Quantum Gaussian processes for prediction of channel observations

Jonas Jäger; Yaroslav Khmelnitskiy; Paolo Braccia; Artur Miroszewski; Diego García-Martín; M. Cerezo; Piotr Czarnik

quant-phAug 19, 2026

Quantum Gaussian processes for prediction of channel observations

Authors: Jonas Jäger, Yaroslav Khmelnitskiy, Paolo Braccia, Artur Miroszewski, Diego García-Martín, M. Cerezo, Piotr Czarnik

Organizations: Department of Computer Science and Institute of Applied Mathematics, University of British Columbia, Vancouver, V6T 1Z4 B.C., Canada · Stewart Blusson Quantum Matter Institute, Vancouver, V6T 1Z4 B.C., Canada · Institute of Theoretical Physics, Jagiellonian University, Kraków, Poland. · Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA ·

Abstract

Given a set of input states, we consider the task of predicting the expectation value of a Pauli observable at the output of an unknown quantum evolution, using only a limited number of measurements. Recently, quantum Gaussian process (QGP) regression was introduced for this task across various classes of unitary evolution. Here, we extend the QGP framework beyond unitary dynamics. In particular, we prove convergence of the channel's outputs to a QGP and derive the associated closed-form kernel under a uniform (Lebesgue measure) prior over quantum channels. The kernel's dimensional factor, however, dictates the required observation precision. While manageable when the channel and observable are restricted to small subsystems, exponential suppression precludes learning when the subsystem grows extensively with the system size. Since the Lebesgue prior is overly broad for many applications, we propose an empirical Bayes heuristic that replaces the dimensional factor with a learnable scale parameter while retaining the kernel's state-overlap correlation structure. In numerical simulations of up to 64 qubits, channel QGP regression with the Lebesgue kernel exhibits a strong inductive bias for local channels, enabling faithful extrapolation. For global 64-qubit channels, the rescaled kernel restores learnability, with predictions improving systematically with the shot budget. Results from a noisy quantum computer further demonstrate the robustness of QGP regression under experimental conditions. Beyond regression, we validate QGPs as Bayesian-optimization surrogates for state preparation under noisy XXZ dynamics.

Figures & tables

Figure 1: Quantum Gaussian process regression for quantum channels. (a) A parametrized family of input states

\rho(t)

is acted on by an unknown quantum channel

\mathcal{C}

. Expectation values

\ell_{\mathcal{C}}(t)=\Tr[\mathcal{C}(\rho(t))O]

are measured for a small set of input states and used to predict the observable for new inputs. (b) The kernel encodes correlations between the input states. Together with the measured training data, it defines the QGP posterior used for regression. In this work, we consider both the channel kernel derived under the assumption of uniform channel sampling, and its rescaled heuristic counterpart. (c) Beyond regression, the channel QGP framework can be applied to tasks such as state preparation, quantum control, and Bayesian optimization.

Figure 2: QGP regression with the channel kernel. We learn the expectation value of

X_{1}

(

X_{60}

) for a family of 4-qubit (64-qubit) input states generated by single-qubit rotations parametrized by

t

followed by system unitary dynamics. Joint system-environment time evolution implements the channel. Both are implemented by a transverse-field one-dimensional Ising Hamiltonian and a Trotter product formula. For

n=4

, the channel acts globally on all system qubits, while for

n=64

it acts locally on a 4-qubit subsystem. In (a), we show QGP regression for the global channel with the shot budget

N_{s}^{(p)}=1.32\cdot 10^{8}

for a single prediction. The black curve is the ground truth, the blue circles are the training data (including error bars representing their finite-shot uncertainty), the red pluses are the predictions, and the shaded region denotes the

95\%

prediction confidence interval, with a half-width

2\sigma_{P}(t)

. In (b), we show the corresponding root mean squared error (RMSE) versus

N_{s}^{(p)}

for

k=4

and

k=8

training observations, together with the RMSE of row-PTM learning as a reference. In (c) and (d), we show the analogous results for the local channel, with

N_{s}^{(p)}=2.7\cdot 10^{7}

in (c) and

k\in\{3,6\}

in (d). To account for finite-shot fluctuations, all experiments are evaluated across 100 independent runs. Hence, (a,c) display regression instances with the median RMSE, while (b,d) show the mean RMSE (the markers) and sample standard deviation (the error bars).

Figure 3: QGP regression with the channel kernel on IBM’s quantum computer ibm_boston . We predict

\langle X_{1}\rangle(t)

for a family of 4-qubit input states generated by single-qubit rotations parametrized by

t

. The channel is a real-device implementation of a circuit performing Trotterized transverse-field Ising dynamics coupling the system to a 4-qubit environment. The ground truth, indicated by the black markers, are the results obtained from the quantum device with 34400 shots per data point. The error bars, computed as twice the standard deviation, quantify the finite shot effects. The blue markers are the training data obtained from the same device and with the same number of shots. The red pluses are the QGP predictions and the shaded region denotes the 95% prediction confidence interval. The kernel is evaluated numerically without shot noise effects, neglecting the hardware noise on the input states.

