Comparing Machine Learning Strategies for Quantum Noise Reduction

The convolutional autoencoder follows the architecture introduced by (Karan Kendre, 2025) for density-matrix denoising. The model treats each 32×32 complex-valued density matrix as a two-channel image (real and imaginary components), enabling the network to exploit the local spatial patterns created by different quantum noise channels. The encoder consists of a sequence of convolutional blocks with ReLU activations and 2×2 max-pooling, progressively reducing spatial resolution while expanding the channel dimension (1→32→64). The decoder mirrors the encoder through nearest-neighbor upsampling and transposed convolutions, reconstructing the full resolution density matrix. A final sigmoid activation constrains the reconstructed values to a normalized range.

The Transformer autoencoder models each 32×32 complex-valued density matrix as a sequence of token embeddings, where each token corresponds to a row or column vectorized from the matrix’s real and imaginary parts. This architecture allows every element in the input to attend globally to every other element, preserving the non-local structure characteristic of entangled quantum states. The encoder consists of a stack of Transformer encoder layers with multi-head self-attention, enabling the model to capture global correlations across the entire state. The encoded sequence is compressed through a symmetric bottleneck module that projects the embedding dimension down and back up, serving as a latent representation. The decoder comprises Transformer decoder layers that apply both self-attention within the output sequence and cross-attention to the encoder memory. A linear projection and sigmoid activation generate the reconstructed sequence, which is reshaped into a 2D matrix.

The reconstruction loss incorporates the Uhlmann fidelity, the standard measure of similarity between mixed quantum states. For each predicted–target pair of density matrices (\(\hat{\rho}\)) and (\(\rho\)), the fidelity is computed by diagonalizing (\(\hat{\rho}\)), forming its matrix square root, and evaluating the trace of the geometric mean (\(\sqrt{\hat{\rho}}\), \(\rho\), \(\sqrt{\hat{\rho}}\)). This procedure captures discrepancies across eigenvalues and coherences, making it substantially more informative than elementwise losses when evaluating density-matrix reconstructions.

The Uhlmann fidelity is defined as \[ F(\hat{\rho}, \rho) = \left( \mathrm{Tr}\sqrt{\sqrt{\hat{\rho}},\rho,\sqrt{\hat{\rho}}}\right)^2 . \]

The physics-informed loss enforces the basic structural axioms of a valid density matrix during reconstruction. Given a predicted state (\(\hat{\rho}\)), the loss penalizes violations of Hermiticity, trace normalization, and positive semidefiniteness, which are the three core constraints defining legitimate quantum states. Hermiticity is enforced by measuring the deviation between (\(\hat{\rho}\)) and its conjugate transpose, trace preservation is encouraged by penalizing deviations of (\(\mathrm{Tr}(\hat{\rho})\)) from unity, and positivity is promoted by penalizing negative eigenvalues of (\(\hat{\rho}\)). Together, these terms guide the model toward producing physically consistent outputs, irrespective of the noise model or architecture.

Formally, the loss takes the form \[ \mathcal{L}_{\mathrm{phys}}(\hat{\rho}) = \lambda_{\mathrm{herm}} \left|\hat{\rho} - \hat{\rho}^\dagger\right|_2^2 + \lambda_{\mathrm{trace}},\big(\mathrm{Tr}(\hat{\rho}) - 1\big)^2 + \lambda_{\mathrm{psd}},\big| \min(0,, \lambda_i(\hat{\rho})) \big|_2^2 , \]

where (\(\lambda_i(\hat{\rho})\)) are the eigenvalues of (\(\hat{\rho}\)).

Clean states are generated by sampling random pure states and evolving them under randomly drawn single and two qubit unitaries. Each state is then corrupted by a stochastic noise channel that includes a mixture of depolarizing, amplitude damping, and phase damping noise with randomly sampled strengths. This produces paired datasets of noisy inputs and clean targets for supervised training. The full dataset contains one million simulated examples drawn uniformly from twenty noise type and noise level combinations. Each of the four model and loss configurations receives the same training budget of 100,000 circuits, with 80% set aside for training and the other 20% set aside for testing.