Task-Resolved Fisher Spectroscopy for Quantum Reservoir Computing

Yang Peng

quant-phAug 26, 2026

Task-Resolved Fisher Spectroscopy for Quantum Reservoir Computing

Authors: Yang Peng

Organizations: Department of Physics and Astronomy, California State University, Northridge, California 91330, USA · Institute of Quantum Information and Matter and Department of Physics, California Institute of Technology, Pasadena, California 91125, USA

Abstract

Quantum reservoir computing (QRC) uses fixed quantum dynamics to encode a time series and trains only a classical readout, but a benchmark capacity alone does not reveal whether task information is lost in the reservoir, the measurement, feature compression, or finite sampling. We introduce task-resolved Fisher spectroscopy, in which prediction targets define orthonormal score coordinates on the stationary distribution of input histories. Reweighting labeled histories along these scores generates an exactly affine family of reservoir states and measurement outcomes. In the same coordinates, we obtain a many-body Fisher-information hierarchy relating the state quantum Fisher information, the Fisher information of the complete measurement record, and moment matrices retained through many-body order $r$ . The quadratic form of each moment matrix is exactly the stationary capacity of the optimal linear readout, while a finite-measurement extension predicts the approach from one-shot records to ideal expectation-value features. The measurement-level quantities require only stationary labeled records and measured outcome strings, not an input model or quantum-state tomography. In a five-spin open reservoir, interactions route fourth-order temporal information into higher-body correlations, so low-order compression can incur orders-of-magnitude sampling overhead even when a conventional parity target remains accessible. For correlated inputs and outputs, record-defined task scores predict held-out capacities, the required feature order, and measurement-budget dependence; optimizing the local measurement axis recovers otherwise hidden task information. The framework connects benchmark performance to experimentally actionable diagnoses of reservoir encoding, measurement choice, classical representation, and shot allocation.

Figures & tables

Figure 1: (a) Quantum-reservoir workflow. A stationary input history drives a fixed open quantum reservoir. Measurement produces a bit string, products through many-body order $r$ form the retained feature vector, and only the final linear readout is trained. (b) Task-resolved Fisher spectroscopy. The target labels define orthonormal task scores $s_{A}$ . Their response tangents determine the reservoir-state QFIM $\mathsf{H}_{\rm task}$ and the complete-record CFIM $\mathsf{F}$ ; compressing the same measurement record to moments through order $r$ gives $\mathsf{B}_{r}$ . The hierarchy separates measurement loss from feature-compression loss. The lower box shows the finite-measurement extension: $\mathsf{P}_{r}^{(1)}=\mathsf{B}_{r}$ , while $\mathsf{P}_{r}^{(M)}$ determines the usable capacity $\mathcal{C}_{r}^{(M)}[y]$ for $M$ measurements of each history-conditioned reservoir state.

Figure 2: Interaction-dependent correlation depth of fourth-order temporal information. Correlation depth is the minimum measured many-body order required for a specified fraction of complete local- $X$ Fisher information. The plotted ensemble contains 100 paired coupling realizations at each $J_{s}/h$ . Headline comparisons are restricted to the full-rank regime $J_{s}/h\leq 10$ .

Figure 3: Sampling penalty from low-order data compression. Median worst-mode sampling overhead $\kappa=1/\lambda_{\min}$ when the complete local- $X$ bit string is compressed to correlation products through order $r$ . All correlations are extracted from the same local- $X$ bit strings; the overhead arises from discarding higher-order classical statistics, not from using additional measurement settings.

Figure 4: Task-specific accessibility versus geometric worst case. The standard PC4 coordinate remains mostly visible in two-body moments even when another fourth-order temporal mode is almost absent from that low-order subspace. A single nonlinear benchmark can therefore underestimate the measurement complexity of the reservoir’s full temporal representation.

Figure 5: Covariance estimate versus trained QRC performance. Filled markers show the optimal linear-readout capacity estimated from an independent labeled reference trajectory. Open markers show capacities obtained by fitting an ordinary least-squares readout on a separate training trajectory and evaluating it on a held-out trajectory. Small horizontal offsets separate the paired markers visually. Results are shown only at the five available retained correlation orders $r=1,\ldots,5$ . The close agreement of each pair shows that the covariance estimate predicts held-out linear-readout performance.

