Distributionally Robust Variational Quantum Algorithms with Shifted Noise

TL;DR

This work tackles the sensitivity of variational quantum algorithms (VQAs) to real-time shifts in hardware noise by formulating a distributionally robust optimization (DRO) problem. It treats the noise level as a random variable with an unknown PDF and defines a distributional uncertainty set via Maximum Mean Discrepancy, leading to a min-max objective that protects performance against shifted noise. The authors solve the resulting problem with a distributionally robust Bayesian optimization (DRBO) framework that uses Gaussian-process surrogates and LCB acquisitions, and they validate the approach on QAOA for MaxCut and VQE for a 1D Heisenberg model, showing improved robustness under shifted noise while maintaining reasonable performance under the reference scenario. The work offers a principled method to enhance VQA reliability in NISQ devices and provides a foundation for integrating shift-aware optimization with error mitigation and calibration strategies.

Abstract

Given their potential to demonstrate near-term quantum advantage, variational quantum algorithms (VQAs) have been extensively studied. Although numerous techniques have been developed for VQA parameter optimization, it remains a significant challenge. A practical issue is that quantum noise is highly unstable and thus it is likely to shift in real time. This presents a critical problem as an optimized VQA ansatz may not perform effectively under a different noise environment. For the first time, we explore how to optimize VQA parameters to be robust against unknown shifted noise. We model the noise level as a random variable with an unknown probability density function (PDF), and we assume that the PDF may shift within an uncertainty set. This assumption guides us to formulate a distributionally robust optimization problem, with the goal of finding parameters that maintain effectiveness under shifted noise. We utilize a distributionally robust Bayesian optimization solver for our proposed formulation. This provides numerical evidence in both the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE) with hardware-efficient ansatz, indicating that we can identify parameters that perform more robustly under shifted noise. We regard this work as the first step towards improving the reliability of VQAs influenced by shifted noise from the parameter optimization perspective.

Figures (8)

• Figure 1: Overview of the distributionally robust variational quantum algorithms. Given an ideal ansatz and noise model, we assume the noise level is a random variable that can change in real time. We have samples of the noise level variable $\xi$ from a reference distribution, ansatz parameter $\boldsymbol{\theta}$, and the corresponding VQA performance $f(\boldsymbol{\theta},\xi)$. With the shifted noise, the VQA landscape and its optimum $\boldsymbol{\theta}$ can potentially change. Specifically, the optimal $\boldsymbol{\theta}$ under a certain noise level may not perform well under another noise level. Likewise, an optimal $\boldsymbol{\theta}$ under a reference noise level PDF may not perform well under another noise level PDF. To address the landscape shift, we reformulate the parameter optimization problem as a min-max formulation to find a robust parameter $\boldsymbol{\theta}$. In other words, we aim to optimize the performance under the worst-case noise level PDF. We use a distributionally robust Bayesian optimization solver to solve the new parameter optimization formulation, which is still handled by classical computers.
• Figure 2: The results for solving $N=14$$3$-regular graph MaxCut problems via QAOA. The $x$ axis denotes the significance of noise shift, where the noise level PDF is the reference one at $x=0$. The $y$ axis is the expectation of the approximation ratio of the QAOA solution evaluated at different noise PDFs. We report the average result over $10$ non-isomorphic graphs. As we can see, the standard BO-LCB and BO-EI solutions have the best performance under the reference noise. However, under an increasingly shifted noise, the DRBO solution begins to outperform the standard BO solutions. Meanwhile, the BO-Stable solution is over-conservative with respect to the noise. It significantly scarifies the performance under the reference PDF and the slightly shifted PDFs to gain an improvement under significant shifts. These observations are consistent in the experiments with different QAOA depths.
• Figure 3: One example of evolving of solution in different BO algorithms. The $x$ axis is the iterations in a BO algorithm, and the $y$ axis is the expectation of cost function evaluated over noise level at a $\boldsymbol{\theta}$. The evaluated $\boldsymbol{\theta}$ at one iteration is obtained by maximizing the model posterior, which is unnecessary to be the explored $\boldsymbol{\theta}$ at that iteration. Under the reference noise PDF, the BO-LCB algorithm converges to a better solution, while the DRBO converges to a better solution under the shifted noise. The rightmost figure shows the example PDFs of the reference noise level, shifted noise level, and the estimated worst-case noise level from the DRBO algorithm.
• Figure 4: An example of explored $\boldsymbol{\theta}$ in $p=1$ noisy QAOA cost landscape in a $N=8$ MaxCut problem. The optimum $\boldsymbol{\theta}$ differs from the noiseless optimum. Compared to the BO-LCB and BO-Stable, the DRBO explores the parameter space that performs well under the shifted noise.
• Figure 5: More results on the Max-Cut experiments with graph sizes $N=8,10,12$ and QAOA depth $p=1,2,3$. While potentially sacrificing the performance under the reference noise a little, the DRBO solution performs better than standard BO methods under the significantly shifted noise. Meanwhile, BO-Stable solutions are over-conservative.