QuantumNAT: Quantum Noise-Aware Training with Noise Injection, Quantization and Normalization

1 INTRODUCTION

• PQC contain trainable weights, and QNNs use networks of parameterized gates to embed data and perform machine-learning tasks on quantum computers.The QNN architecture includes encoding, trainable quantum, and measurement components.Source: §1 · p. 1, Figure 2 · p. 2
• NISQ quantum operations have 10−2 to 10−3 error rates, and QNN accuracy on real hardware falls by up to 64% versus noise-free simulation.The same QNN can achieve different accuracy across devices; IBMQ-Yorktown has five times the error rate of IBMQ-Santiago.Source: §1 · p. 1
• Existing mitigation methods do not account for PQC characteristics and are limited to inference, while prior PQC work does not consider noise.QuantumNAT is proposed to optimize robustness in both training and inference.Source: §1 · p. 1
• QuantumNAT combines post-measurement normalization, hardware-noise-model error-gate injection during training, and post-measurement quantization.Normalization addresses measurement-distribution shift; quantization reduces measurement precision to achieve a denoising effect.Source: §1 · p. 1
• Across 8 ML tasks, 5 design spaces, and 6 quantum devices, accuracy improved up to 42%, 43%, and 23% for 2-, 4-, and 10-class tasks.The reported real-hardware accuracies exceed 94%, 80%, and 34% for 2-, 4-, and 10-class tasks, respectively.Source: §1 · p. 2

2 BACKGROUND AND RELATED WORK

• Quantum machine learning includes QNNs using variational circuits; formulations include quantum Boltzmann machines and quantum classifiers [5] [25] [26] [28].QML tasks cited include metric learning, data analysis, and principal component analysis.Source: §2 · p. 2
• Quantum error mitigation includes extrapolation [23], which measures circuits at different error rates, and PQC training with a noisy simulator using reinforcement learning [15].The passage also cites broader error-mitigation techniques [29].Source: §2 · p. 2
• QuantumNAT differs from prior methods by optimizing training and inference, using PQC error tolerance to motivate quantization, and incurring under 2% overhead.The passage contrasts this overhead with methods involving high measurement counts or circuit-complexity costs.Source: §2 · p. 2
• In classical neural networks, low-precision quantization can improve generalization and robustness, while training noise can smooth the loss landscape.The passage also describes quantization as reducing complexity and mitigating potential overfitting through parameter-space sparsification.Source: §2 · p. 2

3.1 Post-Measurement Normalization

• Quantum noise transforms a QNN measurement outcome through input-independent scaling and an input-dependent shift, motivating batch normalization of each qubit’s outcomes.In the small-batch regime, when the input-specific shifts have low variance, their effect is approximated by a shift by their mean; normalization is used during training and inference.Source: §3.1 · pp. 2–3
• Post-measurement normalization reduces the mismatch between noise-free and noisy distributions and improves SNR on each qubit and individual measurement outcome.Unlike Batch Normalization, testing uses its own batch statistics rather than training statistics, and normalization has no trainable affine parameter.Source: §3.1 · p. 3, Figure 4 · p. 3

3.2 Quantum Noise Injection

• Hardware-informed noise injection inserts sampled Pauli error gates during training and emulates readout error using device-specific noise probabilities.The method also explores direct perturbations of outcomes or rotation angles; the paper reports gate insertion performs better than these direct perturbations.Source: §3.2 · pp. 3–4

3.3 Post-Measurement Quantization

• For Fashion-4 on IBMQ-Santiago, quantization reduced MSE from 0.235 to 0.167 and increased SNR from 4.256 to 6.455.The method clips normalized outcomes, uniformly quantizes them, and adds a quadratic penalty to encourage outcomes near quantization centroids.Source: §3.3 · p. 4, Figure 6 · p. 4

4.1 Experiment Setups

• Experiments cover eight classification tasks, six quantum devices, and 8192 shots per experiment; resource limits restrict most image-task tests to 300 test images.The QNN benchmarks use four qubits for 2- and 4-class tasks and ten for 10-class tasks.Source: §4.1 · p. 4

4.2 Main Results

• Table 1 reports that QuantumNAT achieves the highest accuracy on 26 benchmarks, with an average improvement of 22%.The experiments selected noise factor and quantization level using validation loss before testing.Source: §4.2 · p. 4, Table 1 · p. 5
• QuantumNAT improves accuracy in 13 of 16 design-space settings, supporting its applicability across QNN model sizes and circuit designs.The evaluated spaces include ZZ+RY, RXYZ, ZX+XX, and RXYZ+U1+CU3.Source: §4.2 · p. 5, Table 2 · p. 5
• Noise-aware training cost scales linearly with qubit number, and combining extrapolation with normalization further improves QNN accuracy.Normalization and quantization also scale linearly because they operate on measurement outcomes.Source: §4.2 · p. 5, Table 3 · p. 5, Table 4 · p. 5

