Exploiting biased noise in variational quantum models

TL;DR

The paper addresses how quantum noise influences variational quantum algorithms (VQAs) on NISQ devices and challenges the default use of noise twirling for mitigation. It develops a Pauli-transfer-matrix (PTM) based channel framework and uses data re-uploading circuits to separate expressivity (Fourier spectrum with frequencies set by $\omega$ from eigenvalue differences $\lambda_j-\lambda_k$) from trainability (gradient magnitudes). The authors find that non-unital, biased noise can preserve or even enhance optimisation by maintaining spectral components and gradient richness, while symmetric Pauli/twirled noise tends to degrade performance; coherent errors can be absorbed via reparameterisation. Numerical results on a variational eigensolver for the transverse-field Ising model corroborate that biased noise yields better ground-state energies than twirled channels, suggesting noise-tailoring as a design principle for VQAs on real hardware.

Abstract

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.

Figures (7)

• Figure 1: (a) Sampled data points from a degree-three truncated Fourier series, used as the target function for supervised learning. (b) Ideal (noiseless) single-qubit data re-uploading circuit, where $S$ are data-encoding blocks and $W$ are trainable unitaries with parameters $\theta = \{\theta_1, \theta_2, \ldots\}$. (c) Fourier spectrum of the target function, showing the frequency components the model must learn. (d) Output of the quantum model, $f(x, \boldsymbol{\theta})$. With sufficient training, the model closely reproduces the target function. (e) Schematic of the noise model: quantum channels $\mathcal{N}$ are inserted after each $S$ and $W$ block to study the impact of different noise types on learning performance.
• Figure 2: Impact of noise on model expressivity. (a) Output range of a single-qubit data re-uploading circuit under different noise channels, determined by increasing the amplitude of target functions until the model fails to fit the data. Error bars represent the mean and standard deviation across multiple samples, and lines of best fit were computed using linear regression. The comparison highlights the difference between biased noise (amplitude damping) and more uniform noise (Pauli noise, here simulated as the equivalent of Pauli- and Clifford-twirled amplitude damping), with the former preserving a broader output range and enabling more expressive behaviour. Coherent unitary errors, by contrast, show no degradation in expressivity, as they can be absorbed into the trainable parameters. (b) Upper and lower bounds of the model's output under different noise conditions. The same three example circuits are evaluated across multiple noise types to illustrate how each type deforms the achievable output range. (c) Output range comparison across circuits of increasing depth, where the degree of the target function matches the number of layers. As in (a), error bars denote the mean and standard deviation, and lines of best fit were obtained via linear regression. Results show that deeper circuits amplify the impact of noise, with more uniform channels causing more severe degradation.
• Figure 3: Effect of noise on gradient magnitudes during training. Box plots show distributions of absolute gradients $\left| \frac{\partial \tilde{f}}{\partial \theta_i} \right|$ for a two-layer data re-uploading circuit under varying strengths of amplitude damping and Pauli-twirled amplitude damping. Each distribution uses 10,000 random parameter sets and inputs. Pauli-twirled noise causes a sharp decay in gradient magnitudes, indicating reduced trainability, while amplitude damping preserves a richer gradient profile, enabling more effective optimisation.
• Figure 4: Overview of the VQE setup and the impact of noise on solution quality. (a) Parameterised VQE circuit with $T$ Trotter steps, where $\boldsymbol{\theta}$ denotes the set of trainable parameters and $t$ indexes the individual Trotter steps. The circuit is optimised to minimise the loss function $\mathcal{L}(\boldsymbol{\theta}) = \mathrm{Tr}(H \rho(\boldsymbol{\theta}))$, where $\rho(\boldsymbol{\theta})$ is the pre-measurement density matrix prepared by the circuit and $H$ is the problem Hamiltonian. (b) Implementation of the $R_{ZZ}$ gate acting on qubits $i$ and $i+1$, along with the noise injection strategy, where noise is introduced following each two-qubit gate (CNOT). (c) Effect of noise on VQE solutions for a transverse-field 2D Ising model with periodic boundary conditions. As noise strength increases, solutions deviate from the true ground-state energy ($E_0 = -3.232$) and approach zero. Error bars, shown for all data points, represent the mean and standard deviation of the relative error across 100 samples; most are too small to be visible at this scale. Coherent noise (modelled via a controlled-$Y$ rotation) is absorbed into the trainable parameters, showing no degradation. Amplitude damping approximates the true solution, with one outlier where the optimiser was trapped in a local minimum. Pauli- and Clifford-twirled amplitude damping exhibit exponential decay towards zero, indicating severe quality loss under twirling. Lines of best fit were obtained using either linear regression (for coherent noise) or non-linear curve fitting to an asymptotic decay model of the form $y = a e^{-b x} + c$ (for decoherence-based noise channels).
• Figure 5: Effect of noise on gradient magnitudes during training. Box plots show the distribution of absolute gradient values, $\left| \frac{\partial \tilde{f}}{\partial \theta_i} \right|$, for a two-layer quantum circuit subjected to varying strengths of Pauli-twirled and Clifford-twirled amplitude damping noise. Each distribution is computed from 10,000 randomly sampled parameter sets and input values. At higher noise strengths, Pauli-twirled noise yields slightly larger maximum gradients; however, the inner quartiles are consistently higher under Clifford-twirled noise, indicating a more broadly robust gradient landscape for optimisation.