QFlowNet: Fast, Diverse, and Efficient Unitary Synthesis with Generative Flow Networks

Quantum unitary synthesis addresses the problem of translating abstract quantum algorithms into sequences of hardware-executable quantum gates. Solving this task exactly is infeasible in general due to the exponential growth of the underlying combinatorial search space. Existing approaches suffer from misaligned optimization objectives, substantial training costs and limited generalization across different qubit counts. We mitigate these limitations by using supervised learning to approximate the minimum description length of residual unitaries and combining this estimate with stochastic beam search to identify near optimal gate sequences. Our method relies on a lightweight model with zero-shot generalization, substantially reducing training overhead compared to prior baselines. Across multiple benchmarks, we achieve faster wall-clock synthesis times while exceeding state-of-the-art methods in terms of success rate for complex circuits.

We demonstrate that AI-driven program synthesis can autonomously discover fundamental strategies for decomposing unitary matrices in photonic networks. By extending DreamCoder to complex-valued linear algebra, the system generates decomposition programs achieving the minimal $N(N-1)/2$ Mach-Zehnder interferometers, distinct from both Reck and Clements architectures. Learned programs encode dimension-agnostic invariants: strategies discovered for $5 \times 5$ matrices generalize to higher dimensions such as $64 \times 64$. The discovered programs encode interpretable, dimension-agnostic construction rules. These rules generalize across matrix sizes without retraining, demonstrating that autonomous program synthesis can serve as a scalable paradigm for algorithm discovery and the automated design of universal unitary operators. Beyond universal decompositions, the system automatically exploits matrix structure to reduce the interferometer count below the universal theoretical bound. For instance, for Householder matrices, it discovers a dimension-independent rule that requires only $2N-3$ MZIs. This achieves linear, rather than quadratic, scaling and generalizes to arbitrary $N$ without retraining. For matrices obtained from the singular value decomposition of sparse matrices, reductions generally increase with sparsity, reaching up to 38% fewer MZIs than the universal theoretical bound $N(N-1)/2$ at 95% sparsity. These MZI reductions translate directly into practical hardware benefits for scalable photonic implementations. Taken together, the system functions as a single unified engine that discovers both universal decomposition rules and matrix-specific optimizations, without being provided with the structural or analytical properties of the input matrices.

Can Large Language Models (LLMs) understand and reason about quantum operators? Despite their remarkable capabilities in mathematics and symbolic reasoning, LLMs remain inherently blind to quantum representations such as unitary matrices. In this work, we take a step toward bridging this gap by introducing an approach that maps unitary operators into the latent space of an LLM, enabling unified modeling over quantum and linguistic inputs. We instantiate this idea on Clifford+T circuit synthesis over a Pauli rotation gate set, where our model achieves results competitive with state-of-the-art methods and scales consistently with training data, with no signs of saturation. Our approach further enables language-conditioned synthesis, allowing gate constraints unseen during training to be specified directly in natural language. This work suggests a path toward quantum--aware foundation models that can natively interpret and reason about quantum operations, which could have broader implications reaching across quantum compilation and algorithm discovery.