Expressibility and trainability of a two-dimensional pairwise quantum-circuit ansatz

As quantum hardware evolves toward planar 2D layouts, researchers must determine how to design parameterized quantum circuits (PQCs) that effectively utilize this connectivity. This paper investigates whether a native 2D pairwise ansatz—designed to match the physical grid of superconducting processors—outperforms traditional 1D chain and ring-based ansatze in terms of expressibility (the ability to explore Hilbert space) and trainability (the ease of parameter optimization).

The authors construct a 16-qubit 2D pairwise ansatz on a 4x4 grid and compare it against three 1D benchmarks: a 1D pairwise ring, a 1D circular ansatz, and a Shifted Circular Alternating (SCA) ansatz. The comparison is performed by matching the number of repeated rotation-entanglement layers. To evaluate performance, the authors employ three metrics: Kullback–Leibler (KL) divergence and second-order frame potential for expressibility, and gradient variance of a global Pauli-Z-string expectation value as a proxy for trainability.

The 2D ansatz consistently outperforms the 1D variants in shallow circuits (layers L=1 and 2). It achieves the smallest KL divergence and approaches the theoretical lower bound of the second-order frame potential faster than the 1D models, indicating higher expressibility. Regarding trainability, the 2D circuit exhibits smaller gradient variance for the tested observable at depths L=1–4. However, these performance advantages diminish as the circuit depth increases; by L=6, the gradient variances of all four ansatze become statistically indistinguishable.

This work highlights that circuit topology is a critical design parameter for variational quantum algorithms. By aligning the ansatz structure with the native connectivity of the hardware, researchers can achieve greater expressibility in shallow-depth regimes, which is vital for current noisy intermediate-scale quantum (NISQ) devices where minimizing circuit depth is essential to mitigate noise accumulation.