quant-phApr 7, 2026

Shot-based quantum encoding: a data-loading paradigm for quantum neural networks

Authors: Basil Kyriacou, Viktoria Patapovich, Maniraman Periyasamy, Alexey Melnikov

Organizations: Terra Quantum AG, 9000 St. Gallen, Switzerland

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Abstract

Efficient data loading remains a bottleneck for near-term quantum machine learning. Existing schemes (angle, amplitude, and basis encoding) either underuse the exponential Hilbert-space capacity or require circuit depths that exceed the coherence budgets of noisy intermediate-scale quantum hardware. We introduce shot-based quantum encoding (SBQE), a data embedding strategy that distributes the hardware's native resource, shots, according to a data-dependent classical distribution over multiple initial quantum states. By treating the shot counts as a learnable degree of freedom, SBQE produces a mixed-state representation whose expectation values are linear in the classical probabilities and can therefore be composed with nonlinear activation functions. We show that SBQE is structurally equivalent to a multilayer perceptron whose weights are realized by quantum circuits, and we describe a hardware-compatible implementation protocol. Benchmarks on three image datasets, with 10 independent initializations per model, show that SBQE achieves 89.1% +- 0.9% test accuracy on Semeion (reducing error by 5.3% relative to amplitude encoding and matching a width-matched classical network), 80.95% +- 0.10% on Fashion MNIST (exceeding amplitude encoding by +2.0% and a linear multilayer perceptron by +1.3%), and 90.25% +- 0.18% on MNIST (exceeding amplitude encoding by 2.1 percentage points and the width-matched classical network by 0.3), all without any data-encoding gates.

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Figure 1: Encoding a datum into a mixed state through the learned probability map ( q=8q=8 qubits, trained model). For two Fashion-MNIST images of different classes (a trouser and a bag), from left to right: the input image; its d=8d=8 principal-component features x\boldsymbol{x} ; the learned probability vector p(x)\boldsymbol{p}(\boldsymbol{x}) over the 28=2562^{8}=256 computational-basis states [Eq. ( 7 )]; and the induced mixed state ρ(x)=diag⁡(p(x))=∑jpj∣j⟩ ⁣⟨j∣\rho(\boldsymbol{x})=\operatorname{diag}(\boldsymbol{p}(\boldsymbol{x}))=\sum_{j}p_{j}\ket{j}\!\bra{j} , which is diagonal in the computational basis. Images of different classes yield distinct probability vectors, and hence distinct mixed states.
Figure 2: Learning curves on the Semeion benchmark ( 88 qubits, 44 layers). Left panel: cross-entropy loss; right panel: classification error ( 1−1- accuracy). Within each panel, solid lines are training and dashed lines validation, both averaged over 10 runs; shaded bands show one standard deviation, and colors distinguish the three models as in the legend.
Figure 3: Learning curves on the Fashion-MNIST benchmark ( 88 qubits, 44 layers). Left panel: cross-entropy loss; right panel: classification error ( 1−1- accuracy). Within each panel, solid lines are training and dashed lines validation, both averaged over 10 runs; shaded bands show one standard deviation, and colors distinguish the three models as in the legend.
Figure 4: Learning curves on the MNIST benchmark ( 88 qubits, 44 layers). Left panel: cross-entropy loss; right panel: classification error ( 1−1- accuracy). Within each panel, solid lines are training and dashed lines validation, both averaged over 10 runs; shaded bands show one standard deviation, and colors distinguish the three models as in the legend.
Figure 5: Test accuracy of the SBQE quantum layer (stars) compared with same-shape classical low-rank maps of p(x)\boldsymbol{p}(\boldsymbol{x}) ( 2nq ⁣→ ⁣nq2^{n_{\mathrm{q}}}\!\to\!n_{\mathrm{q}} ) as a function of the mixing layer’s trainable-parameter count ( nq=8n_{\mathrm{q}}=8 qubits, 4 layers; 10 seeds, best-validation weights). The quantum layer reaches its accuracy with far fewer parameters than a classical map of comparable performance, indicating that the circuit (not the front-end or the PCA reduction) carries the class separation.
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