Quantum approximate optimization via learning-based adaptive optimization

TL;DR

QAOA optimization on noisy quantum processors is hampered by local minima and barren plateaus; this work introduces Double Adaptive-Region Bayesian Optimization (DARBO) that uses a Gaussian-process surrogate and two adaptive regions to efficiently navigate the QAOA landscape at depths $p$ and problem size $n$. DARBO is validated across analytic simulations, shot-noise simulations, and superconducting hardware, delivering faster convergence, higher stability, and better final approximation ratios than Adam, COBYLA, and SPSA, with the performance gap increasing under measurement and quantum noise. The end-to-end pipeline includes quantum error mitigation, enabling meaningful optimization on real devices and suggesting a route to practical quantum advantage in classical tasks. The approach is extensible to higher-dimensional variational quantum algorithms and can benefit from further BO advances and compiling techniques.

Abstract

Combinatorial optimization problems are ubiquitous and computationally hard to solve in general. Quantum approximate optimization algorithm (QAOA), one of the most representative quantum-classical hybrid algorithms, is designed to solve combinatorial optimization problems by transforming the discrete optimization problem into a classical optimization problem over continuous circuit parameters. QAOA objective landscape is notorious for pervasive local minima, and its viability significantly relies on the efficacy of the classical optimizer. In this work, we design double adaptive-region Bayesian optimization (DARBO) for QAOA. Our numerical results demonstrate that the algorithm greatly outperforms conventional optimizers in terms of speed, accuracy, and stability. We also address the issues of measurement efficiency and the suppression of quantum noise by conducting the full optimization loop on a superconducting quantum processor as a proof of concept. This work helps to unlock the full power of QAOA and paves the way toward achieving quantum advantage in practical classical tasks.

Figures (4)

• Figure 1: The proof-of-concept workflow for error mitigated QAOA on the superconducting quantum processor with DARBO. We compile and deploy the 5-qubit QAOA program for given objective functions on a 20-qubit real superconducting quantum processor and evaluate the objective value with quantum error mitigation methods. DARBO treats the QEM-QAOA as a black-box, and optimizes the circuit parameters by fitting the surrogate model with constraints. The constraints are provided by the two adaptive regions, which are responsible for surrogate model building and acquisition function sampling, respectively.
• Figure 2: QAOA optimization for MAX-CUT problem on w3R graphs (exact simulations). All the optimizations are performed on $n=16$ w3R graphs. (a)-(e) The optimization trajectories from different optimizers, i.e., Adam (in blue color), COBYLA (in red color), and DARBO (in green color) are plotted versus the number of circuit evaluations. Results on different circuit depths from $p=2$ to $p=10$ are reported, respectively. (f) The final converged approximation gap $1-r$ after sufficient numbers of optimization iterations. For each circuit evaluation, we query the exact expectation of the objective function via numerical simulation. Each line is averaged over five w3R graph instances, where the shaded range shows the standard deviation of the results across different graph instances. For each graph instance, the best optimization result among the 20 independent optimization trials is reported. The error bar in (f) shows the standard deviation across different graph instances.
• Figure 3: QAOA optimization for MAX-CUT problem on w3R graph (simulation with measurement shot noise). All the optimizations are performed on $n=16$ w3R graphs with $p=10$. (a)-(c) The optimization trajectories from different optimizers, i.e., Adam (in blue color), COBYLA (in red color), SPSA (in brown color), and DARBO (in green color) in terms of the number of circuit evaluations. Results for different shot numbers from $\text{shots}=200$ to $\text{shots}=5000$ are reported, respectively. (d) The final converged approximation ratio $1-r$ after sufficient numbers of optimization iterations. For each circuit evaluation, we collect the number of shot measurements to further reconstruct the loss expectation value. Each line is averaged over five w3R graph instances where the shaded range shows the standard deviation of the results across different graph instances. For each graph instance, the best optimization result from 20 independent optimization trials is kept. The error bar in (d) shows the standard deviation across different graph instances.
• Figure 4: Quantum optimization of a five-variable QUBO problem on real quantum hardware. $\text{Measurement shots}=10000$. (a)-(b) show results from two circuit depths $p=1$ and $p=2$ QAOA, respectively. The line is the average optimization trajectory of five independent optimization trials, while the shaded area represents the standard deviation across five independent optimization trials. Averaged loss refers to the expectation value of the problem QUBO Hamiltonian. Raw (in orange color): at each step, we obtain the loss expectation directly from measurement results on the real quantum processor. Mitigation (in blue color): at each step, we obtain the loss expectation from measurement results integrated with QEM techniques. Ideal (in red color): at each step, we obtain the loss from numerical simulation. (c) The success ratio when we run inference on the trained QAOA program, i.e., the probability that we can obtain a correct bitstring answer for the problem on real quantum hardware. The dashed line is the random guess baseline with a probability of $1/16$. We report the best success ratio of the five optimization trials.