Demonstration of Controlled-Phase Gates between Two Error-Correctable Photonic Qubits

To realize fault-tolerant quantum computing, it is necessary to store quantum information in logical qubits with error correction functions, realized by distributing a logical state among multiple physical qubits or by encoding it in the Hilbert space of a high-dimensional system. Quantum gate operations between these error-correctable logical qubits, which are essential for implementation of any practical quantum computational task, have not been experimentally demonstrated yet. Here we demonstrate a geometric method for realizing controlled-phase gates between two logical qubits encoded in photonic fields stored in cavities. The gates are realized by dispersively coupling an ancillary superconducting qubit to these cavities and driving it to make a cyclic evolution depending on the joint photonic state of the cavities, which produces a conditional geometric phase. We first realize phase gates for photonic qubits with the logical basis states encoded in two quasiorthogonal coherent states, which have important implications for continuous-variable-based quantum computation. Then we use this geometric method to implement a controlled-phase gate between two binomially encoded logical qubits, which have an error-correctable function.

Figure 1

Geometric manipulation of a photonic cat state. (a) Schematic of the nonadiabatic AA phase of a qubit. Two successive $\pi$ rotations of the qubit produce a geometric phase $\gamma =\pi +\phi$, where $\phi$ is the angle between the two rotation axes. (b) Experimental sequence to manipulate the cat state. A cavity is dispersively coupled to the qubit and initialized in a cat state $(|0⟩+{|2\alpha ⟩}_c)/\sqrt{2}$ with the help of an ancillary qubit $Q_2$. The AA phase produced by the rotations of $Q_1$ conditional on the cavity’s vacuum state is encoded in the probability amplitude of $|0⟩$, resulting in a phase gate. (c) Measured Wigner function of the cavity state before the phase gate, corresponding to fidelity of 0.980 to the ideal cat state. (d) Wigner function of the cavity state after the gate with $\phi =0$. The slight rotation and deformation of the Wigner function is due to the self-Kerr effect of the cavity. (e) Measured parity of the cavity state as a function of $\phi$ after a displacement $D(-\alpha e^{i\delta })$ for different values of $\delta$. Symbols are experimental data, in excellent agreement with numerical simulations (solid lines).

Figure 2

Quantum process tomography (QPT) of single-cavity geometric phase gates. (a) Experimental sequence. (b) The Pauli transfer process $R$ matrix fidelity as a function of $m$, the number of the Z gate on the cavity state. The insets show the measured $R$ matrices after one and nine Z gates, respectively. A linear fit of the process fidelity decay gives the Z gate fidelity $F_{\mathrm{Z}}=0.987\pm 0.001$. (c) The measured and ideal Pauli transfer $R$ matrices of the S gate and T gate with fidelities $F_S=0.968$ and $F_T=0.964$.

Figure 3

Two-cavity geometric phase gate. (a) A 3D view of device B. A superconducting transmon qubit $Q_3$ at the center couples to two coaxial cavities $S_1$ and $S_2$, which couple to two other individual ancillary transmon qubits $Q_1$ and $Q_2$, respectively. Each of these transmon qubits independently couples to a stripline readout resonator used to perform simultaneous single-shot readout. (b) Schematic of the experimental sequence. (c) Measured individual Wigner functions of storage cavity $S_1$ and $S_2$. When the control cavity $S_2$ prepared in ${|\alpha ⟩}_c$ (${|-\alpha ⟩}_c$), the even cat state $({|\alpha ⟩}_c+{|-\alpha ⟩}_c)/\sqrt{2}$ in target cavity $S_1$ evolves to even (odd) cat state under the two-cavity CZ gate. The slight rotation and deformation of the Wigner functions after gate are due to the Kerr effect of the cavities. (d) Ideal (left) and measured (right) Pauli transfer $R$ matrices of the two-cavity CZ gate with the coherent encoding $\{{|0⟩}_L={|\alpha ⟩}_c,{|1⟩}_L={|-\alpha ⟩}_c\}$. The corresponding process fidelity $F_{\text{CZ\_ED}}$ ($F_{\mathrm{ED}}$) is 0.859 (0.954).

Figure 4

Two-cavity CZ gate with binomial encoding. (a) Measured Pauli transfer $R$ matrix of the two-cavity CZ gate with the binomial encoding. The corresponding process fidelity $F_{\text{CZ\_ED}}$ ($F_{\mathrm{ED}}$) is 0.816 (0.922). (b) The measured and ideal joint Wigner function of the entangled logical Bell state $|{\mathrm{\Phi }}_{+}⟩=({|01⟩}_L+{|10⟩}_L)/\sqrt{2}$ on the Im-Re and Im-Im planes, respectively.