One of the major challenges hindering advancement of quantum computing is the sensitive nature of the physical systems used to build a quantum computer. One suggestion for improving reliability is a particular type of logic gates, based on Berry's geometric phase, showing improved robustness to external disturbance of the quantum system over the course of a calculation. Such logic gates have previously been shown for the smallest possible two-level qubits. Using the method of adiabatic shortcut we endevour to discover similarly realistic and robust logic gates for units of quantum information in higher dimensions. The example shown in this paper discusses three-level qutrits, but is expected to apply to theoretically unlimited higher dimensions since new geometric complications are expected to arise primarily when moving from a two-level to a multi-level problem.  We here present a set of primitive single-qutrit gates able to perform universal quantum computations if supplemented by a two-qutrit gate. We also present a set of condensed single-qutrit gates for commonly needed operations. By detailing the underlying mathematical framework, relying on the multi-dimensional generalisation of Berry's phase describing the time evolution of degenerate quantum states, we also suggest an easily scalable geometric interpretation of quantum gates in higher dimensions along with visual representation of logic gates using parameters of the physical system to sequentially unlock and manipulate subspaces of the quantum information unit.

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. / Kvantdatorer kan teoretiskt utföra vissa uppgifter som klassiska datorer vid realistiska tider inte kan. Att köra en kvantdator kräver exakt kontroll över systemet, till exempel genom adiabatisk utvecking, och isolering från omgiviningen för att behålla koherens. Denna rapport kombinerar dessa två, något motsägelsefulla, tekniker för felhantering. För att minska körtiden används en övergångsfri kvantkörningsalgoritm, också kallad adiabatisk genväg. Konceptet brusfria delsystem, en generalisering av dekoherensfria underrum, används för robusthet mot sammanflätning med omgivningen genom att skapa logiska kvantbitar som fungerar som en brusfri kod. Vidare tillämpas den adiabatiska genvägen för den brusfria koden på ett spinn-eko för att eliminera den dynamiska fasen, som är känslig för felpropagering, så att endast Berrys fas, som är okänslig för felpropagering, effektivt plockas upp. Motsvarande Hamiltonian härleds uttryckligen för de enda två fallen av tvådimensionella brusfria delsystem: 3 eller 4 kvantbitar med respektive totalspinn j = 1/2 och 0. Detta möjliggör beräkning med en geometrisk kvantdator baserad på en universell en-kvantbitsgrind, som också härleds explicit.