Trotter-based simulators of holonomic quantum gates on decoherence-free subspaces and subsystems

In quantum information, one strives to have a system that is resistant against environmental error, while maintaining the ability to adequately control gates. Holonomic Quantum Computation (HQC) is robust against imprecisions in gate control and encoding information into noiseless subsystems prevents the system from decoherence. In this paper we consider encoding information in a Decoherence-Free Subsystem (DFS) which protects against dephasing errors and encoding information in a noiseless subsystem which protects against collective decoherence. Combining these with HQC is theoretically a very logical choice. However, it turns out that the Hamiltonians used to create such systems contain three- and four-qubit interactions, which are generally hard to achieve experimentally. An attempt was made to simulate these Hamiltonians using the Trotter equation with only two-qubit interactions. For both cases, we found that the single-qubit case was achievable, even with a relatively low amount of Trotter steps in the case of the DFS. In the case of the noiseless subsystem, we managed to reduce three-qubit interactions to two-qubit interactions only. However, for the DFS we could not find an implementation that remains inside the DFS during the whole circuit evaluation, losing the advantage of the DFS. Similarly for the noiseless subsystem case, we found the correct four-qubit terms, but together with a noise term that we were not able to cancel using only operators that remain inside the noiseless subsystem. Further investigation in this term is needed. One could look if there is a fundamental limit here, or consider two more practical approaches: perhaps one could strive for a different two-qubit gate. Another solution could be to use gates that leave the DFS/noiseless subspace within a Trotter step and evaluate the error this introduces to the system.