Homological Quantum Rotor Codes: Logical Qubits from Torsion (2303.13723v4)

Abstract: We formally define homological quantum rotor codes which use multiple quantum rotors to encode logical information. These codes generalize homological or CSS quantum codes for qubits or qudits, as well as linear oscillator codes which encode logical oscillators. Unlike for qubits or oscillators, homological quantum rotor codes allow one to encode both logical rotors and logical qudits in the same block of code, depending on the homology of the underlying chain complex. In particular, a code based on the chain complex obtained from tessellating the real projective plane or a M\"{o}bius strip encodes a qubit. We discuss the distance scaling for such codes which can be more subtle than in the qubit case due to the concept of logical operator spreading by continuous stabilizer phase-shifts. We give constructions of homological quantum rotor codes based on 2D and 3D manifolds as well as products of chain complexes. Superconducting devices being composed of islands with integer Cooper pair charges could form a natural hardware platform for realizing these codes: we show that the $0$-$\pi$-qubit as well as Kitaev's current-mirror qubit -- also known as the M\"{o}bius strip qubit -- are indeed small examples of such codes and discuss possible extensions.

• The paper introduces a new class of quantum rotor codes that extend error-correction techniques by encoding logical qubits from torsion elements in chain complexes.
• It utilizes a chain complex framework with integer matrix operators to integrate both continuous rotor and discrete qudit logical information.
• Practical implications include potential realization in circuit-QED platforms, suggesting a scalable pathway for noise-resilient quantum computing.

An Expert Overview of "Homological Quantum Rotor Codes: Logical Qubits from Torsion"

The paper "Homological Quantum Rotor Codes: Logical Qubits from Torsion" by Vuillot, Ciani, and Terhal introduces a novel class of quantum codes that extend the traditional concepts associated with quantum error-correction to systems involving quantum rotors. By integrating homological techniques, these rotor codes exhibit distinct properties when compared to established qubit and oscillator codes, particularly in their ability to encode logical information using both rotors and qudits in the same code structure.

The central proposition of the paper is the definition of homological quantum rotor codes, which offer a generalized framework to encode logical qubits from torsion elements within the underlying chain complex. This allows for the encoding of both continuous (rotor) and discrete (qudit) logical information. In contrast to traditional qubit and oscillator codes, the rotor codes described can accommodate logical operators that exhibit different scaling behaviors due to the continuous nature of rotors.

Key contributions are:

1. Chain Complex Framework: The paper defines homological rotor codes on chain complexes with integer coefficients. The chain complex features integer matrix operators, $H_X$ and $H_Z$, satisfying a specific commutation condition ($H_X H_Z^T = 0$), typical in homological constructions. These matrices define the constraints of the code, i.e., the stabilizer group.
2. Encoding Logical Information: Logical information in rotor codes is encoded using both the homology and cohomology of the chain complex, with $k$ logical operators corresponding to logical rotors and qudits dictated by the torsion of the homology group. Crucially, the rotor code can encode more intricate types of logical operations compared to qubit codes, capturing manifold orientation and torsion attributes uniquely.
3. Numerical and Theoretical Results: The authors present a thorough analysis of the distance scaling and resilience of the homological rotor codes via the rotor manifold's homology, linking it to powerful quantum error correction guarantees. Claims are bold yet grounded in mathematical formalism—asserting connections between manifold non-orientability and logical qubit encodings.
4. Implementations in Circuit-QED: Practically, the authors suggest superconducting circuits—specifically islands with integer Cooper pair charges—as a potential realization platform. Highlighted is the realization of small examples, like the $0$-$\pi$ qubit and Kitaev’s M\"obius strip qubit, as manifestations of these codes.
5. Novel Insights and Further Research Paths: The paper opens several avenues for further research, including exploring the construction of codes with system size scaling. The promise of greater scalability and noise resilience positions homological rotor codes as candidates for practical quantum error correction, potentially inspiring new code construction techniques leveraging higher-dimensional topological attributes.

In closing, the paper argues convincingly that homological rotor codes constitute an intriguing bridge between well-studied qubit systems and more complex multi-dimensional quantum systems. Future developments, perhaps exploring greater numerical simulations or new fabrication methodologies in superconducting circuit frameworks, could yield significant practical advances in quantum error correction.