Benchmarking hybrid digitized-counterdiabatic quantum optimization

Hybrid digitized-counterdiabatic quantum computing (DCQC) is a promising approach for leveraging the capabilities of nearterm quantum computers, utilizing parameterized quantum circuits designed with counterdiabatic protocols. However, the classical aspect of this approach has received limited attention. In this study, we systematically analyze the convergence behavior and solution quality of various classical optimizers when used in conjunction with the digitized-counterdiabatic approach. We demonstrate the effectiveness of this hybrid algorithm by comparing its performance to the traditional QAOA on systems containing up to 28 qubits. Furthermore, we employ principal component analysis to investigate the cost landscape and explore the crucial influence of parametrization on the performance of the counterdiabatic ansatz. Our findings indicate that fewer iterations are required when local cost landscape minima are present, and the SPSA-based BFGS optimizer emerges as a standout choice for the hybrid DCQC paradigm.

Figure 1

Energy as a function of iterations for the $p=1$ layer, $N=10$ qubits. Results show simulator data for 50 iterations using the PS-Adam optimizer comparing QAOA and DCQC ansatz. Ten initial parameters are randomly chosen for each ansatz. The shaded area shows the standard deviation.

Figure 2

Energy as a function of a number of function evaluations for (a) PS-based optimizers, (b) SPS-based optimizers, (c) gradient-free optimizers, and (d) the best ones from the previous group. Comparisons are made between different optimizers on $p=1$, $N=10$ qubits system using fully parameterized DCQC ansatz. Ten random initial parameters are chosen to be the same for each optimizer.

Figure 3

Number of operations as a function of optimizer type for 10 instances of SK model when $p=1$ layer, $N=10$ qubits. The figure shows the counts (iterations, function evaluations) needed to reach the fidelity of 90% using SPSA-BFGS with fully parameterized DC ansatz. The gray lines at the tips of the bars represent the width of each standard error.

Figure 4

Approximation ratio as a function of qubit number. The result represents $p=1$ layered, $N=10$ to $N=28$ qubits ansatz. SPSA-BFGS optimizer is considered for both maQAOA and DCQC ansatz.

Figure 5

Principal component (PC) analysis result on the cost landscape of $p=1$, $N=10$ qubits system. The figure shows how the optimization trajectory goes with different hybrid optimizers on the cost landscape. The $x,y$ axes denote how much the first and second PC will affect the variance of the data. The red and blue dots represent the final and initial points respectively.