Kernel-based optimization of measurement operators for quantum reservoir computers

Markus Gross; Hans-Martin Rieser

quant-phFeb 16, 2026

Kernel-based optimization of measurement operators for quantum reservoir computers

Authors: Markus Gross, Hans-Martin Rieser

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

Abstract

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.

Figures & tables

Symbol	Meaning
$\mathcal{H}$	Hilbert space
$\mathcal{M}(\mathcal{H})$	Space of Hermitian operators on $\mathcal{H}$
$N$	Total number of qubits
$N_{I}$ , $N_{A}$	Number of input/ancilla qubits
$D$	Hilbert space dimension ( $D=2^{N}$ )
$d$	Input data dimension ( $\mathbf{x}\in\mathbb{R}^{d}$ ) ⋆

Table 1: Summary of notation used in this work. ⋆ For QELMs,

\mathbf{x}=\mathbf{z}_{t}

and

d=d^{\prime}

, while for stateful QRCs, the identification in Eq. 3 is necessary.

	(A) Fixed measurements (constrained primal)	(B) Free observable (unconstrained primal)	(C) Kernel form (dual of B)
Given data	$\{(\mathbf{x}_{i},y_{i})\}_{i=1}^{P}$ , reservoir states $\rho_{i}:=\rho(\mathbf{x}_{i})\in\mathcal{M}(\mathcal{H})$
Trainable object	$\mathbf{w}\in\mathbb{R}^{m}$	$M\in\mathcal{M}(\mathcal{H})$	$\boldsymbol{\alpha}\in\mathbb{R}^{P}$
Allowed observables	$M=\sum_{k=1}^{m}w_{k}M_{k}$ $\in S:=\mathrm{span}\{M_{k}\}$	any $M\in\mathcal{M}(\mathcal{H})$	$M=\sum_{i=1}^{P}\alpha_{i}\rho_{i}$ (representer form)
Prediction	$f(\mathbf{x})=\sum_{k=1}^{m}w_{k}\mathrm{tr}(M_{k}\rho(\mathbf{x}))$	$f(\mathbf{x})=\mathrm{tr}(M\rho(\mathbf{x}))$	$f(\mathbf{x})=\sum_{i=1}^{P}\alpha_{i}K(\mathbf{x}_{i},\mathbf{x})$
Training objective	$\displaystyle\min_{\mathbf{w}}\frac{1}{2}\sum_{i=1}^{P}\Bigl(y_{i}-\sum_{k=1}^{m}w_{k}\mathrm{tr}(M_{k}\rho_{i})\Bigr)^{2}+\frac{\lambda}{2}\\|\mathbf{w}\\|_{2}^{2}$	$\min_{M}\frac{1}{2}\sum_{i=1}^{P}\bigl(y_{i}-\mathrm{tr}(M\rho_{i})\bigr)^{2}+\frac{\lambda}{2}\,\mathrm{tr}(M^{2})$	$\displaystyle\min_{\boldsymbol{\alpha}}\frac{1}{2}\\|\mathbf{y}-\mathbf{K}\boldsymbol{\alpha}\\|_{2}^{2}+\frac{\lambda}{2}\,\boldsymbol{\alpha}^{\top}\mathbf{K}\boldsymbol{\alpha}$
Matrices	$\Phi_{ik}:=\mathrm{tr}(M_{k}\rho_{i})$	$K_{ij}:=\mathrm{tr}(\rho_{i}\rho_{j})$	$K_{ij}:=\mathrm{tr}(\rho_{i}\rho_{j})$

Table 2: Three training formulations for a QRC. The training objective and solution are specialized here to the case of least-squares loss and assuming an orthonormal operator basis. For other kinds of losses, usually no analytic expressions for the weights

\mathbf{w}^{*}

exist. If the QRC carries an internal memory state, the above formulations remain intact but the data

\mathbf{x}_{i}\hat{\equiv}x_{t}

represents an input history [see Eq. 3 ] and

\rho(\mathbf{x})\hat{\equiv}\rho(x_{t})

is the state resulting by sequentially processing that history (see Sections III.1 and F ).

Panel of Figure 1 — Figure 1: Optimization of the measurement operator for a QELM trained on the MNIST (first $2N$ principal components) classification task (left) without and (right) with a reservoir unitary (transverse field Ising model). The dashed line represents the test accuracy obtained using the full operator $M^{*}$ (i.e., one per digit class), while the connected symbols correspond to the Pauli decomposition and diagonalization approaches [ Eqs. 13 and 14 ]. The notation ord( $k$ ) and mon( $p$ ) refers to the observable order $k$ and monomial order $p$ in Eq. 31 . The curve labeled ‘primal: Z,ZZ+mon(4)’ represents primal optimization using only the Pauli- $Z$ and -ZZ observables enhanced by readout monomials up to 4th order, while CSVM refers to a classical SVM trained on (non-)linear features constructed from the principal components. Since there is no intrinsic randomness in the model, error bars arise only from the selection of samples and are negligible here. † Here, we optimized the kernel only over a single class (to limit combinatorial explosion in the number of readout terms), but used the resulting operator set to re-train the QELM for all classes.

