Adiabatic Shortcut to Geometric Quantum Computation in Noiseless Subsystems

Abstract: Quantum computers can theoretically perform certain tasks which classical computers at realistic times could not. Operating a quantum computer requires precise control over the system, for instance achieved by adiabatic evolution, and isolation from the environment to retain coherence. This report combines these two, somewhat contradicting, error preventing techniques. To reduce the run-time a transitionless quantum driving algorithm, or, adiabatic shortcut, is employed. The notion of Noiseless Subsystems (NS), a generalization of decoherence free subspaces, are used for robustness against environmental decoupling, by creating logical qubits which act as a noiseless code. Furthermore, the adiabatic shortcut for the NS code is applied to a refocusing scheme (spin-echo) in order to remove the dynamical phase, sensitive to error propagation, so that only the Berry phase is effectively picked up. The corresponding Hamiltonian is explicitly derived for the only two cases of two-dimensional NS: N=3,4 qubits with total spin of j=1/2,0, respectively. This constitutes geometric quantum computation (GQC) enacting a universal single-qubit gate, which is also explicitly derived.