Measurement-based Simulation of Geometric Gates in Topological Qubits on NISQ Devices

While the adiabatic exchange of Majorana zero modes (MZMs) enables a non-universal set of geometrically protected gates, realising an experimental implementation of MZM braiding remains challenging. In an alternative proposal, charge-parity measurement of two neighboring MZMs supports braiding by teleportation. Moreover, owing to the lack of definitive evidence of MZMs in semiconducting systems, there have been several simulations of MZMs on NISQ devices which more naturally lend themselves to braiding. In this work, measurement-based braiding about MZM Y-junctions are simulated by multi-qubit Pauli-parity measurements of a logical qubit. Logical single-qubit geometric $S^{(\dagger)}$-gates and entangling two-qubit gates is shown using two-physical-qubit joint measurements alone, whilst partial phase rotations such as a $T^{(\dagger)}$-gates require at least one three-qubit joint measurement. These relatively small scale circuits offer both novel measurement-based geometric gates as well as a measurement-based demonstration of quantum Hamiltonian simulation.