Exploiting biased noise in variational quantum models

Variational quantum algorithms (VQAs) are promising tools for demonstrating quantum utility on near-term quantum hardware, with applications in optimisation, quantum simulation, and machine learning. While researchers have studied how easy VQAs are to train, the effect of quantum noise on the classical optimisation process is still not well understood. Contrary to expectations, we find that twirling, which is commonly used in standard error-mitigation strategies to symmetrise noise, actually degrades performance in the variational setting, whereas preserving biased or non-unital noise can help classical optimisers find better solutions. Analytically, we study a universal quantum regression model and demonstrate that relatively uniform Pauli channels suppress gradient magnitudes and reduce expressivity, making optimisation more difficult. Conversely, asymmetric noise such as amplitude damping or biased Pauli channels introduces directional bias that can be exploited during optimisation. Numerical experiments on a variational eigensolver for the transverse-field Ising model confirm that non-unital noise yields lower-energy states compared to twirled noise. Finally, we show that coherent errors are fully mitigated by re-parameterisation. These findings challenge conventional noise-mitigation strategies and suggest that preserving noise biases may enhance VQA performance.

💡 Research Summary

This paper investigates the impact of quantum noise on the classical optimisation loop of variational quantum algorithms (VQAs). While error‑mitigation protocols typically employ twirling to symmetrise noise—converting arbitrary errors into stochastic Pauli or depolarising channels—the authors demonstrate that such symmetrisation can be detrimental in the variational setting.

The authors first consider a universal quantum regression model based on data‑reuploading circuits. These circuits alternate data‑encoding gates (S(x)=e^{-i x H}) with trainable unitaries (W(\theta)) and are known to exactly represent any truncated Fourier series when the number of layers matches the series degree. By vectorising states and observables and using the Pauli‑Transfer‑Matrix (PTM) formalism, the noisy circuit is expressed as a sequence of super‑operators interleaved with noise channels (\mathcal N). This representation isolates the effect of noise on the Fourier coefficients (c_\omega(\theta)) that determine the model output (f(x,\theta)=\sum_\omega c_\omega(\theta) e^{i\omega x}).

Analytically, the authors show that uniform Pauli channels—exactly what twirling produces—multiply every coefficient by the same attenuation factor. Consequently, the output range (\mathrm{Range}(\langle M\rangle)) shrinks and the gradient magnitudes (|\partial f/\partial \theta_i|) are uniformly suppressed, leading to pronounced barren‑plateau behaviour. In contrast, non‑unital, asymmetric channels such as amplitude‑damping or biased Pauli noise introduce direction‑dependent attenuation. Some Fourier components (typically low‑frequency ones) retain larger amplitudes, while others decay more strongly. This asymmetry creates a bias in the loss landscape that classical gradient‑based optimisers can exploit, yielding larger gradient norms and a more favourable optimisation trajectory.

The paper also proves that coherent (unitary) errors can be fully absorbed by re‑parameterising the circuit, i.e., redefining the trainable angles to incorporate the unitary error. Hence, coherent noise does not fundamentally limit trainability.

Numerical simulations corroborate the theory. For the regression task, circuits subjected to amplitude‑damping noise achieve significantly lower mean‑square loss and higher gradient variance than those subjected to Pauli‑twirled noise. The authors then extend the analysis to a variational eigensolver (VQE) for the transverse‑field Ising model. When the same noise models are applied, amplitude‑damping (non‑unital) noise leads to a smaller percentage error in the estimated ground‑state energy compared with twirled Pauli noise, confirming that the directional bias of the noise can be beneficial.

Overall, the work challenges the prevailing assumption that “more symmetric = better” for error mitigation in VQAs. It suggests that preserving or even engineering biased, non‑unital noise may improve expressivity (the ability to represent a broad class of functions) and trainability (the ease with which gradients can be found). The authors argue that future hardware‑aware VQA design should consider the specific noise bias of the device, possibly tailoring ansätze or initial parameters to align with that bias, rather than indiscriminately applying twirling.

In conclusion, the paper provides a comprehensive theoretical framework (PTM‑based analysis), analytical results on how different noise types affect Fourier coefficients, and extensive numerical evidence that biased noise can be an asset rather than a liability in variational quantum computing. It opens new avenues for noise‑aware algorithm design, targeted error‑mitigation strategies that retain useful bias, and deeper studies of hardware‑specific noise profiles in near‑term quantum processors.