Error Threshold for Color Codes and Random Three-Body Ising Models

Abstract

We study the error threshold of color codes, a class of topological quantum codes that allow a direct implementation of quantum Clifford gates suitable for entanglement distillation, teleportation, and faulttolerant quantum computation. We map the error-correction process onto a statistical mechanical random three-body Ising model and study its phase diagram via Monte Carlo simulations. The obtained error threshold of p c ¼ 0:109ð2Þ is very close to that of Kitaev's toric code, showing that enhanced computational capabilities do not necessarily imply lower resistance to noise. DOI: 10.1103/PhysRevLett.103.090501 PACS numbers: 03.67.Pp, 03.67.Lx, 75.40.Mg, 75.50.Lk Protecting quantum states from external noise and errors is central for the future of quantum information technology. Because interaction with the environment is unavoidable, active quantum error-correction techniques based on quantum codes have been devised to restore the damaged quantum states from errors caused by decoherence [1,2]. These approaches are, in general, cumbersome and require many additional quantum bits, thus making the system more error prone. An imaginative and fruitful approach to quantum protection is to exploit topological properties of a system, e.g., by using the nontrivial topology of a surface to encode quantum states at the logical level [3]. Topology is thus considered as a resource, much like entanglement is a resource for quantum information tasks. Topological quantum computation is the combination of these two resources with the aim of winning the battle against decoherence. These topological quantum errorcorrecting codes are instances of stabilizer quantum codes [4], in which errors are diagnosed by measuring certain check operators or stabilizers. In topological codes these check operators are local, which, in practice, is an important advantage. Moreover, error correction has a deep connection to random spin models in statistical mechanics and lattice gauge theories [5].One of the original motivations for introducing surface codes was to achieve error protection at the physical level through energy barriers that would remove the need for external recovery actions. Only the application of strong magnetic fields (compared to the topological coupling) destabilizes the topological phase [6]. However, several studies [5,[7][8][9][10] and a rigorous proof [8] have shown that the toric code (TC) is not stable against thermal excitations, except in four dimensions [5,9].Therefore, the study of active error correction in topological codes [5] is fully justified. Ultimately, the goal is not only to achieve good quantum memories but also to perform quantum computations with them. In this regard, the TC [3] is somehow limited since it allows only for a convenient (transversal) implementation of a limited set of quantum gates: Pauli gates of X and Z type and the CNOT gate. To overcome this limitation, topological color codes (TCCs) have been introduced [11,12]. Using TCCs, it is possible to implement the whole Clifford group of...

Topological subsystem codes

“…This is directly connected to an order-disorder phase transition in a model with random interactions. An analogous transition is observed for the random model that corresponds to color codes [22,33,34]. It is then natural to expect a similar connection in other topological codes, as we describe next.…”

“…Similar mappings exist also for color codes [22], in this case to 2D random 3-body Ising models. In both cases, the CSS structure of these codes is an important ingredient in the constructions: they are subspace codes with S = S X S Z in such a way that S σ is generated by products of σ operators, σ = X, Z.…”

“…To connect the results of the previous section with the works [13,21,22] CSS codes must be considered. These are codes with G = i1 G X G Z for some G σ generated by products of σ operators, σ = X, Z.…”

“…Then [35], in the random model the average of the free energy difference (22) diverges with the system size, [∆ i (K, τ )] Kp → ∞, for p < p c along the Nishimori line. This is exemplified [13] by surface codes and the corresponding random 2d Ising models, where [∆ i (K, τ )] Kp is the domain wall free energy.…”

Qudit color codes and gauge color codes in all spatial dimensions

“…There are many examples of decoding algorithms for 2D qubit color codes that have been developed recently [44][45][46][47][48][49][50], but very little is known for the case of higher qudit and spatial dimensions.…”