Quantum Logic Gates for Coupled Superconducting Phase Qubits

Based on a quantum analysis of two capacitively coupled current-biased Josephson junctions, we propose two fundamental two-qubit quantum logic gates. Each of these gates, when supplemented by single-qubit operations, is sufficient for universal quantum computation. Numerical solutions of the time-dependent Schrödinger equation demonstrate that these operations can be performed with good fidelity.

Figure 1

Capacitively coupled Josephson junctions: (a) ideal circuit diagram (possible experimental parameters include $C_J=6\mathrm{p}\mathrm{F}$, $C_C=60.6\mathrm{f}\mathrm{F}$, and $I_c=21\mu \mathrm{A}$, $J_0=0.988$, ${\omega }_0/2\pi =6.45\mathrm{G}\mathrm{H}\mathrm{z}$); (b) time-dependent ramp of bias currents, specified through the detuning $ϵ$ (see text).

Figure 2

Normalized energy levels, $N_s=4$, $\zeta =0.01$, as a function of detuning parameter $ϵ$: (a) the energy levels $E_1(ϵ)$ and $E_2(ϵ)$ with avoided level crossing at $ϵ=0$; (b) the energy levels $E_3(ϵ)$ through $E_5(ϵ)$ with avoided crossings of 4 and 5 at ${ϵ}_{\pm }\cong \pm 0.04$, and 3 and 4 at $ϵ=0$. The ground state energy $E_0$ (not shown) is approximately $0.981\hslash {\omega }_0$. The natural two-qubit states of this system are $|0;{ϵ}_A)$, $|1;{ϵ}_A)$, $|2;{ϵ}_A)$, and $|4;{ϵ}_A)$, with ${ϵ}_A=-0.1$ (see text).

Figure 3

Entanglement, $N_s=4$, $\zeta =0.01$: The entanglement of the energy eigenstates as a function of detuning parameter $ϵ$: (a) $|1;ϵ)$; (b) $|3;ϵ)$; (c) $|4;ϵ)$; and (d) $|5;ϵ)$.

Figure 4

Dynamical evolution of state with initial condition $|4;{ϵ}_A)$. The probability $P(t)=|(4;{ϵ}_A|\psi (t)⟩{|}^2$ is shown for (a) the phase gate $U_1$ and (b) the swaplike gate $U_2$.

Figure 5

Dynamical evolution of state with initial condition $|1;{ϵ}_A)$ under the swaplike gate. The contours represent the numerically computed wave function (modulus squared) evolving from what is nearly $|10;{ϵ}_A⟩$ to $|01;{ϵ}_A⟩$.