Special-Unitary Parameterization for Trainable Variational Quantum Circuits

Abstract

We propose SUN-VQC, a variational-circuit architecture whose elementary layers are single exponentials of a symmetry-restricted Lie subgroup, $\text{SU}\left(2^{k}\right) \subset \text{SU}\left(2^{n}\right)$ with $k \ll n$. Confining the evolution to this compact subspace reduces the dynamical Lie-algebra dimension from $\mathcal{O}\left(4^{n}\right)$ to $\mathcal{O}\left(4^{k}\right)$, ensuring only polynomial suppression of gradient variance and circumventing barren plateaus that plague hardware-efficient ansätze. Exact, hardware-compatible gradients are obtained using a generalized parameter-shift rule, avoiding ancillary qubits and finite-difference bias. Numerical experiments on quantum auto-encoding and classification show that SUN-VQCs sustain order-of-magnitude larger gradient signals, converge $\mathbf{2} \boldsymbol{-} \mathbf{3} \times$ faster, and reach higher final fidelities than depth-matched Paulirotation or hardware-efficient circuits. These results demonstrate that Lie-subalgebra engineering provides a principled, scalable route to barren-plateau-resilient VQAs compatible with nearterm quantum processors.