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arXiv:2511.06354 · Physical Review Letters

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

Yifang Xu, Yilong Zhou, Lida Sun, Hongwei Huang, Zi-Jie Chen, Lintao Xiao, Bo Zhang, Chuanlong Ma, Ziyue Hua, Weiting Wang, Guangming Xue, Haifeng Yu, Weizhou Cai, Chang-Ling Zou, Luyan Sun

quant-phSubmitted Nov 9, 2025• Announced Nov 11, 2025• 0 citations
arXivHTMLPDF

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.

Journal reference
Physical Review Letters 136, 210602 (2026)
DOI
10.1103/yv9x-1sqp
Comments
13 pages, 7 figures, 1 table

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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.
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.
Geometric quantum computation offers a practical strategy toward robust quantum computation due to its inherent error tolerance. However, the rigorous geometric conditions lead to complex and/or error-disturbed quantum controls, especially for logical qubits that involve more physical qubits, whose error tolerance is effective in principle, but their experimental demonstration is still demanding. Thus, how to best simplify the needed control and manifest its full advantage has become the key to widespread applications of geometric quantum computation. Here we propose a fast and robust geometric scheme, with decoherence-free subspace encoding, and present its physical implementation on superconducting quantum circuits, where we only utilize the experimentally demonstrated parametrically tunable coupling to achieve high-fidelity geometric control over logical qubits. Numerical simulation verifies that it can efficiently combine the error tolerance from both the geometric phase and logical-qubit encoding, displaying our gate-performance superiority over the conventional dynamical one without encoding, in terms of both gate fidelity and robustness. Therefore, our scheme can consolidate both error suppression methods for logical-qubit control, which sheds light on the future large-scale quantum computation.
The binomial code is renowned for its parity-mediated loss immunity and loss-error recoverability, while geometric phases are widely recognized for their intrinsic resilience against noise. Capitalizing on their complementary merits, we propose a noise-resilient protocol to realize Nonadiabatic geometric quantum computation with binomial codes in a superconducting system composed of a microwave cavity %off-resonantly dispersively coupled to a %three-level qutrit. The control field %geometric quantum computation is designed by %combining geometric phases, integrating reverse engineering and optimal control. This design provides a customized control protocol featuring strong error-tolerance and inherent noise-resilience. Using experimentally accessible parameters in superconducting systems, numerical simulations show that the protocol yields relatively high average fidelity for geometric quantum gates based on binomial code, even in the presence of parameter fluctuations and decoherence. Thus, this protocol may provide a practical approach for realizing reliable Nonadiabatic geometric quantum computation with binomial codes in current technology.
Geometric gates that use the global property of the geometric phase is believed to be a powerful tool to realize fault-tolerant quantum computation. However, for singlet-triplet qubits in semiconductor quantum dot, the low Rabi frequency of the microwave control leads to overly long gating time, and thus the constructing geometric gate suffers more from the decoherence effect. Here we investigate the key issue of whether the fast geometric gate can be realized for singlet-triplet qubits without introducing an extra microwave-driven pulse, while maintaining the high-fidelity gate operation at the same time. We surprisingly find that both the single- and two-qubit geometric gates can be implemented via only modulating the time-dependent exchange interaction of the Hamiltonian, which can typically be on the order of $\ensuremath{\sim}\mathrm{GHz}$, and thus the corresponding gate time is of several nanoseconds. Furthermore, the obtained geometric gates are superior to their counterparts, i.e., the conventional dynamical gates for singlet-triplet qubits, with a relatively high fidelity surpassing 99%. Therefore our scheme is particularly applied to singlet-triplet qubits to obtain fast and high-fidelity geometric gates. Our scheme can also be extended to other systems without a microwave drive.
We experimentally demonstrate fast and high-fidelity geometric control of a quantum system with the brachistochrone method on hybrid spin registers in diamond. Based on the time-optimal universal geometric control, single geometric gates with fidelities over 99.299.2%99.2 on the spin state of the nitrogen-vacancy center are realized with average durations shortened by 21.521.5%21.5 compared with the conventional geometric method. The fidelity of the fast geometric two-qubit gate exceeds 96.596.5%96.5 on the hybrid spin registers. With the fast and high-fidelity universal set of geometric gates available, we implement a quantum entanglement-enhanced phase estimation algorithm and demonstrate the Heisenberg quantum limit of phase estimation at room temperature. Hence our results show that high-fidelity quantum control based on a fast geometric route will be a versatile tool for broad applications of quantum information processing in practice.
We propose a scheme to realize error-resistant nonadiabatic binomial-code geometric quantum computation using reverse engineering. A strong Kerr nonlinearity restricts the evolution in a computational subspace of the binomial code and a two-photon squeezing drive provides the connections between the logical states. The effective Hamiltonian possesses SU(2) dynamic structure and is analyzed through reverse engineering based on a dynamic invariant. By combining reverse engineering with the optimal control method, we find the evolution paths for nonadiabatic geometric quantum computation and derive the control field robust against the systematic error. Numerical simulations show that the scheme holds excellent resistance to the systematic error and is still well implemented in the presence of resonator leakage with the current superconducting nonlinear resonator technology. Therefore, the scheme may provide a promising approach for accurate nonadiabatic binomial-code geometric quantum computation.