Figure 4: QGP regression with the rescaled channel kernel . In (a), we show learning of

\langle X_{1}\rangle

for the same 4-qubit setup as in Fig. 2 , with shot budget

N_{s}^{(p)}=1.32\cdot 10^{8}

per prediction. In (c),

\langle X_{31}\rangle

was learned for a 64-qubit family of input states prepared by single-qubit rotations parametrized by

t

, and a global channel generated by one-dimensional Trotterized transverse-field Ising dynamics coupling the system to a 64-qubit environment. Here,

N_{s}^{(p)}=1.32\cdot 10^{6}

. In (a) and (c), we plot the ground truth (the black lines), the training data (the blue circles), the predictions (the red crosses) and their

95\%

confidence intervals (the shaded areas). In (b) and (d), we compare the RMSE of the rescaled channel QGP learning and the channel QGP learning for

n=4

and

n=64

, respectively. For both methods, we plot

k=4

and

k=8

RMSE values versus

N_{s}^{(p)}

. For

n=4

, the channel QGP (the red circles and the cyan asterisks) gives very similar results to the rescaled channel QGP (the blue squares and the green triangles), as shown by the overlapping markers. For

n=64

, the channel QGP remains at approximately the same RMSE for both training-set sizes and does not improve as the shot budget is increased. In contrast, the rescaled channel QGP reproduces the observable dependence across the prediction interval and its RMSE decreases systematically with increasing shot budget. To account for the shot noise effects, in (a) and (c) we show learning instances with median RMSE chosen from a sample of

100

learning instances. The error bars of the training data, the RMSE estimates and the prediction confidence intervals are computed as in Fig. 2 . We also note that reference channel QGP RMSE data for

n=4

are taken from Fig. 2 .

Figure 5: Bayesian optimization of noisy XXZ dynamics with channel GP surrogates. Here we show BO of staggered magnetization ( 37 ) for the dynamics of a 10-qubit XXZ Hamiltonian with transverse and disordered longitudinal fields ( 35 , 36 ) and single-qubit depolarizing noise. As a reference, we show gradient descent (GD) optimization. We optimize over a family of input spin-spiral states parametrized by parameters

\theta

and

q

( 33 ). In (a), the contour plot shows our loss function landscape ( 38 ) on an 80-by-80 grid. The overlapping red crosses show the best parameters found by 9 BO runs, indicating that all the runs converge to the same minimum. The BO uses channel QGPs with rescaled kernels as surrogate models. The blue lines show trajectories of 9 GD optimization runs. Each trajectory shows an initial point (circle) and an end point (star). The GD runs have the same initial points as the BO runs. Each GD (BO) run consists of

800

(

100

) loss evaluations performed numerically using a full density matrix simulator. In (b), we plot the best loss value versus the number of loss evaluations for the BO (the red curves) and gradient descent (the blue lines) runs.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Performance of the row-PTM learning versus shot distribution ratio used for the row-PTM learning. Here, we use a setup from Fig. 2 with

n=n_{A}=4

. We plot root mean squared error (RMSE) versus a ratio of the row learning shot cost

N_{s}^{(1)}

to the total shots required for a prediction

N_{s}^{(p)}

. Here

N_{s}^{(p)}

includes both shots for the row learning and state tomography of the initial state. To account for randomness due to the shot noise, for each shot division we compute RMSE

100

times, with each repetition using a different instance of numerically simulated shot noise. In the plot, for each ratio, we show the mean (the marker) and the standard deviation (the error bar) of these RMSE estimates.

Figure 7: QGP regression with the channel kernel for low and moderate shot budgets. We learn an expected value of

X_{1}

(

X_{60}

) versus an input state parameter

t

for the setups from Fig. 2 . In (a, b), we plot QGP learning for the

n=4

case, and shot budgets required for a prediction

N^{(p)}_{s}=1.4\cdot 10^{6}

N^{(p)}_{s}=4.4\cdot 10^{6}

. The black curves are ground truth, the blue circles are QGP training data, the red pluses are QGP predictions, and the shaded regions denote the 95% prediction confidence interval. In (c) and (d) we plot the results for the local channel (

n=64

) case, for

N_{s}^{(p)}=2.7\cdot 10^{4}

and

N_{s}^{(p)}=9.0\cdot 10^{5}

, respectively. As in Fig. 2 , we choose QGP learning instances with median RMSE in a set of 100 learning instances. The training data error bars represent their standard deviations of the mean.

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Despite rapid recent advances in quantum machine learning, the field is in many ways stuck. Existing approaches can exhibit serious limitations, and we still lack learning frameworks that are simple, interpretable, scalable, and naturally suited to quantum data. To address this, here we introduce quantum Gaussian processes, a Bayesian framework for learning from quantum systems through priors over unknown quantum transformations. We show that, under suitable conditions, unitary quantum stochastic processes define Gaussian processes, thereby enabling regression, classification, and Bayesian optimization directly on quantum data. The key ingredient in this framework is sufficient knowledge of a quantum process's structure and symmetries to define an informative prior through its corresponding quantum kernel, effectively injecting a strong, physics-informed inductive bias into the learning model. We then prove that matchgate, or free-fermionic, evolutions give rise to provable and scalable quantum Gaussian processes, providing the first family in our framework where the unknown unitary acts non-trivially on all qubits. Finally, we demonstrate accurate long-range extrapolation, phase-diagram learning in many-body systems, and sample-efficient Bayesian optimization in a quantum sensing task. Our results identify quantum Gaussian processes as a promising route toward simpler and more structured forms of quantum learning.

Abstract

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