Topological order in an exactly solvable 3D spin model (1006.4871v1)

Abstract: We study a 3D generalization of the toric code model introduced recently by Chamon. This is an exactly solvable spin model with six-qubit nearest neighbor interactions on an FCC lattice whose ground space exhibits topological quantum order. The elementary excitations of this model which we call monopoles can be geometrically described as the corners of rectangular-shaped membranes. We prove that the creation of an isolated monopole separated from other monopoles by a distance R requires an operator acting on at least R^2 qubits. Composite particles that consist of two monopoles (dipoles) and four monopoles (quadrupoles) can be described as end-points of strings. The peculiar feature of the model is that dipole-type strings are rigid, that is, such strings must be aligned with face-diagonals of the lattice. For periodic boundary conditions the ground space can encode 4g qubits where g is the greatest common divisor of the lattice dimensions. We describe a complete set of logical operators acting on the encoded qubits in terms of closed strings and closed membranes.

Analysis of Topological Order in a 3D Spin Model

The paper by Bravyi, Leemhuis, and Terhal explores topological quantum order in a 3D generalization of the toric code introduced by Chamon. This model is predicated on a face-centered cubic (FCC) lattice of qubits interacting via nearest-neighbor six-qubit terms. The intricate features of this model revolve around its unique means of expressing topological quantum order and encoding logical information. The elementary excitations, or monopoles, and their peculiar method of creation suggest a fundamentally different dynamical behavior compared to previously investigated models.

Topological Characteristics

The authors present a formalized 3D spin model which expands on the well-regarded 2D toric code. In this 3D setting, monopoles can only be created at the corners of rectangular-shaped membranes, suggesting a profoundly different mechanism of confinement and propagation for excitations compared to string-like operators in 2D systems. Remarkably, they demonstrate that isolated monopoles cannot be created by string-like operators, as their creation necessitates an operator acting on $\Omega(R^2)$ qubits, where $R$ is their separation distance. Consequently, monopoles, dipoles, and quadrupoles (composed of multiple monopoles) form distinct classifications of excitations, each possessing unique topological charges.

Ground State Properties

The degenerate ground space, analyzed under periodic boundary conditions, is found to encode $4g$ qubits, where $g$ is the greatest common divisor of the lattice dimensions. The work includes an innovative classification of logical operators, constructing them as closed string-nets from closed strings and membranes — a fact that is further complicated by the model's unique alignment and rigidity properties of the strings. The configuration results in logical qubits intertwined with the lattice dimensions' parity properties, revealing latent complexities in generating logical operations without resorting to the intervening stabilizer operators.

Implications for Quantum Computing

The model introduces a challenging error correction potential, with the encoded logical qubits safeguarded by a macroscopic energy barrier akin to the 3D surface codes. Notably, the rigid nature of the strings implies an inherent robustness to local perturbations, making them candidates for fault-tolerant quantum computation under certain conditions. The model raises questions about the thermal stability akin to the 4D toric code and whether such states can manifest resilience at non-zero temperatures in realistic quantum devices.

Future Directions

The research sets a noteworthy precedent for further exploration of 3D topological phases, especially considering their applicability in quantum memory devices. Questions remain regarding the extensibility of these frameworks to non-Abelian anyons and possibly crafting similar models with different lattice geometries or non-trivial topological attributes. A pivotal avenue is understanding whether, or how, glassiness and dynamics at non-zero temperatures could lead to feasible long-term quantum storage strategies, potentially paralleling or surpassing the well-noted 2D topological codes.

This paper provides critical insights into reconstructing quantum codes beyond conventional topologies, suggesting an intriguing landscape of possibilities for quantum computational frameworks. Its balanced treatment of the mathematics and physical implications marks a significant advancement in theoretical quantum information science. The discussions and conclusions herein are not only pivotal for fundamental understandings but also potentiate practical implications in designing resilient quantum architectures.