## Geometrical quantum computation in Josephson nanocircuits

In most of the implementations proposed so far quantum gates are obtained by varying in time in a controlled way the Hamiltonian of the individual qubits as well as their mutual coupling. An alternative design makes use of quantum geometric phases, obtained by adiabatically varying the qubits' Hamiltonian in such a way to describe a suitably chosen closed loop in its parameter space. Geometric quantum computation may offer considerable advantages since it may be intrinsically fault-tolerant. Area preserving errors do not change the accumulated Berry phases around a given closed loop in the parameter space and therefore will not affect one- and two-qubit gates. A first proposal to implement geometric quantum computation has been put forward theoretically and verified experimentally in NMR quantum computation. The possibility to measure Berry phases in super-conducting nano-circuits has been put forward in G. Falci et al. Nature 407, 355 (2000).

The proposed set-up consists of a super-conducting electron box formed by an asymmetric SQUID, pierced by a magnetic flux and with an applied gate voltage. As in the conventional charge qubit, the device operates in the charging regime. When restricted to the space spanned by two charge states |0>, |1> differing by one Cooper pair, the Hamiltonian takes the form of a spin-1/2 in an external fictitious magnetic field pointing in an arbitrary direction. The possibility to have an asymmetric SQUID is crucial to perform non-trivial loops in the parameter space.

Non-Abelian phases can also appear in the quantum dynamics of super-conducting nano-circuits, this is what was the result of this part of the work of the consortium. There are various interesting aspects associated with this analysis. In addition to their possible detection, which is intriguing by itself, the existence of non-Abelian holonomies in superconducting nanocircuits leads to a new scheme for adiabatic charge pumping and allows to implement solid state holonomic quantum computation. The adiabatic manipulation of degenerate subspaces and the degeneracy condition itself is non-trivial to achieve for an artificially fabricated device.

In the work by Faoro et al. the network proposed to realize solid-state non-abelian quantum computation consists of three super-conducting islands each of which is connected to a fourth island. Gate voltages are applied to the three bottom islands via gate capacitances. Also in this case the device operates in the charging regime each coupling is designed as a Josephson interferometer (a loop interrupted by two junctions and pierced by a magnetic field). Thus the effective Josephson energies can be tuned by changing the flux in the corresponding loop. Electrostatic energies can be varied by changing the gate voltages. One-qubit operation can be performed by changing the magnetic fluxes adiabatically. In some particular case, this adiabatic manipulation corresponds to a charge pumping between different parts of the super-conducting nano-circuit. When the two qubits (realize with the network described above) are connected via a Josephson junction, it is possible to realize a two-qubit operation as well.

Some caution is required to apply this scheme. In an experimental realization it will be difficult to achieve perfect degeneracy of all states. Thus the question is imposed to which extend incomplete degeneracy of the qubit states is permissible. The adiabatic condition requires the inverse operation time to be smaller than the minimum energy difference to the neighbouring states. There is another important constraint on the operational time. As the degenerate states are different from the ground state of the system, the time must not be too large in order to prevent inelastic relaxation.

The work by Cholashinski, motivated also by the recent experiment with Josephson-junction system composed of two coupled charge qubits, consider a different design as a potential candidate for observation of quantum holonomies. As compared to the previous proposal, where the simplest two-dimensional holonomies, are constructed using four coupled charge qubits, there is a simplification since only two qubits are employed. Moreover the transformations are realized within a twofold degenerate ground state, rather than excited state. In this way we avoid the problem of depopulation of the subspace, mentioned before, is alleviated. Assuming the perfect performance not affected by the noise one may rely on a quite simple design without strong constraints on the system parameters. Further studies in this second set-up require the implementation of two-qubit gates. Finally one may also use the device to study the process of the adiabatic charge transport.

The proposed set-up consists of a super-conducting electron box formed by an asymmetric SQUID, pierced by a magnetic flux and with an applied gate voltage. As in the conventional charge qubit, the device operates in the charging regime. When restricted to the space spanned by two charge states |0>, |1> differing by one Cooper pair, the Hamiltonian takes the form of a spin-1/2 in an external fictitious magnetic field pointing in an arbitrary direction. The possibility to have an asymmetric SQUID is crucial to perform non-trivial loops in the parameter space.

Non-Abelian phases can also appear in the quantum dynamics of super-conducting nano-circuits, this is what was the result of this part of the work of the consortium. There are various interesting aspects associated with this analysis. In addition to their possible detection, which is intriguing by itself, the existence of non-Abelian holonomies in superconducting nanocircuits leads to a new scheme for adiabatic charge pumping and allows to implement solid state holonomic quantum computation. The adiabatic manipulation of degenerate subspaces and the degeneracy condition itself is non-trivial to achieve for an artificially fabricated device.

In the work by Faoro et al. the network proposed to realize solid-state non-abelian quantum computation consists of three super-conducting islands each of which is connected to a fourth island. Gate voltages are applied to the three bottom islands via gate capacitances. Also in this case the device operates in the charging regime each coupling is designed as a Josephson interferometer (a loop interrupted by two junctions and pierced by a magnetic field). Thus the effective Josephson energies can be tuned by changing the flux in the corresponding loop. Electrostatic energies can be varied by changing the gate voltages. One-qubit operation can be performed by changing the magnetic fluxes adiabatically. In some particular case, this adiabatic manipulation corresponds to a charge pumping between different parts of the super-conducting nano-circuit. When the two qubits (realize with the network described above) are connected via a Josephson junction, it is possible to realize a two-qubit operation as well.

Some caution is required to apply this scheme. In an experimental realization it will be difficult to achieve perfect degeneracy of all states. Thus the question is imposed to which extend incomplete degeneracy of the qubit states is permissible. The adiabatic condition requires the inverse operation time to be smaller than the minimum energy difference to the neighbouring states. There is another important constraint on the operational time. As the degenerate states are different from the ground state of the system, the time must not be too large in order to prevent inelastic relaxation.

The work by Cholashinski, motivated also by the recent experiment with Josephson-junction system composed of two coupled charge qubits, consider a different design as a potential candidate for observation of quantum holonomies. As compared to the previous proposal, where the simplest two-dimensional holonomies, are constructed using four coupled charge qubits, there is a simplification since only two qubits are employed. Moreover the transformations are realized within a twofold degenerate ground state, rather than excited state. In this way we avoid the problem of depopulation of the subspace, mentioned before, is alleviated. Assuming the perfect performance not affected by the noise one may rely on a quite simple design without strong constraints on the system parameters. Further studies in this second set-up require the implementation of two-qubit gates. Finally one may also use the device to study the process of the adiabatic charge transport.