The state-of-the-art of the Innsbruck trapped-ion quantum computer [1] is briefly reviewed. We present an overview on the available quantum toolbox and discuss the scalability of the approach.
With up to 50 fully controlled ion qubits we perform quantum simulations investigating quantum transport [2] and emerging hydrodynamics features [3]. Employing the quantum toolbox for entanglement-enhanced Ramsey interferometry, we find optimal parameters for quantum metrology [4]. Quantum computers can be protected from noise by encoding the logical quantum information redundantly into multiple qubits using error-correcting codes. Manipulating logical quantum states by imperfect operations requires that all operations on the quantum register obey a fault-tolerant circuit design to avoid spreading uncontrolled errors. We demonstrate a fault-tolerant universal set of gates on two logical qubits in the trapped-ion quantum computer [5].

[1] I. Pogorelov et al., PRX Quantum 2, 020343 (2021)
[2] C. Maier et al., Phys. Rev. Lett. 122, 050501 (2019)
[3] M. K. Joshi et al., Science 376, 720 (2022)
[4] C. D. Marciniak et al., Nature 603, 604 (2022)
[5] L. Postler et al., Nature 605, 675 (2022)