jij are uniformly distributed between −γ and +γ, where γ is the decay rate.
- The model: A bipartite bosonic input state is represented by ρin and couples its two optical modes to the reservoir sequentially.The sequential coupling models wave packets incident one after another.
- The model: The joint reservoir-and-input state evolves according to a quantum master equation using cascaded-formalism terms.
- The model: Input modes couple during separate brief time windows, while the reservoir first evolves for a duration t1 toward its steady state.For the simulations, τ = ℏ/γ.
- The model: Fermionic site occupations measured at t = t1 + 2τ form the readout, whose linear combination is fitted by optimized output weights.The output weights are trained against known task-specific data.
Recognition of quantum entanglement
The processor is trained to classify squeezed-thermal inputs as entangled or separable using output weights optimized by ridge regression. It generalizes this separability criterion to several non-Gaussian entangled states beyond the training class.
- Recognition of quantum entanglement: The training set contains bipartite squeezed-thermal states generated by applying a two-mode squeezing operator to thermal states.
- Recognition of quantum entanglement: Random squeezing parameters are chosen so that approximately half the states are entangled and half are separable.
- Recognition of quantum entanglement: The processor assigns output vectors (1, 0) to entangled states and (0, 1) to separable states, with output weights learned by minimizing prediction error through ridge regression.
- Recognition of quantum entanglement: During testing, the first output component determines the predicted class, while logarithmic negativity independently verifies whether the input is entangled.
- Recognition of quantum entanglement: The learned separability criterion applies beyond squeezed-thermal inputs to photon-added or photon-subtracted squeezed states, superpositions c0|00⟩ + c1|11⟩, and bound entangled states.
- Recognition of quantum entanglement: Although trained only on squeezed-thermal states, the QRP recognizes the listed non-Gaussian entangled states efficiently.
- Recognition of quantum entanglement: Figure 3 reports correctly identified separability properties as bar heights, averaged over 10 random coupling and input-weight configurations.
Quantitative estimations and multiprocessing
The quantum reservoir processor simultaneously estimates multiple non-linear properties of an input state, with accuracy and precision improving for larger reservoirs. This can replace multiple-observable measurements with one reservoir measurement followed by post-processing.
- Multiprocessing: The processor outputs M parameters simultaneously by training an M-dimensional readout vector and learning the output weight matrix.Each output element estimates one parameter once the optimal weights are obtained.
- Quantitative estimations: The demonstrated six parameters are logarithmic negativity, von Neumann entropy, and Tr(ρ^n) for n = 2 . . . 5.Logarithmic negativity retains negative values in this example.
- Quantitative estimations: Parameters with a series expansion in powers of the input state can be estimated similarly.The paper states that arbitrary parameters of the form ⟨P⟩ = Σ_n c_nρ^n can be treated this way.
- Quantitative estimations: Predictions become more precise and accurate as the fermionic reservoir grows from 2 to 4 sites.Figure 4 compares true and predicted values for six quantities using 2-fermion and 4-fermion reservoirs.
- Experimental simplification: A trained processor can obtain different parameters after one reservoir measurement, whereas conventional estimation may require multiple observables or full state tomography.This simplification requires additional training resources and sufficient reservoir size for the desired precision.
DISCUSSION
The architecture is presented as both programmable quantum hardware and software for quantum machine-learning tasks. Several physical platforms could realize the fermionic-lattice reservoir.
- DISCUSSION: The platform can be programmed through training as quantum hardware or used as software for otherwise hard quantum-machine-learning tasks.The paper specifically mentions recognition of quantum entanglement and estimation of non-linear input-state functions.
- DISCUSSION: Bound-entangled-state identification is given as an example software application.The discussion distinguishes this software use from hardware realization of the reservoir.
- DISCUSSION: The 2D fermionic lattice could be realized using semiconductor quantum dots, NV centres in diamond, or trapped atoms.The model may also be implemented with driven-dissipative arrays of fermionized photons and related photonic systems.
METHODS
The method uses randomly parameterized squeezed and non-Gaussian bipartite states as inputs and a 2D fermionic reservoir as the processing substrate. The supplementary methods describe several state families and their parameter ranges.
- Squeezed-thermal states: The thermal-state density matrix is expressed in the Fock basis using occupation-number weights indexed by the two mode occupations.The average occupation number per mode is denoted n̄.
- Squeezed-thermal states: The squeezed-thermal training states are generated by applying a two-mode squeezing operator to bipartite thermal states.The squeezing parameter is α = |α|e^iθ, with θ, s, and φ sampled over specified intervals so about 50% are Gaussian entangled.
- Non-Gaussian states: Photon-added squeezed states are formed by applying creation operators to a squeezed vacuum state.For |α| < 0.378, the Simon criterion does not always detect entanglement in these states.
- Non-Gaussian states: Photon-subtracted squeezed states are formed by applying annihilation operators to a squeezed vacuum state.The squeezing magnitude and phase are sampled so the prepared states have an average occupation close to the squeezed-thermal training states.
- Additional state families: The supplementary set also includes states c0|00⟩ + c1|11⟩ sampled uniformly on a Bloch sphere and a family of bound-entangled states.The bound-entangled family uses parameters satisfying 0 < a < c < 1 and a small a/c in a truncated Fock space.
SUPPLEMENTARY MATERIAL FOR “QUANTUM RESERVOIR PROCESSING”
The supplementary material details balanced training data, pump robustness, broader purity estimation, and examples showing both generalization and limits of quantum reservoir processing.
- Distribution of the training set: Training states are chosen so approximately 50% are entangled and 50% separable, avoiding class bias.The squeezed-thermal-state parameter ranges are selected to produce this balanced distribution.
- Function of the incoherent pump: Except near P ≈ γ, separability-recognition performance remains very high across incoherent pump strengths.The data uses 200 random squeezed-thermal states and averages over 10 three-fermion reservoir realizations.
- Function of the incoherent pump: At P = 0, the reservoir processor loses only 1% of its success rate, so the incoherent pump can be removed without major performance change.This conclusion is reported for recognizing separability of squeezed-thermal states.
- Wider sampling of the training set: With wider sampling, purity-related predictions Tr[ρ^n] cover the full range [0, 1] and improve as the reservoir grows from 2 to 4 fermions.Panels (a)–(d) correspond to n = 2...5, with red denoting 2 fermions and blue 4 fermions against ideal black lines.
- Wider sampling of the training set: The wider training set improves Tr[ρ^n] prediction accuracy but is inefficient for entanglement recognition.Its inputs are not equally distributed between entangled and separable states.
- Entangled cat states: Entangled cat states are not recognized when the processor is trained only on squeezed-thermal states, but same-class entangled cat states achieve success rates above 90% with randomized coherence.Here |±β⟩ are coherent states and p is the degree of coherence.