Protected Quantum Computation with Multiple Resonators in Ultrastrong Coupling Circuit QED

We investigate theoretically the dynamical behavior of a qubit obtained with the two ground eigenstates of an ultrastrong coupling circuit-QED system consisting of a finite number of Josephson fluxonium atoms inductively coupled to a transmission line resonator. We show a universal set of quantum gates by using multiple transmission line resonators (each resonator represents a single qubit). We discuss the intrinsic “anisotropic” nature of noise sources for fluxonium artificial atoms. Through a master equation treatment with colored noise and many-level dynamics, we prove that, for a general class of anisotropic noise sources, the coherence time of the qubit and the fidelity of the quantum operations can be dramatically improved in an optimal regime of ultrastrong coupling, where the ground state is an entangled photonic “cat” state.

Figure 1

Description of the considered system. The building block is a superconducting transmission line resonator embedding $N$ Josephson atoms ($N=2$ in the sketch here). By choosing judiciously the type of artificial atom (the depicted circuit represents fluxonium atoms inductively coupled to the resonator), the first two ground levels of the resonator are entangled states ($|\alpha ⟩$ is a photon coherent state, $|\pm ⟩$ is a Josephson junction state “polarized” along the pseudospin $x$ direction). One resonator represents a single qubit: a register of $M$ qubits is given by $M$ resonators.

Figure 2

Coherence time in units of $1/{\omega }_{eg}$ calculated via the master equation (6) for ${\omega }_{eg}={\omega }_{\mathrm{cav}}$ and with the initial state $|{\Psi }_0⟩=\cos (\theta )|{\Psi }_E⟩+\sin (\theta )e^{i\varphi }|{\Psi }_G⟩$. Results are averaged over the possible initial values for $\theta$ and $\varphi$. Left-hand panel: Coherence time versus the normalized vacuum Rabi frequency ${\Omega }_0/{\omega }_{eg}$ for one atom ($N=1$) with Josephson loss rates $\{{\Gamma }_x,{\Gamma }_y,{\Gamma }_z\}={\omega }_{eg}\{{10}^{-6},{10}^{-3},{10}^{-3}\}$. The different cavity loss rates, ${\Gamma }_r/{\omega }_{\mathrm{eg}}={10}^{-6},{10}^{-7},0$, correspond [1] to different quality factors $Q={\omega }_{eg}/(4\pi {\Gamma }_r)\simeq {10}^5,{10}^6,\infty$. Inset: The number of photons $⟨n⟩=|\alpha {|}^2=⟨a^{†}a⟩$ is plotted versus ${\Omega }_0/{\omega }_{eg}$ for $N=1$. Top right-hand panel: Coherence time for $N=1$, 2, and 3 atoms for ${\Gamma }_r/{\omega }_{eg}={10}^{-6}$ and with a lower anisotropy in the atomic loss rates: $\{{\Gamma }_x,{\Gamma }_y,{\Gamma }_z\}={\omega }_{eg}\{{10}^{-5},{10}^{-3},{10}^{-3}\}$ as a function of the photonic amplitude $\alpha =\sqrt{N}{\Omega }_0/{\omega }_{eg}$. Bottom right-hand panel: Maximum coherence time as a function of the number of atoms $N$ for different values of ${\Gamma }_x$, hence for different noise anisotropy.

Figure 3

Left-hand panel: Fidelity of the single-qubit $X$ rotation gate (for ${\theta }_x=\pi /2$) versus ${\Omega }_0/{\omega }_{eg}$. Master equation (6) was used with a time-dependent Hamiltonian, $\widehat{H}(t)=\widehat{H}(t=0)+C(t){\widehat{\sigma }}_x^1$, with $\widehat{H}(t=0)$ the initial spin-boson Hamiltonian (3) with $N=1$ (black solid line), $N=2$ (blue dashed line), and $N=3$ [gray (red) solid line] Josephson atoms in the resonator. Inset: Time evolution of $C(t)$. Right-hand panel: Fidelities of the 2-qubit gate $e^{-i{\theta }_{x_{12}}{\widehat{\Sigma }}_{x_1}\otimes {\widehat{\Sigma }}_{x_2}}$ for ${\theta }_{x_{12}}=\pi /2$ with respect to $\frac{{\Omega }_0}{{\omega }_{eg}}$ the vacuum Rabi frequency in each resonator in which there is 1 atom embedded. The 2-qubits coupling constant $C^{12}(t)$ follows the same time evolution as $C(t)$ in the inset. Note that we have included noise in the mutual coupling, via the jump operator ${\widehat{S}}_{x12}={\widehat{\sigma }}_{x,1}^1{\widehat{\sigma }}_{x,2}^1$ and the loss rate ${\Gamma }_{x12}={\omega }_{eg}\times {10}^{-6}$.