High-Fidelity Controlled-Phase Gate for Binomial Codes via Geometric Phase Engineering

Abstract

High-fidelity two-logical-qubit gates are essential for realizing fault-tolerant quantum computation with bosonic codes, yet experimentally reported fidelities have rarely exceeded 90\%. Here, we propose a geometric phase engineering approach for implementing controlled-phase gates for binomially encoded logical qubits. This method leverages the structural simplicity of geometric drives to reduce the numerical optimization dimensionality while fully incorporating system nonlinearities, enabling fast and high-fidelity logical operations. As an example, we experimentally demonstrate a process fidelity of 97.4$\pm$0.8\% for a controlled-Z gate between two binomial codes, surpassing all previously reported two-logical-qubit gates in bosonic codes. This work demonstrates that geometric phase engineering provides an effective and experimentally feasible route to fast, high-fidelity logical operations in bosonic quantum processors.

Figures (4)

• Figure 1: Approaches for implementing controlled-phase gate between binomial codes. (a) Geometric phase gate. A frequency-selective drive is applied to the coupler qubit, targeting the $\left|22\right\rangle$ state in the two cavities. This completes a full rotation on the Bloch sphere, imparting a pure geometric phase of $\pi$ to the $\left|22\right\rangle$ component (green). The drive must be sufficiently slow to minimize undesired off-resonant effects on other states (red). (b) Hamiltonian engineering. This general approach implements arbitrary unitary operations, such as the CZ gate, by driving all relevant modes with waveforms optimized via quantum optimal control techniques. (c) Geometric phase engineering. It adopts the geometric phase concept while employing optimization algorithms to further compress the gate duration and achieve higher fidelity. The coupler qubit undergoes a complex evolution and all Fock states accumulate geometric phases. The control is tailored so that the coupler returns to its ground state, yielding a $\pi$ phase only on the target $\left|22\right\rangle$ state (green) while the phases on all other states are strictly zero (red).
• Figure 2: Quantum process tomography (QPT) for performance characterization. (a) Device layout and the QPT sequence. The device consists of two cavities (blue) and three transmon qubits, with the middle qubit serving as the coupler (green) and the two side qubits serving as ancillas (yellow). Measurement of the coupler qubit (gray) is used to filter out unwanted excitations arising from experimental imperfections, thereby improving the gate fidelity. (b) Results for the encode/decode processes. To exclude leakage, the post-selection probability for the coupler remaining in the ground state $\left|g\right\rangle$ is 96.5%. (c) Results for the full encode-CZ-decode sequence. The extracted CZ gate fidelity is 94.9$\pm$0.7%. By post-selecting on the coupler ground state $\left|g\right\rangle$ after the sequence (92.2% success probability), the fidelity is improved to 97.4$\pm$0.8%.
• Figure 3: Preparation of logical Bell state and joint Wigner tomography. (a) Pulse sequence. The logical superposition states $(\left|0_\mathrm{L}\right\rangle+\left|1_\mathrm{L}\right\rangle)/\sqrt{2}$ are first prepared in both cavities, followed by the application of the CZ gate and a Hadamard gate on S$_2$ using $\mathrm{Q_C}$ as the ancilla. Quantum states are characterized using joint Winger tomograph. (b) Joint Wigner tomography of the initial product state. (c) Joint Wigner tomography of the entangled state after applying the CZ gate. (d) Joint Wigner tomography of the final Bell state. Normalization factors are omitted in (b-d) for clarity.
• Figure 4: Error budget of the logical CZ gate. The dominant error source is the dephasing of S$_2$. "Control infidelity" is obtained from the simulation without decoherence, mainly due to imperfection in the optimized control waveform.