Panel of Figure 2 — Figure 2: Optimization of the measurement layer for a memoryless QRC without ancilla qubits (QELM), for the cases without and with a (TFIM-based) reservoir unitary. The QRC is trained on a 3-dimensional chaotic time series [ Eq. 36 ]. The error bars on the forecast horizon represent the standard deviation (not error of the mean) after averaging over different initial conditions. (Their magnitude is comparable to classical RC results for chaotic systems.) The dashed line represents the forecast horizon when using the kernel-based optimal measurement operator $M^{*}$ directly, while the connected points correspond to decompositions into subsets of observables [ Eqs. 13 and 14 ]. The dash-dotted line gives the forecast horizon (with its standard deviation indicated by the gray area) for the primal optimization over the complete operator basis. $E_{\text{train}}$ denotes the (RMS) training error. The horizontal axis represents the number of observables ranked by their relevance (left = least relevant; maximum number $=64\times 3=192$ for right panel).

Figure 3: Optimization of the measurement operator for a QRC with internal memory ( $N_{A}=1$ ancilla qubit coupled via a TFIM unitary to $N_{I}=3$ input qubits), trained to predict the Lorenz-63 time series [ Eq. 36 ] encoded via amplitude encoding [ Eq. 28 ]. The dashed line represents the forecast horizon using the optimal operator $M^{*}$ directly, while the connected points correspond to decompositions into subsets of observables [ Eqs. 13 and 14 ]. The forecast horizon obtained from the primal optimization over the full operator basis is shown by the dashed-dotted line, with the gray area representing the standard deviation.

Panel of Figure 4 — Figure 4: Optimization of the measurement layer for a QRC with internal memory ( $N_{A}=3$ ancilla qubits, coupled to $N_{I}=1$ input qubits via a TFIM-based unitary). In (a), the QRC is trained on a random harmonic signal [ Eq. 37 ] with $n=9$ frequencies, encoded via amplitude encoding ( 29 ). In (b), the Mackey-Glass time series [ Eq. 38 ] is encoded via Eq. 28 . The dashed line represents the forecast horizon (relative to the maximum tested time in (a)) using the optimal measurement operator $M^{*}$ (as computed by the kernel-based optimization), while the connected points correspond to decompositions into subsets of observables ( Eqs. 13 and 14 , ranked by their relevance, along horizontal axis). The error bars on the forecast horizon represent the standard deviation after averaging over different initial conditions. The forecast horizon obtained from the primal optimization over the full operator basis is shown by the dashed-dotted line, with the gray area representing the standard deviation. $E_{\text{train}}$ denotes the (RMS) training error.

Step	Stateless (QELM)	Stateful (QRC)
1. Data	Input pairs $\{(\mathbf{x}_{i},y_{i})\}_{i=1}^{P}$	Time series $\{\mathbf{z}_{t}\}_{t=0}^{T}$ ; Targets $y_{t}=\mathbf{z}_{t+1}$
		Effective inputs $x_{t}:=(\mathbf{z}_{t-L},\ldots,\mathbf{z}_{t})$
		$\{(x_{t},y_{t})\}_{t=L}^{T-1}$ can be treated as unordered samples
2. Encoding	Map each $\mathbf{x}_{i}$ to state:	To obtain $\rho(x_{t})$ , evolve for $\tau=t-L,\ldots,t$ :
	$\rho_{i}\equiv\rho(\mathbf{x}_{i})$	$\eta(\tau)=U(\eta_{I}(\mathbf{z}_{\tau})\otimes\eta_{M}(\tau))U^{\dagger}$ , $\eta_{M}(\tau+1)=\mathrm{tr}_{I}\eta(\tau)$ ;
		Then $\rho_{t}\equiv\rho(x_{t})=\eta(t)$

Table 3: Algorithm flow for the kernel-based (dual) optimization of the measurement operator in stateless (memoryless) and stateful (recurrent) QRCs. Note that

\mathbf{y}\in\mathbb{R}^{P}

is the vector of target samples and the expression for the dual coefficients

\alpha^{*}

thus applies componentwise. In the stateful case,

P=T-L

is the number of time steps available for training. For details, see Appendices A and F .

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

	$\mathrm{tr}(\sigma^{p}\rho_{j})$
$\sigma^{p}$	rotational	amplitude
$\mathbb{I}$	$1$	$1$
$\sigma^{x}$	$\sin(2\pi z)$	$2\sqrt{z(1-z)}$
$\sigma^{y}$	$0$	$0$
$\sigma^{z}$	$\cos(2\pi z)$	$2z-1$

Appendix

Table 4: Single-qubit expectation values for rotational and amplitude encoding.

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