Figure 6: A measurement-budget design diagnostic. Stationary PC2 capacity from pair observables for representative $J_{s}/h=1$ and $10$ reservoirs as a function of independent measurement shots $M$ per replayed time step. Dashed levels are the ideal expectation-value limits; crosses are direct finite-shot estimates from trained and held-out trajectories at $M=1000$ . The strongly interacting reservoir appears excellent in the noiseless limit but requires far more measurements to expose the same temporal feature.

Figure 7: Task-resolved information hierarchy for correlated multi-output prediction. A stationary binary Markov source drives one fixed $J_{s}/h=1$ reservoir. Each task score has unit variance, so its curve starts at one and then shows the reservoir QFI, the complete-record CFI for one measurement shot, and the information retained by products through many-body order $r$ . Solid lines use the local- $X$ measurement; dashed lines use the optimized common measurement axis introduced below.

Figure 8: Covariance prediction for correlated multi-output QRC. Filled markers show capacities calculated from a labeled stationary calibration record, and open markers show capacities from independently trained and held-out linear readouts. Circles and squares denote the two original, nonorthogonal outputs. Small horizontal offsets separate the paired markers at each available retained order

r=1,\ldots,5

. The first output approaches its plateau at lower order, whereas the second continues to benefit from higher-body features. The close agreement within each pair shows that the covariance calculation predicts ordinary linear-readout performance.

Figure 9: Finite measurement budget for correlated outputs. Task capacities from local-

X

products through

r=2

versus the number

M

of independent measurements of each history-conditioned state. Curves evaluate Eq. ( 31 ); crosses are direct finite-shot train/test checks at

M=100

and

1000

Figure 10: Task-guided optimization of a common local measurement axis. Normalized order-two task accessibility $\eta_{2}(\bm{n})$ over the upper Bloch hemisphere for the fixed correlated-source reservoir realization. The antipodal axes $\bm{n}$ and $-\bm{n}$ differ only by relabeling the binary outcomes. Markers identify local $X$ and the optimized order-two axis near $+Z$ .

Figure 11: Measurement optimization adapted to the experimentally retained many-body order. The normalized task accessibility

\eta_{r}

defined in Eq. ( 46 ) is shown versus the maximum retained correlation order

r

. The fixed local-

X

measurement is compared with a single axis optimized for the complete record (

r=N

) and with the order-aware optimum

\bm{n}_{\rm opt}^{(r)}

obtained by maximizing

\eta_{r}(\bm{n})

over the measurement axis separately at each

r

. In this realization, the complete-record optimum is already close to the order-specific optima, but both strongly outperform local

X

Reservoir Computing

Reservoir computing uses recurrent dynamical systems to transform temporal inputs into features that support efficient learning with simple readouts.

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Feb 16, 2026J/K move · Enter open · S save

Kernel-based optimization of measurement operators for quantum reservoir computers

Markus Gross, Hans-Martin Rieser

Institute for AI Safety and Security, German Aerospace Center (DLR), Sankt Augustin and Ulm, Germany

Quantum Reservoir Computing Quantum Kernel Methods

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Finding optimal measurement operators is crucial for the performance of quantum reservoir computers (QRCs), since they employ a fixed quantum feature map. We formulate the training of both stateless (quantum extreme learning machines, QELMs) and stateful (memory dependent) QRCs in the framework of kernel ridge regression. We thus extend the kernel viewpoint of supervised quantum models to recurrent QRCs by deriving an exact Hilbert--Schmidt kernel representation of the optimal readout observable on history space. This approach renders an optimal measurement operator that minimizes prediction error for a given reservoir and training dataset. For large qubit numbers, this method is more efficient than the conventional training of QRCs. We discuss efficiency and practical implementation strategies, including Pauli basis decomposition and operator diagonalization, to adapt the optimal observable to hardware constraints. To demonstrate the effectiveness of this approach, we present numerical experiments on image classification and time series prediction tasks, including chaotic and strongly non-Markovian systems. The developed method can also be applied to other quantum machine learning models.

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