Exploring Entanglement and Parameter Sensitivity in QAOA through Quantum Fisher Information

TL;DR

This work uses Quantum Fisher Information (QFI) to diagnose parameter sensitivity and entanglement structure in QAOA applied to Max-Cut on cyclic and complete graphs (N = 4–10). It compares RX-only and RX-RY mixers across multiple depths and entanglement patterns, revealing that complete graphs yield larger QFI while no configuration reaches the Heisenberg limit; entanglement shifts information from diagonal to off-diagonal terms with diminishing returns after the first stage. A QFI-Informed Mutation (QIm) heuristic leveraging precomputed QFI improves optimization performance over baselines on 7- and 10-qubit instances, demonstrating QFI’s value as a lightweight preconditioner for variational quantum algorithms. Together, these findings establish QFI as both a diagnostic tool and a practical guide for circuit design and parameter-tuning in near-term quantum optimization.

Abstract

Quantum Fisher Information (QFI) can be used to quantify how sensitive a quantum state reacts to changes in its variational parameters, making it a natural diagnostic for algorithms such as the Quantum Approximate Optimization Algorithm (QAOA). We perform a systematic QFI analysis of QAOA for Max-Cut on cyclic and complete graphs with $N = 4 - 10$ qubits. Two mixer families are studied, RX-only and hybrid RX-RY, with depths $p = 2, 4, 6$ and $p = 3, 6, 9$, respectively, and with up to three entanglement stages implemented through cyclic- or complete-entangling patterns. Complete graphs consistently yield larger QFI eigenvalues than cyclic graphs; none of the settings reaches the Heisenberg limit ($4N^2$), but several exceed the linear bound ($4N$). Introducing entanglement primarily redistributes QFI from diagonal to off-diagonal entries: non-entangled circuits maximize per-parameter (diagonal) sensitivity, whereas entangling layers increase the covariance fraction and thus cross-parameter correlations, with diminishing returns beyond the first stage. Leveraging these observations, we propose, as a proof of concept, a QFI-Informed Mutation (QIm) heuristic that sets mutation probabilities and step sizes from the normalized diagonal QFI. On 7- and 10-qubit instances, QIm attains higher mean energies and lower variance than equal-probability and random-restart baselines over 100 runs, underscoring QFI as a lightweight, problem-aware preconditioner for QAOA and other variational quantum algorithms.

Figures (16)

• Figure 1: One-layer QAOA circuits on 4 qubits with RX-only and RX-RY mixers.
• Figure 2: One-layer QAOA circuits on 4 qubits with different entanglement patterns.
• Figure 3: Cyclic and complete 7-node Max-Cut instances used as representative test cases.
• Figure 4: RX-only mixing operators for cyclic max-cut problems with 4, 7, and 10 nodes, using different depths. ME is the maximum eigenvalue of the model and LE is the lowest eigenvalue of the model.
• Figure 5: RX-only mixing operators for complete max-cut problems with 4, 7, and 10 nodes, using different depths. ME is the maximum eigenvalue of the model and LE is the lowest eigenvalue of the model.