Large-scale Greenberger-Horne-Zeilinger states through a topologically protected zero-energy mode in a superconducting qutrit-resonator chain

We propose a superconducting qutrit-resonator chain model, and analytically work out forms of its topological edge states. The existence of the zero-energy mode enables one to generate a state transfer between two ends of the chain, accompanied with state flips of all intermediate qutrits, based on which $N$-body Greenberger-Horne-Zeilinger (GHZ) states can be generated with great robustness against disorders of coupling strengths. Three schemes of generating large-scale GHZ states are designed, each of which possesses the robustness against loss of qutrits or of resonators, meeting a certain performance requirement of different experimental devices. With experimentally feasible qutrit-resonator coupling strengths and available coherence times of qutrits and resonators, it has a potential to generate large-scale GHZ states among dozens of qutrits with a high fidelity. Further, we show the experimental consideration of generating GHZ states based on the circuit QED system, and discuss the prospect of realizing fast GHZ states.

Figure 1

(a) A superconducting qutrit-resonator chain. The $n\mathrm{th}$ unit cell contains one flux qutrit and one single-mode resonator, labeled as $A_n$ and $B_n$, respectively, and holds an intracell qutrit-resonator coupling strength $J_1$. Between two adjacent cells, a qutrit $A_{n+1}$ is coupled to the resonator $B_n$ with an intercell coupling strength $J_2$. The resonator $B_n$ drives the transition $|L\rangle \leftrightarrow |e\rangle$ ($|R\rangle \leftrightarrow |e\rangle$) of the two nearest-neighbor qutrits $A_n$ and $A_{n+1}$ when $n$ is odd (even). (b) Circuit schematic of one unit cell in the superconducting qutrit-resonator chain. The coupling strength can be dynamically tuned by a coupler of SQUID. The energy level space of the qutrit can be tuned by FBL. (c) Schematics of energy level transitions for qutrits $A_1$, $A_n$ ($1<n<N$), and $A_N$. The energy level structure of a flux qutrit holds two ground states ($|L\rangle$ and $|R\rangle$) and one excited state ($|e\rangle$). We denote that coupling strengths $J_L=J_1$ and $J_R=J_2$ ($J_L=J_2$ and $J_R=J_1$) when $n$ is even (odd).

Figure 2

(a) Energy spectra of the chain versus $J_1/J_2$. (b) Distribution of the zero-energy mode versus $J_1/J_2$. (c) Energy spectra of the chain versus $g_{0}t$. (d) Distribution of the zero-energy mode versus $g_{0}t$. The size of the chain is $L=2N-1=49$.

Figure 3

Time evolution of populations for $|{\varphi }_0\rangle$, $|{\varphi }_{\text{ideal}}\rangle$, $|l_A\rangle$, and $|r_A\rangle$ with the size of the superconducting qutrit-resonator chain $L=49$. We choose $g_{0}T=3600$ as a total evolution time.

Figure 4

(a) Time evolution of ${log}_{10}(1-F)$ for generating $N$-body GHZ states with $N$ from 10 to 60 at intervals of 5. (b) The fitting function and numerical scatters between the number of qutrits and evolution time with $99.9\%$ fidelity. (c) The fidelity of the GHZ state against the coupling strength with disorder $\delta$ corresponding to 11, 18, and 25 qutrits. (d) The fidelity of the GHZ state versus the varying $\delta /g_0$ and $g_{0}t/{10}^3$ for an 11-body GHZ state.

Figure 5

Fidelities for GHZ states versus decay rates of qutrits or resonators for different $N$, where $\gamma$ ($\kappa$) in the legend represents decoherence involving only the qutrit (resonator) decay. The evolution time $T$ is chosen as $g_{0}T=6.9416N^2+2.455N-59.8933$ that ensures a $99.9\%$ fidelity for generating a lossless $N$-body GHZ state.

Figure 6

(a) and (b) Schematics of energy level transitions for qutrits $A_1$ and $A_N$ in Scheme B. (c) and (d) Schematics of energy level transitions for qutrits $A_1$ and $A_N$ in Scheme C. The interaction diagram of other cells is the same as Fig. 2 in both Scheme A and Scheme B. The choice of coupling strengths and driving fields of last qutrits $A_N$ depends on the odevity of $N$.

Figure 7

Distributions of the zero-energy mode for (a) Scheme B and (b) Scheme C, respectively, on component states $|{\psi }_n\rangle$ and $|{\varphi }_n\rangle$. The size of the superconducting qutrit-resonator chain is $L=2N-1=49$.

Figure 8

(a) and (c) for Scheme B and Scheme C, respectively: Time evolution of population for $|{\mathrm{\Phi }}_0\rangle$, $|{\mathrm{\Phi }}_{\text{ideal}}\rangle$, $|{\phi }_0\rangle$, $|{\phi }_{\text{ideal}}\rangle$, and edge states $|l_{B\left(C\right)}\rangle$ and $|r_{B\left(C\right)}\rangle$ for the size of superconducting qutrit-resonator chain $L=2N-1=49$. (b) and (d) for Scheme B and Scheme C, respectively: Fidelity for the GHZ states versus the decay rate $\gamma$ or $\kappa$ of the superconducting qutrit-resonator chain for different number of qutrits, where $\gamma$ ($\kappa$) in the legend represents decoherence involving only the qutrit (resonator) decay. Parameters for (a) and (b): $J_{1\left(2\right)}^{\prime }=20J_{1\left(2\right)}$, ${\mathrm{\Delta }}_{12}=400g_0$, and ${\mathrm{\Omega }}_{1\left(2\right)}=20g_0$.

Figure 9

Fidelity of the GHZ state with the scale of entanglement, $N$, under the feasible coupling strengths (e.g., $g_0/2\pi =50$, 10, and 1 MHz) and coherence times of qutrits and resonators in Scheme A. We choose coherence time of the qutrits and resonators as ${\tau }_a={\tau }_b=1$ ms.