Quantum-Circuit Algorithms for Many-Body Topological Invariant and Majorana Zero Mode

Topological states of matter are promising for long-term fault-tolerant quantum computing (FTQC), surpassing noisy intermediate-scale quantum (NISQ) devices. However, quantum-circuit (QC) algorithms for probing topological properties are still underdeveloped. We propose three novel QC algorithms to: (i) identify ground states in parity subspaces, (ii) compute the many-body topological invariant, and (iii) visualize zero-energy edge modes [1]. Using the Kitaev chain, a 1D topological superconductor, we demonstrate their effectiveness.

(i) Ground states are located using parity-preserving unitary operators within the variational quantum eigensolver (VQE) framework. (ii) We adapt the Green-function formalism for topological invariants into a QC-compatible time-evolution model, introducing damping and cutoff times. Simulations with the QC simulator (qulacs) show robustness even for NISQ devices. (iii) Majorana zero modes (MZMs) are visualized by calculating the excitation transfer amplitude between parity ground states, confirming edge localization and interchange at topological transitions. These algorithms are applicable to various topological systems, including multi-dimensional ones.