Improving fermionic variational quantum eigensolvers with Majorana swap networks
Simulating computationally hard fermionic systems is a promising application of quantum computing. However, mapping nonlocal fermionic operators to qubits often produces deep circuits, rendering such simulations impractical on near-term hardware. We introduce two Majorana swap network compilation strategies for variational quantum eigensolvers that reduce circuit depth and two-qubit gate count. First, we develop a cyclic compilation algorithm that localizes all two-particle interaction terms in a general fermionic Hamiltonian containing up to $\mathcal{O}(M^4)$ such terms using only $\mathcal{O}(M^3)$ auxiliary Majorana-swap transpositions, where $M$ is the number of fermionic modes. Here, the cubic scaling refers to auxiliary routing; a complete UCCGSD ansatz still contains $\mathcal{O}(M^4)$ double-excitation rotations. Second, we design a Majorana swap network for the $k$-UpCCGSD variational ansatz, which is already more compact than UCCGSD. In this setting, our network yields constant-factor reductions of approximately $50$ % in circuit depth and $20$ % in two-qubit gate count under all-to-all connectivity. For the more restricted $2\times N$ connectivity, the reductions are larger --- about $55$ % in circuit depth an $40$ % in gate count. These structural improvements are accompanied by improved robustness in numerical noise simulations on the small molecular instances tested.