LieQAS: Differentiable Quantum Architecture Search via Continuous Lie-Algebraic Parameterization

Quantum Architecture Search (QAS) aims to automate variational circuit design. While classical differentiable Neural Architecture Search (NAS) can continuously relax candidate operations, its standard weighted operation-mixing relaxation does not directly transfer to quantum circuits. Existing differentiable QAS methods therefore introduce differentiability over discrete circuit-operation choices through sampling, categorical relaxation, or mixed-state representations. These mechanisms make architecture optimization indirect: sampling yields stochastic gradient estimates, while surrogate or ensemble objectives may mismatch with the discrete circuit ultimately selected. We propose LieQAS, a differentiable QAS framework that instead parameterizes the searched unitary directly in continuous Lie-algebraic coordinates using Pauli strings. Gradients are taken directly with respect to these Pauli-string coefficients, while the architecture is defined by the active Pauli-string support and updated through progressive pruning. We further introduce hardware-cost-aware regularization and non-commutativity-aware search with singleton finalization, so that the finalized architecture admits exact direct extraction. Experiments on molecular VQE benchmarks show that LieQAS achieves chemical accuracy on all four VQE instances while reducing compiled gate count by up to 64% relative to QuantumDARTS, with lower circuit depth and no more CX gates. On MNIST 0/1 classification, LieQAS reduces CX count by up to 89% while maintaining competitive mean accuracy.