Protocol for Universal Gates in Optimally Biased Superconducting Qubits

We present a new experimental protocol for performing universal gates in a register of superconducting qubits coupled by fixed on-chip linear reactances. The qubits have fixed, detuned Larmor frequencies and can remain, during the entire gate operation, biased at their optimal working point where decoherence due to fluctuations in control parameters is suppressed to first order. Two-qubit gates are performed by simultaneously irradiating two qubits at their respective Larmor frequencies with appropriate amplitude and phase, while one-qubit gates are performed by the usual single-qubit irradiation pulses.

Figure 1

Superconducting two-qubit circuits for performing universal quantum gates at optimal bias point. (a) Charge qubits (Saclay style) coupled by capacitor. (b) Flux qubits (Delft style) coupled by mutual inductance.

Figure 2

Energy levels of $\mathrm{\text{qubit}}+\mathrm{rf}$ photon systems with (inner levels) and without (outer levels) qubit-photon coupling. Outer: Systems have an infinite ladder of doubly degenerate levels corresponding to products of a photon number state (green, orange) and a qubit state (red, blue). Inner: Photon-qubit coupling lifts degeneracy in each manifold by Rabi frequency ${\omega }_{1,2}^R$. Transitions between adjacent manifolds (wavy arrows) correspond to absorption or emission of a photon from a dressed qubit system. The off-resonant qubits can be put on speaking terms by adjusting Rabi frequencies such that ${\omega }_1^R+{\omega }_2^R=\delta$. Shown: ${\omega }_{1,2}^R=\delta /2$.

Figure 3

(a) Polar representation of pulse sequence for universal two-qubit gate $(Y_1{Y}_2{)}^{1/2}$ using FLICFORQ. Two-qubit pulses (black) have amplitude $\delta /2$ and duration $t^{\mathrm{ent}}$, and are initiated only at times $t_m^{\mathrm{sync}}=4\pi m/\delta$ (gray dashed lines) when the doubly and quadruply rotating frames coincide. One-qubit pulses (gray) have amplitude $(\frac{\delta \pi }{{\omega }^{xx}}\mathrm{mod}2\pi )/t^{\mathrm{sync}}$ and duration $t^{\mathrm{sync}}$. (b) Sample simulation of pulse sequence using the full time-dependent Hamiltonian (1) and parameters ${\omega }_1^L=1.1060$, ${\omega }_2^L=1.0527$, ${\omega }^{xx}=0.003679$. Initial state is $|00⟩$. Simultaneous vanishing of each reduced density operator indicates generation of maximally entangled state. Final state is $(|00⟩-i|11⟩)/\sqrt{2}$; gate fidelity is $>99\%$, with errors due to Bloch-Siegert shift. Plotted are the components of each reduced density operator (${\rho }_i^{\alpha }=⟨{\sigma }_i^{\alpha }⟩$). (c) Schematic experimental set-up for producing pulses to implement universal gates with FLICFORQ.