ABS 042: Layer-Resolved Sensitivity and Locality in Finite-Depth Variational Quantum Circuits

Janish Bhat ¹, Angad Juneja Gupta ¹, Zhaoting Hu ¹, George Edakkara ²

¹ UC San Diego
² Birla Institute of Technology and Science, Pilani - Goa Campus

The Van Wickle Journal (2026) Volume 2, ABS042

Introduction: Understanding the propagation of information in variational quantum circuits is essential for assessing their trainability at finite-depth. A variational quantum circuit is a type of parametrized circuit whose loss landscapes vary depending on the circuit depth and entanglement structure. Although prior studies show that factors, such as locality and operator spreading, control how these circuit parameters affect the expected values, these effects are often discussed qualitatively. This study provides a quantitative comparison between the sensitivity of local and global observables in finite-depth variational circuits with simple one-dimensional entanglement gates. When analyzing the gradients across layers, we find that local observables show gradient magnitudes that are consistently 6-8× larger than the global observables at all depths. The global to local gradient norm ratio, which is called the relative depth sensitivity, increases gradually to nearly double its original value over the circuit depth that we have considered in this study. Our findings offer a quantitative insight for locality and propagation of information in quantum variational circuits, which can help inform improvements and new innovations in circuit architecture as well as trainability in near-term quantum algorithms.

Methods: We designed our experiment using a variational quantum circuit with 11 qubits and 10 layers arranged in the one-dimensional nearest neighbor architecture. Each layer has trainable single-qubit rotation parameters and two-qubit entangling gates. In order to base our research around metrics of trainability, we find the layer sensitivity by calculating the gradient norm of the cost function for each layer's parameters. These gradients are computed using many random parameter initializations in order for us to compute mean sensitivity for each layer. In this study, the observables we compared were the local Pauli-Z observable and global mean Pauli-Z observable. This same experimental setup was repeated for CNOT, CZ and iSWAP entangling gates.

Results: We have proven that the gradient magnitudes of local observables for finite-depth VQCs are 6–8× larger than those of global variables across all 10 layers. This shows that local observables show a much stronger training signal. The ratio of global-to-local sensitivity increases gradually to nearly double its original value over the circuit depth considered. This ratio was maximized when iSWAP gates were used and was minimized with CNOT gates.

Discussion: Our results show that trainability in variational quantum circuits depends on circuit depth, measurement locality, and the entangling gate used. The 6–8× advantage that the local observables have shown suggests that designing cost functions based on local observables can help reduce the possibility of barren plateau-like behavior. Future work should mainly focus on applying this idea to larger systems, two-dimensional topologies, and noisy hardware setups to make our research applicable to developing quantum algorithms in the near future.