4.3 Ablation Studies

• Post-measurement normalization consistently increases signal-to-noise ratio across four QNN architectures and three devices on MNIST-4.Table 5 compares accuracy and SNR before and after normalization.Source: §4.3 · p. 5, Table 5 · p. 5
• With quantization, gate insertion beats measurement-outcome perturbation by 11% on average; without quantization, both outperform rotation-angle perturbation.The passage attributes the difference under quantization to direct measurement perturbations being canceled by quantization.Source: §4.3 · p. 5, Figure 7 · p. 6
• On Fashion-4 with IBMQ-Athens, accuracy is best at noise factor 0.2 and five quantization levels; extremes in either setting reduce accuracy.Too few levels limit model capacity, too many weaken denoising, and excessive noise can destabilize training; feature visualizations show normalization expanding the distribution and injection enlarging class margins.Source: §4.3 · p. 6, Figure 8 · p. 6
• Noise injection and quantization each improve accuracy by 9%, while combining them yields a 17% gain.The passage describes the combined result as evidence of synergistic benefits within QuantumNAT.Source: §4.3 · p. 6, Figure 9 · p. 6

• The conclusion focuses on overcoming quantum noise, which the paper identifies as one of two requirements on the road to practical quantum advantages.Source: §5 · p. 6
• The proposed pipeline combines post-measurement normalization, noise injection, and post-measurement quantization.Source: §5 · p. 6
• The authors conclude that robust PQCs may be useful in exploring practical quantum-computing applications.Source: §5 · p. 6

• A qubit carries information as a superposition of two basis states, and an n-qubit system has 2^n basis states.The circuit state is represented by a 2^n-length complex statevector.Source: §A.1 · p. 6
• Quantum gates transform a state encoding input data and trainable parameters, while measurement yields qubit outcomes and their expectation value.The described QNN forms a nonlinear network by cascading gate-and-measurement blocks.Source: §A.1 · p. 6
• Imperfect controls, unwanted qubit interactions, and environmental interference can cause decoherence and gate errors; noise models support realistic simulations [2] [12] [18].Exact modeling is challenging, so approximate strategies based on Pauli/Clifford Twirling have been developed [18] [24].Source: §A.1 · p. 7

• The framework analyzes how to characterize and mitigate quantum-noise effects on quantum neural networks.Source: §A.2 · p. 7
• The analysis assumes a general mixed quantum state, single-qubit measurement output, and fixed but unknown quantum noise.Source: §A.2 · p. 7
• For multi-qubit QNNs, the analysis can be applied qubit by qubit.Source: §A.2 · p. 7

A.2.1 Measurement of Quantum Neural Networks.

• For each qubit, computational-basis outcomes are coded +1 for |0⟩ and −1 for |1⟩, then repeated-shot measurements are averaged.The empirical mean estimates the expectation value.Source: §A.2.1 · p. 7
• The analysis models hardware noise as a state map and assumes its operators have no explicit classical-input dependence when the architecture is fixed.The measurement expectation for each input is the quantity whose noise impact is analyzed.Source: §A.2.1 · p. 7
• Noisy measurement expectation is a linear map of the noiseless expectation, with scale γ and input-dependent shift β_i.The theorem states that such parameters exist for valid quantum noise; the proof identifies the scale and shift terms.Source: §A.2.1 · p. 7

A.3.1 Importance of hardware-specific noise model.

• Best Fashion-2 accuracy came when the training noise model matched the inference device; Yorktown’s gate error was 5× Santiago’s, and model updates may require retraining.The authors identify faster finetuning for a new noise setting as future work.Source: §A.3.1 · p. 7

A.3.2 Compatibility with existing noise-adaptive compilation.

• With noise-adaptive compilation, QuantumNAT still provides over 10% accuracy improvements on MNIST-2.The comparison uses Qiskit optimization level 3, which enables noise-adaptive qubit mapping and instruction scheduling.Source: §A.3.2 · p. 8

A.3.3 Experiments on Fully Quantum Models.

• 7.4% average baseline improvement persisted for fully quantum models across six tasks on two machines.The experiments used two fully quantum models with three and six U3+CU3 layers.Source: §A.3.3 · p. 8

A.3.4 Experiments on Effect of Number of Intermediate Measurements.

• On Santiago, 2 blocks × 3 layers gave the best deployment accuracy; intermediate measurements trade noise mitigation against state collapse, while QuantumNAT is architecture-agnostic.Under the total 6-layer setting, 2 Block × 3 Layer had better accuracy in most cases.Source: §A.3.4 · p. 8

A.3.5 Accuracy Gap between Using Noise Model and Real QC.

• Accuracy gaps between noise-model evaluation and real quantum computers were typically smaller than 5%.Source: §A.3.5 · p. 8

A.3.6 Accuracy improvements comparison as number of classes increases.

• 230% relative accuracy improvement on 10-class tasks exceeded 4-class and 2-class improvements; 4-class improvement also exceeded 2-class improvement.Source: §A.3.6 · p. 8

A.3.7 Experiments on Using Validation Set Statistics for Test Set.

• 0.67 versus 0.65 average accuracy came from test-set versus validation-set statistics across nine benchmarks, supporting validation statistics for small test batches.The authors report similar validation and test-set measurement statistics.Source: §A.3.7 · p. 8

A.4 Hyperparameters for main results

• Table 14 records task-specific noise factors and quantization levels for the main results.Source: §A.4 · p. 8