Single-step implementation of a hybrid controlled-not gate with one superconducting qubit simultaneously controlling multiple target cat-state qubits

Hybrid quantum gates have recently drawn considerable attention. They play significant roles in connecting quantum information processors with qubits of different encoding and have important applications in the transmission of quantum states between a quantum processor and a quantum memory. In this work, we propose a single-step implementation of a multitarget-qubit controlled-not gate with one superconducting (SC) qubit simultaneously controlling $n$ target cat-state qubits. In this proposal, the gate is implemented with $n$ microwave cavities coupled to a three-level SC qutrit. The two logic states of the control SC qubit are represented by the two lowest levels of the qutrit, while the two logic states of each target cat-state qubit are represented by two quasi-orthogonal cat states of a microwave cavity. This proposal operates essentially through the dispersive coupling of each cavity with the qutrit. The gate realization is quite simple because it requires only a single-step operation. There is no need of applying a classical pulse or performing a measurement. The gate operation time is independent of the number of target qubits, thus it does not increase as the number of target qubits increases. Moreover, because the third higher energy level of the qutrit is not occupied during the gate operation, decoherence from the qutrit is greatly suppressed. As an application of this hybrid multitarget-qubit gate, we further discuss the generation of a hybrid Greenberger-Horne-Zeilinger (GHZ) entangled state of SC qubits and cat-state qubits. As an example, we numerically analyze the experimental feasibility of generating a hybrid GHZ state of one SC qubit and three cat-state qubits within present circuit QED technology. This proposal is quite general and can be extended to implement a hybrid controlled-not gate with one matter qubit (of different type) simultaneously controlling multiple target cat-state qubits in a wide range of physical systems, such as multiple microwave or optical cavities coupled to a three-level natural or artificial atom.

Figure 1

Schematic circuit of a hybrid multitarget-qubit gate. Here, $c$ represents the control SC qubit, while $1,2,...,n$ represent the $n$ cat-state qubits. The two logic states of the SC qubit are represented by the two lowest levels $|g\rangle$ and $|e\rangle$ of a three-level SC qutrit, while the two logic states of each target cat-state qubit are represented by the two quasi-orthogonal cat states $|0\rangle$ and $|1\rangle$ [see equation (1) in the text]. In the circuit, the symbol $\oplus$ represents a not gate on each target cat-state qubit. If the control SC qubit is in the state $|e\rangle$, then the state at $\oplus$ for each target cat-state qubit is bit flipped as $|0\rangle \to |1\rangle$ and $|1\rangle \to |0\rangle$. However, when the control SC qubit is in the state $|g\rangle$, the state at $\oplus$ for each target cat-state qubit remains unchanged.

Figure 2

(a) Schematic circuit of $n$ microwave cavities coupled to a SC qutrit. Each square represents a one-dimensional (1D) or three-dimensional (3D) microwave cavity. The circle $A$ represents the SC qutrit, which is inductively or capacitively coupled to each cavity. (b) Illustration of $n$ cavities $(1,2,...,n)$ dispersively coupled to the $|e\rangle \leftrightarrow |f\rangle$ transition of the qutrit. In panel (b), the level spacing between the upper two levels is smaller than that between the two lowest levels, which applies to a SC phase, transmon, Xmon, or fluxonium qutrit. Alternatively, the level spacing between the upper two levels can be larger than that between the two lowest levels, which applies to a SC charge, flux, or fluxonium qutrit, etc.

Figure 3

Schematic of setup of $m$ SC qubits ($1,2,...,m$) and $n$ microwave cavities ($1,2,...,n$). Here, the $m\mathrm{th}$ SC qubit is formed by the two lowest levels of a SC qutrit $A$ [see Fig. 2], which is coupled to the $n$ cavities. Note that the remaining $m-1$ SC qubits ($1,2,...,m-1$) are decoupled from the $n$ cavities. The $m-1$ circles on the left represent the $m-1$ SC qubits ($1,2,...,m-1$), while the rightmost circle represents the $m\mathrm{th}$ SC qubit or the coupler qutrit $A$. Each square represents a 1D or 3D microwave cavity

Figure 4

Diagram of three 1D microwave cavities capacitively coupled to a SC transmon qutrit. Each cavity here is a one-dimensional transmission line resonator. The transmon qutrit consists of two Josephson junctions and a capacitor.

Figure 5

Illustration of three cavities (1, 2, 3) coupled to the $|e\rangle \leftrightarrow |f\rangle$ transition of the qutrit, as well as the unwanted couplings of the three cavities with the $|g\rangle \leftrightarrow |f\rangle$ transition and the $|g\rangle \leftrightarrow |e\rangle$ transition of the qutrit.

Figure 6

Fidelity versus $\delta$ for ${\kappa }^{-1}=45\mu \mathrm{s}$, $60\mu \mathrm{s}$, and $100\mu \mathrm{s}$.

Figure 7

Fidelity versus $c$ for ${\kappa }^{-1}=45\mu \mathrm{s}$, $60\mu \mathrm{s}$, and $100\mu \mathrm{s}$.