Three-dimensional (3D) color codes have advantages for fault-tolerant quantum computing, such as protected quantum gates with relatively low overhead and robustness against imperfect measurement of error syndromes. Here we investigate the storage threshold error rates for bit-flip and phase-flip noise in the 3D color code (3DCC) on the body-centered cubic lattice, assuming perfect syndrome measurements. In particular, by exploiting a connection between error correction and statistical mechanics, we estimate the threshold for 1D stringlike and 2D sheetlike logical operators to be p_{3DCC}^{(1)}≃1.9% and p_{3DCC}^{(2)}≃27.6%. We obtain these results by using parallel tempering Monte Carlo simulations to study the disorder-temperature phase diagrams of two new 3D statistical-mechanical models: the four- and six-body random coupling Ising models.
Three-dimensional (3D) color codes have advantages for fault-tolerant quantum computing, such as protected quantum gates with relatively low overhead and robustness against imperfect measurement of error syndromes. Here we investigate the storage threshold error rates for bit-flip and phase-flip noise in the 3D color code (3DCC) on the body-centered cubic lattice, assuming perfect syndrome measurements. In particular, by exploiting a connection between error correction and statistical mechanics, we estimate the threshold for 1D stringlike and 2D sheetlike logical operators to be p_{3DCC}^{(1)}≃1.9% and p_{3DCC}^{(2)}≃27.6%. We obtain these results by using parallel tempering Monte Carlo simulations to study the disorder-temperature phase diagrams of two new 3D statistical-mechanical models: the four- and six-body random coupling Ising models.
We present and analyze protocols for fault-tolerant quantum computing using color codes. We present circuit-level schemes for extracting the error syndrome of these codes fault-tolerantly. We further present an integer-program-based decoding algorithm for identifying the most likely error given the syndrome. We simulated our syndrome extraction and decoding algorithms against three physically-motivated noise models using Monte Carlo methods, and used the simulations to estimate the corresponding accuracy thresholds for fault-tolerant quantum error correction. We also used a self-avoiding walk analysis to lower-bound the accuracy threshold for two of these noise models. We present and analyze two architectures for fault-tolerantly computing with these codes: one with 2D arrays of qubits are stacked atop each other and one in a single 2D substrate. Our analysis demonstrates that color codes perform slightly better than Kitaev's surface codes when circuit details are ignored. When these details are considered, we estimate that color codes achieve a threshold of 0.082(3)%, which is higher than the threshold of 1.3×10−5 achieved by concatenated coding schemes restricted to nearest-neighbor gates in two dimensions but lower than the threshold of 0.75% to 1.1% reported for the Kitaev codes subject to the same restrictions. Finally, because the behavior of our decoder's performance for two of the noise models we consider maps onto an order-disorder phase transition in the three-body random-bond Ising model in 2D and the corresponding random-plaquette gauge model in 3D, our results also answer the Nishimori conjecture for these models in the negative: the statistical-mechanical classical spin systems associated to the 4.8.8 color codes are counterintuitively more ordered at positive temperature than at zero temperature.
Practical large-scale quantum computation requires both efficient error correction and robust implementation of logical operations. Three-dimensional (3D) color codes are a promising candidate for fault-tolerant quantum computation due to their transversal non-Clifford gates, but efficient decoding remains challenging. In this work, we extend previous decoders for two-dimensional color codes [1], which are based on the restriction of the decoding problem to a subset of the qubit lattice, to three dimensions. Including boundaries of 3D color codes, we demonstrate that the 3D restriction decoder achieves optimal scaling of the logical error rate and a threshold value of 1.55(6)% for code-capacity bit- and phase-flip noise, which is almost a factor of two higher than previously reported for this family of codes [2, 3]. We furthermore present qCodePlot3D, a Python package for visualizing 2D and 3D color codes, error configurations, and decoding paths, which supports the development and analysis of such decoders. These advancements contribute to making 3D color codes a more practical option for exploring fault-tolerant quantum computation.
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 fault-tolerant 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) = 0.109(2) is very close to that of Kitaev's toric code, showing that enhanced computational capabilities do not necessarily imply lower resistance to noise.
In quantum computers, each logical qubit must be encoded in several physical qubits to protect against noise in physical qubits. The tetrahedral color code is such an encoding. In the primal lattice, the three-dimensional tetrahedral color code is a bitruncated octahedral lattice, and in the dual lattice, the tetrahedral color code is a four-colorable body-centered cubic lattice. This color code can be utilized for measurement-based quantum computing, for which all entanglement is present in a cluster state at the beginning of the computation and gate operations are performed by local measurements and classical operations. Moreover, this color code admits a gate set that is both transversal, i.e., realized by qubit-wise operations, and universal when supplemented by measurement and classical computing. During cluster state preparation and computation, errors may occur on the physical qubits. Therefore, it is necessary to have a decoder that, using the syndrome of these errors, finds a set of qubits to correct and performs the operations needed to correct those qubits. The decoder is considered successful if and only if the correction does not cause a logical error on the logical qubit. In this thesis, I present a decoder for Pauli Z, or phase-flip, errors on the three-dimensional tetrahedral color code with nonperiodic boundaries. This decoder for Z-errors includes as a subroutine a decoder for Pauli X errors. The decoder uses a restriction procedure to map the tetrahedral color code to a toric code. The toric code is another quantum error correcting code. Then, an existing toric code decoder interprets the error syndrome. This research on the bounded color code builds upon previous work on the unbounded color code and the two-dimensional color code. Under independently identically distributed noise, evidence indicates an error probability threshold for Z-errors between 0.01% and 0.02% and for X-errors between 2.5% and 3.3%. I also present example errors and illustrate how the decoder attempts to correct those errors.
This thesis is devoted to studying a class of quantum error-correcting codes — topological quantum codes. We explore the question of how one can achieve fault- tolerant quantum computation with topological codes. We treat quantum error-correcting codes not only as a compelling ingredient needed to build a quantum computer, but also as a useful theoretical tool in other areas of physics. In particular, we explore what insights topological codes can provide into challenging questions, such as the classification of quantum phases of matter. In this thesis, we focus on a family of topological codes — color codes, which are particularly intriguing due to the rich physics they display and their computational power. We start by introducing color codes and explaining their basic properties. Then, we show how to perform fault-tolerant universal quantum computation with three-dimensional color codes by transverse gates and code switching. We later compare the resource overhead of the code-switching approach with that of a state distillation scheme. We discuss how to perform error correction with the toric and color codes, as well as introduce local decoders for those two families of codes. By exploiting a connection between error correction and statistical mechanics we estimate the storage threshold error rates for bit-flip and phase-flip noise in the three-dimensional color code. We finish by showing that the color and toric code families in d dimensions are equivalent in a sense of local unitary transformations and explore implications of this equivalence.
We compute the error threshold of color codes, a class of topological quantum codes that allow a direct implementation of quantum Clifford gates, when both qubit and measurement errors are present. By mapping the problem onto a statistical-mechanical three-dimensional disordered Ising lattice gauge theory, we estimate via large-scale Monte Carlo simulations that color codes are stable against 4.5(2)% errors. Furthermore, by evaluating the skewness of the Wilson loop distributions, we introduce a very sensitive probe to locate first-order phase transitions in lattice gauge theories.