MG-Net: Learn to Customize QAOA with Circuit Depth Awareness

TL;DR

The Mixer Generator Network (MG-Net) is introduced, a unified deep learning framework adept at dynamically formulating optimal mixer Hamiltonians tailored to distinct tasks and circuit depths, highlighting MG-Net's superior performance in terms of both approximation ratio and efficiency.

Abstract

Quantum Approximate Optimization Algorithm (QAOA) and its variants exhibit immense potential in tackling combinatorial optimization challenges. However, their practical realization confronts a dilemma: the requisite circuit depth for satisfactory performance is problem-specific and often exceeds the maximum capability of current quantum devices. To address this dilemma, here we first analyze the convergence behavior of QAOA, uncovering the origins of this dilemma and elucidating the intricate relationship between the employed mixer Hamiltonian, the specific problem at hand, and the permissible maximum circuit depth. Harnessing this understanding, we introduce the Mixer Generator Network (MG-Net), a unified deep learning framework adept at dynamically formulating optimal mixer Hamiltonians tailored to distinct tasks and circuit depths. Systematic simulations, encompassing Ising models and weighted Max-Cut instances with up to 64 qubits, substantiate our theoretical findings, highlighting MG-Net's superior performance in terms of both approximation ratio and efficiency.

Figures (13)

• Figure 1: Mixer Hamiltonian affects the performance of QAOA. (a) The optimization trajectories of QAOA with varied mixer Hamiltonians $H_M$. Given a fixed circuit depth $p$, a tailored $H_M$ (highlighted in pink) can more effectively steer the quantum state towards the exact solution compared to the original $H_M$ used in QAOA. (b) Transition of the effective dimension $d_{eff}$ required in QAOA with increasing $p$. 'ma-QAOA' denotes a case with independent parameters herrman2022multi, contrasted with 'QAOA' where parameters are fully correlated. The orange line denotes the average effective dimension over all samples.
• Figure 2: Framework of MG-Net. (a) Training Phase. Initially (left), the cost estimator is trained to precisely predict QAOA performance for specific problem instances, circuit depths, and mixer Hamiltonians. In the subsequent stage (right), with the cost estimator fixed, the mixer generator is trained through unsupervised learning to derive the optimal mixer Hamiltonian that minimizes the cost estimator's output. (b) Inference Phase. Given a problem $G$ and circuit depth $p$, the mixer generator produces a mixer Hamiltonian, subsequently utilized in a QAOA solver to find the solution.
• Figure 3: Structure of cost estimator and mixer generator. (a) Cost estimator. The cost estimator is comprised of three distinct branches, each dedicated to processing different types of data: the original problem, the candidate mixer Hamiltonian, and the circuit depth. Their outputs are then integrated to predict the cost value achievable by the QAOA circuit. (b) Mixer generator. The mixer generation is divided into two distinct parts: operator type generation and parameter grouping generation. The former is executed as a node classification task, while the latter is approached as a link prediction task.
• Figure 4: Behavior of cost estimator. (a) The correlation between the estimated cost and the minimum cost for Max-Cut (left) and TFIM (right). Each point represents the result of a problem instance. The dashed line represents that QAOA can find the exact solution $y=x$. (b) The achievable cost under various circuit depth $p$ for Max-Cut (left) and TFIM (right). The label 'CE' is the abbreviation of cost estimator. The dashed lines represent the cost achieved by QAOA, while the solid lines represent the cost estimated by our model.
• Figure 5: The trainability of the quantum circuits generated by MG-Net for Max-Cut and TFIM. (a) The number $\#P$ of trainable parameters of the quantum circuits with mixer Hamiltonian predicted by MG-Net. (b) Comparison of the effective dimension $d_{\mathop{\mathrm{eff}}\nolimits}$ of quantum circuits in standard QAOA and MG-Net driven QAOA (labeled as 'Ours'). The green and grey solid lines denote the average effective dimension $d_{eff}$ of the predicted circuits that can achieve an approximation ratio over $0.995$ for Max-Cut and TFIM, respectively. It assesses circuits achieving an approximation ratio $r$ of at least $0.995$. (c) The convergence of QAOA with FG, NG and mixer Hamiltonian predicted by MG-Net for Max-Cut on $64$-node weighted graphs.

Theorems & Definitions (16)

• Definition 2.1: Effective dimension
• Theorem 3.1: Convergence
• Definition B.1: Dynamical Lie algebra and dynamical Lie group, larocca2022diagnosing
• Definition B.2: Representation of Lie algebra
• Definition B.3: Commutant
• Lemma B.4: The relation between effective dimension and the dimension of DLA
• proof : Proof of Lemma \ref{['lem:eff_dla']}
• Lemma B.5: Convergence, adapted from Corollary 5.4 in you2022convergence
• Lemma B.6
• proof : Proof of Theorem \ref{['thm:main_convergence']}