In a paper published in the journal PRX Quantum, researchers introduced a fault-tolerant hybrid quantum computation using discrete-variable (DV) and continuous-variable (CV) systems. They developed a hybrid qubit with a bosonic cat code and a single photon, enabling error correction without multiqubit encoding.
Study: Fault-Tolerant Quantum Computation by Hybrid Qubits with Bosonic Cat Code and Single Photons. Image Credit: chittakorn59/Shutterstock.com
Numerical simulations showed their scheme to be more resource-efficient by an order of magnitude compared to previous proposals. This approach, applicable to photonic, superconducting, and trapped-ion systems, demonstrated a higher loss threshold than existing CV and hybrid methods.
Related Work
Past work explored various quantum computing platforms, including photons, superconductors, and trapped ions, each with DV or CV qubit encoding. Challenges in quantum computing include error accumulation from environmental interactions and imperfect operations, which worsen as systems scale.
Quantum error correction is crucial but often requires encoding information in larger Hilbert spaces or multiple qubits. Balancing resource efficiency and fault tolerance remains a significant hurdle across quantum platforms.
Efficient Hybrid Qubits Framework
A hybrid qubit, the H-cat qubit, combines a DV qubit, represented by a single photon, with a CV qubit encoded using a cat code. The cat code is a bosonic quantum error correction (QEC) technique that protects qubits from photon loss by detecting changes in parity. In this setup, the code space is defined by even cat states, ensuring that any photon loss can be identified.
The hybrid qubit's logical basis consists of orthogonal states formed from a combination of the DV photon's polarization and the CV part's even cat states. An alternative hybrid qubit, called the hybrid coherent (H-coh) qubit, encodes the CV part with coherent states. These hybrid qubits can be generated on existing photonic platforms and have been successfully demonstrated in quantum computing and communication systems.
Hybrid fusion operations entangle CV-DV hybrid qubits and are essential for building larger entangled states for quantum computing tasks like measurement-based quantum computation. The fusion process involves performing Bell-state measurements on the qubits' CV and DV components.
Two main schemes, the hybrid ancilla (HA) and single-dual rail (SDR), allow these fusion operations using linear optics and photon-number-resolving detectors. The H-cat qubit benefits from the cat-code's error-correction properties, significantly reducing error rates caused by photon loss.
Specifically, the X error rate is exponentially suppressed as the amplitude of the CV qubit increases, while the Z error rate grows more slowly. The HA scheme generally shows lower error rates than SDR, though SDR is preferred for unambiguous measurements. By optimizing the encoding amplitude, the H-cat qubit achieves superior error suppression compared to the H-coh qubit.
Fault-tolerant quantum computation using hybrid qubits can be designed in circuit-based and measurement-based quantum computing (MBQC) models. In the circuit-based approach, gate operations like Pauli-X, Z-rotations, Hadamard, and controlled-Z are implemented with linear optics and hybrid fusion schemes, which utilize entangled resource states for teleportation-based gate operations.
For MBQC, a Raussendorf-Harrington-Goya (RHG) lattice is constructed with hybrid qubits as resources, combining micro-cluster states through hybrid fusion. This structure achieves high fault tolerance by embedding the surface code and can correct photon loss, enhancing loss thresholds.
Analyzing fault tolerance for hybrid qubits, particularly H-cat and H-coh qubits, shows that H-cat qubits perform better regarding loss thresholds and resource efficiency. The H-cat qubits achieve a threshold of 0.89%, significantly outperforming H-coh qubits at 0.22%.
Resource overhead is also reduced, with H-cat qubits requiring 13 times fewer resources than H-coh qubits to achieve a logical error rate of 10-6. This efficiency stems from the bosonic error correction of the H-cat qubits, which reduces error accumulation, making the hybrid measurement-based quantum computing (MBQC) approach more resource-efficient than existing photonic quantum computing proposals.
Hybrid Quantum Resource Generation
The H-coh pair is a logical qubit for generating resource states in hybrid quantum computation. This state can be prepared using DV-CV entanglement, facilitated by detecting a single photon after a weak beam-splitter interaction. The approach is extended to create a four-component cat code utilizing coherent states. Ancilla qubits are also employed to generate cluster states, requiring the fusion of H-cat pairs and coherent-state qubits.
This hybrid framework can be implemented across various platforms, including superconducting circuits and ion traps, enabling efficient quantum error correction and fault-tolerant operations. By leveraging strong nonlinear interactions between DV and CV qubits, the proposed method enhances the robustness of hybrid quantum systems. Furthermore, successfully integrating these resources is crucial for advancing the scalability of quantum computing technologies.
Conclusion
To sum up, a scheme for fault-tolerant quantum computing based on hybrid qubits combining CV and DV qubits was introduced. This architecture, implemented in photonic platforms, demonstrated significant loss thresholds and resource efficiency enhancements through bosonic cat codes.
The analysis highlighted the proposed method's robustness against photon loss while addressing potential errors in logical gate operations. Overall, this approach paved the way for scalable fault-tolerant quantum computing and offered promising applications in various quantum information-processing tasks.
Journal Reference
Lee, J.et al. (2024). Fault-Tolerant Quantum Computation by Hybrid Qubits with Bosonic Cat Code and Single Photons. PRX Quantum, 5:3. DOI:10.1103/prxquantum.5.030322, https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.5.030322
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