Robust Optimal Quantum Gates for Josephson Charge Qubits

Quantum optimal control theory allows us to design accurate quantum gates. We employ it to design high-fidelity two-bit gates for Josephson charge qubits in the presence of both leakage and noise. Our protocol considerably increases the fidelity of the gate and, more important, it is quite robust in the disruptive presence of $1/f$ noise. The improvement in the gate performances discussed in this work (errors $\sim {10}^{-3}–{10}^{-4}$ in realistic cases) allows us to cross the fault tolerance threshold.

Figure 1

(a) A Cooper pair box can implement a charge qubit when tuned in the regime in which only two charge states are relevant. The box between the qubits represents the coupling which is specified below. (b) An extra Josephson junction (right) or of a capacitance (left).

Figure 2

Error ${\epsilon }_{\min }$ as a function of leakage for two-qubit gates: for Josephson-junction coupling (including a residual capacitive coupling with $E_{cc}/{\tilde{E}}_{JJ}=0.05$), with and without optimization (lower and upper curve, respectively) (left panel); for capacitive coupling, with optimization, as a function of the ratio $E_J^{(1)}/E_{cc}$ (which we use here since the two charging energies in the experiment $E_C^{(i)}$ are different) (right panel). The experimental value $E_J^{(1)}/E_{cc}=0.47$ from Ref. 8 is marked.

Figure 3

Optimized gate error ${\epsilon }_{\min }$ under noise having power spectrum $S_{n_g}(\omega )=A/\omega$ as a function of noise strength $A$: for Josephson coupling with $E_J/E_C=0.05$, with and without optimization (lower and upper curves, respectively) (left panel); for capacitive coupling with $E_J^{(1)}/E_{cc}=0.47$ (right panel). Typical experimental value are around $A\sim {10}^{-5}$.

Figure 4

Gate error ${\epsilon }_{\min }$ as a function of the pulse spectral cutoff ${\omega }_c$ for Josephson coupling with $E_J/E_C=1/20$ (left) and for capacitive coupling with $E_J^{(1)}/E_{cc}=0.47$ (right). Insets: Corresponding optimal pulses.