Generative quantum combinatorial optimization by means of a novel conditional generative quantum eigensolver

TL;DR

This work addresses the challenge of applying quantum algorithms to real-world problems in the near term by introducing conditional-GQE, a context-aware quantum circuit generator based on an encoder–decoder Transformer. The approach yields Generative Quantum Combinatorial Optimization (GQCO), which uses graph-encoded Ising inputs and a dataset-free direct preference optimization training regime to generate quantum circuits up to 10 qubits. Empirically, GQCO achieves ~99% accuracy on 1,000 random 3–10-qubit problems and outperforms classical simulated annealing and the standard QAOA in both accuracy and scalability, with shallower, hardware-efficient circuits. On real hardware (IonQ Aria), a GQCO-generated circuit solved a 10-variable max-cut with a single shot, illustrating practical advantages and highlighting areas for improvement such as degeneracy handling and resource-intensive training. The results suggest a scalable, generalizable pathway for AI-assisted quantum circuit design that could accelerate hybrid quantum–classical computation toward fault-tolerant regimes and broader quantum applications.

Abstract

Quantum computing is entering a transformative phase with the emergence of logical quantum processors, which hold the potential to tackle complex problems beyond classical capabilities. While significant progress has been made, applying quantum algorithms to real-world problems remains challenging. Hybrid quantum-classical techniques have been explored to bridge this gap, but they often face limitations in expressiveness, trainability, or scalability. In this work, we introduce conditional Generative Quantum Eigensolver (conditional-GQE), a context-aware quantum circuit generator powered by an encoder-decoder Transformer. Focusing on combinatorial optimization, we train our generator for solving problems with up to 10 qubits, exhibiting nearly perfect performance on new problems. By leveraging the high expressiveness and flexibility of classical generative models, along with an efficient preference-based training scheme, conditional-GQE provides a generalizable and scalable framework for quantum circuit generation. Our approach advances hybrid quantum-classical computing and contributes to accelerate the transition toward fault-tolerant quantum computing.

Figures (12)

• Figure 1: Schematic of differences between VQA, GPT-QE, and conditional-GQE. (a) VQAs such as VQE prepare a parameterized quantum circuit, called ansatz, for each context (i.e., target problem) and optimizes the parameters to minimize the expected value of the observables. (b) GPT-QE optimizes the parameters for each context; however, the parameters are given as weights in a classical neural network instead of being embedded in the quantum circuit. The final results are obtained by sampling circuits from the trained model. In the current version of the algorithm, one needs to retrain the model whenever a new problem is given. (c) This study develops a context-aware quantum circuit generator by using an encoder-decoder structure that enables the model to be conditioned on the problem context. Once trained, the model can be used for any context in the domain and does not necessarily need to be re-trained.
• Figure 2: Overview of generative quantum combinatorial optimization (GQCO). GQCO employs an encoder-decoder Transformer architecture. The target combinatorial optimization problem is represented as a graph derived from the coefficients of the corresponding Ising model. Features are engineered based on domain knowledge, and an encoded feature representation is obtained using a graph neural network. The encoded feature is passed to a decoder Transformer, which sequentially generates token indices and constructs sequences of 1- or 2-qubit quantum gates. The mixture-of-experts (MoE) architecture is used in the model structure to improve the model expressiveness. The solution to the input problem is obtained from the quantum states computed by the generated circuit.
• Figure 3: Performance evaluation of GQCO and two other solvers. (a) Percentage of correct answers of QAOA, SA, and GQCO on 1,000 randomly generated combinatorial optimization problems (3–10 qubits). (b) Runtime required to reach 90% accuracy. The red line represents GQCO, the blue line represents SA, and the gray dashed line represents the brute-force calculations. QAOA is excluded as it did not achieve 90% accuracy. (c) Runtime versus accuracy across problem sizes. As in (b), the red lines correspond to GQCO, the blue lines to SA, and the green lines to QAOA. Gray vertical lines show brute-force execution times; points to the left indicate a faster runtime than brute force. The points for each solver correspond to varying parameter settings: the number of sampling circuits $\{1, 5, 10, 20, 100\}$ for GQCO, the number of sweeps $\{10^2, 10^3, 10^4, 10^5, 10^6, 10^7\}$ for SA, and the number of layers $\{1, 2, 3, 4\}$ for QAOA.
• Figure 4: Cases where the GQCO model failed to identify the correct solution. (a) Heat maps of the Ising Hamiltonian coefficient matrices for four incorrectly solved problems, with diagonal elements representing external fields and off-diagonal elements representing interaction terms. (b) Quantum circuits with the lowest expected energy, selected from 100 circuits generated by the GQCO model for each combinatorial optimization problem. (c) Corresponding quantum states obtained from these circuits. The histograms show observation probability for each computational basis (left y-axis) computed by state vector simulations, while the point plots show the Hamiltonian expectation value (i.e., the cost of the combinatorial optimization problem) computed in each computational basis (right y-axis). The red dot for each plot corresponds to the basis with the lowest expected value, indicating the ground truth.
• Figure 5: Analyses of the generated circuits. (a) Comparison of circuit depth and (b) number of CNOT gates between GQCO-generated and one-layer QAOA circuits for each problem size. (c) A representative example of a generated circuit. Six quantum gates are applied sequentially to the initial state $\ket{000}$ to obtain the final quantum state $e^{-i\frac{\pi}{10}}\ket{001}$. The quantum states at two intermediate stpes are also illustrated.