Ground-state geometric quantum computing in superconducting systems

We present a theoretical proposal for the implementation of geometric quantum computing based on a Hamiltonian which has a doubly degenerate ground state. Thus the system which is steered adiabatically, remains in the ground-state. The proposed physical implementation relies on a superconducting circuit composed of three SQUIDs and two superconducting islands with the charge states encoding the logical states. We obtain a universal set of single-qubit gates and implement a nontrivial two-qubit gate exploiting the mutual inductance between two neighboring circuits, allowing us to realize a fully geometric ground-state quantum computing. The introduced paradigm for the implementation of geometric quantum computing is expected to be robust against environmental effects.

Figure 1

(a) Scheme for the superconducting geometric qubit. The gate voltage and the capacitance of the $i$th island are denoted by $V_{g_i}$ and $C_{g_i}$, respectively. The $i$th SQUID is described by Josephson energy $E_i$, capacitance $C_i$, and flux ${\Phi }_i$. The phase across the device is $\phi$. (b) The hexagonal (honeycomb) stability diagram in the $n_{\mathrm{gs}}$-$n_{\mathrm{gd}}$ plane with all the SQUIDs closed ($E_i=0$). In the inset, the loops that generate the logical gates are provided. The trapezoidal loop (solid line $1\to 2\to 3\to 4\to 1$) generates the Hadamard gate and triangular loop (dashed line $1\to 4\to 5\to 1$) generates the $U_2$ transformation. The circle represents the point in which the SQUIDs are closed and the phase across the device is reversed from $\phi$ to $-\phi$ in the implementation of the $U_2$ transformation. Notice that away from the degeneracy lines the SQUIDs are open ($E_i\ne 0$) and the loops represent the projection on the $n_{\mathrm{gs}}$ and $n_{\mathrm{gd}}$ plane of the degeneracy curves in the whole $\{n_{\mathrm{gs}},n_{\mathrm{gd}},E_i\}$ space. Here, we have chosen $k=1/3$ and a large energy gap $\alpha ={\mathrm{Ẽ}}_C$ to visualize the loop better.

Figure 2

Time evolution (normalized by the adiabatic time $T_{\mathrm{ad}}$) of the physical parameters to implement the $U_2$ transformation. (a) The gate parameters $n_{\mathrm{gs}}$ (solid line) and $n_{\mathrm{gd}}$ (dashed line) and (b) the absolute values of SQUID parameters $\left|E_m\right|$ (solid line), $\left|E_L\right|$ (dashed line), and $\left|E_R\right|$ (dot-dashed line) in units of ${\mathrm{Ẽ}}_C$. The blue dot denotes the point, in which the SQUIDs are closed and the phase across the device is reversed $\phi \to -\phi$. In all the plots the energy gap is constant $\alpha =0.1{\mathrm{Ẽ}}_C$ and $k=1/3$.

Figure 3

Circuit scheme for two qubit gate implementation. The circuits are coupled by mutual inductance, $M$. A current $\pm I_1$ circulating in the first circuit corresponds to an additional magnetic flux $\delta {\Phi }_2=\pm M{I}_1$ through the second.