Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage (1703.00590v2)

Abstract: We show how a hyperbolic surface code could be used for overhead-efficient quantum storage. We give numerical evidence for a noise threshold of 1.3% for the {4,5}-hyperbolic surface code in a phenomenological noise model (as compared to 2.9% for the toric code). In this code family parity checks are of weight 4 and 5 while each qubit participates in 4 different parity checks. We introduce a family of semi-hyperbolic codes which interpolate between the toric code and the {4,5}-hyperbolic surface code in terms of encoding rate and threshold. We show how these hyperbolic codes outperform the toric code in terms of qubit overhead for a target logical error probability. We show how Dehn twists and lattice code surgery can be used to read and write individual qubits to this quantum storage medium.

Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage

The paper "Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage" presents a novel approach to quantum error correction using surface codes derived from hyperbolic geometry. The authors explore how hyperbolic surface codes, specifically the $\{4,5\}$-hyperbolic tiling, could be leveraged to improve the efficiency of quantum storage systems. A key result is the demonstrated noise threshold of $1.3\%$ for these codes under a phenomenological noise model, which is notable in comparison to the $2.9\%$ threshold exhibited by toric codes. The work also introduces semi-hyperbolic codes that interpolate between toric and hyperbolic codes in terms of encoding rate and error tolerance.

Overview of the Findings

1. Noise Thresholds and Code Performance:
   • Hyperbolic codes, specifically the $\{4,5\}$ hyperbolic surface code, are shown to have a threshold similar to the toric code when noisy parity checks are considered.
   • The semi-hyperbolic codes, which vary the subdivision of the faces in the hyperbolic tiling, offer a method to balance encoding rates with logical error thresholds.
2. Encoding Efficiency:
   • Hyperbolic surface codes maintain an asymptotically constant rate while achieving a code distance scaled logarithmically ($c_2 \log n$) with the number of physical qubits, rather than the $\sqrt{n}$ scaling found in the toric code.
   • The examination of semi-hyperbolic codes reveals that they outperform toric codes in qubit overhead for the same logical error probability aiming for practical efficiencies.
3. Code Architectures:
   • The paper discusses embedding these hyperbolic surface codes into bilayer configurations or 3D space, acknowledging various architectural constraints and potential implementations.
4. Computational Techniques and Movement in Storage:
   • Novel techniques are explored for reading and writing qubits, including Dehn twists that leverage the properties of hyperbolic surfaces.
5. Implications and Future Directions:
   • This research suggests that with further optimization, hyperbolic and semi-hyperbolic surface codes could offer significant advantages in quantum storage systems by reducing qubit overheads while maintaining comparable error correction capabilities to existing models.
   • The application of these geometric codes in systems with large qubit capacities opens pathways to efficient storage solutions that could robustly support large-scale quantum computation.

Implications for Quantum Computing

These findings underscore the potential for hyperbolic surface codes to redefine the landscape of quantum error correction, suggesting efficiencies in the encoding of logical qubits that could spare computational resources significantly. By implementing semi-hyperbolic variations, researchers have an opportunity to flexibly balance physical qubit overheads with desired noise thresholds—a vital consideration for the scalability of quantum computers.

The practical applicability extends to various hardware implementations where geometric locality constraints can be relaxed, allowing for innovative qubit connection strategies. Future developments in this area are expected to focus on the integration of hyperbolic codes within tandem quantum computing and storage architectures, augmenting computational capacities and reducing error rates.

The theoretical framework established by these models also invites further exploration into the symmetries and mathematical properties that could be exploited for new classes of quantum codes, pushing the boundaries of current capabilities in quantum error correction.