Non-Abelian Holonomies, Charge Pumping, and Quantum Computation with Josephson Junctions

Non-Abelian holonomies can be generated and detected in certain superconducting nanocircuits. Here we consider an example where the non-Abelian operations are related to the adiabatic charge dynamics of the Josephson network. We demonstrate that such a device can be applied both for adiabatic charge pumping and as an implementation of a quantum computer.

Figure 1

(a) Elementary Josephson network for the investigation of non-Abelian holonomies. Note that an asymmetric SQUID loop cannot be switched off completely [16]. Since $J_{\mathrm{M}}=0$ may be desirable for quantum computation, the SQUID is designed such that this condition can be satisfied. (b) Pumping cycle for three islands. Starting from $P=(1/2,1/2)$ and adiabatically following the drawn path, the gate $U=e^{i(\pi /2){\sigma }_y}$ can be achieved.

Figure 2

Geometric realizations of gates $U_1=e^{i{\Sigma }_1|1{⟩}_{\ell \ell }⟨1|}$ (a) and $U_2=e^{i{\Sigma }_2{\sigma }_y}$ (b). The structure (nonzero elements) of the unitary matrices $U_1$ and $U_2$ is determined by the choice of the plane containing the loop and by the starting/ending point of the closed path. Different values of phase ${\Sigma }_1$ (${\Sigma }_2$) can be obtained by varying the area enclosed by loops $C_1$ ($C_2$).

Figure 3

Interqubit coupling for the implementation of the gate $U_3$ and its geometric realization.