The two-dimensional color code is an alternative to the toric code that encodes more logical qubits while maintaining crucial features of the $\mathbb{Z}_2\times\mathbb{Z}_2$ toric code in the long wavelength limit. However its short range physics include single qubit Pauli operations that violate either three or six stabilisers as opposed to the toric code where single qubit Pauli operations violate two or four stabilisers. Exploiting this fact we construct several non-CSS versions of the two-dimensional color code falling into two families - those where either three, four or five stabilisers are violated and those which violate exactly four stabilisers for all the three types of single qubit Pauli operations. These models are not equivalent to the original color code by a local unitary transformation. Nevertheless the code spaces of the CSS and non-CSS versions are related by local unitaries which reflects the fact that their long range physics coincide. This also implies that the non-CSS versions host transversal Clifford gates and hence support fault-tolerant computations. As a consequence of the non-CSS structure, the logical operators are of a mixed type which in some cases include all the three Pauli operators making them potentially useful for protection against biased Pauli noise.
The two-dimensional color code is an alternative to the toric code that encodes more logical qubits while maintaining crucial features of the Z2×Z2 toric code in the long wavelength limit. However its short range physics include single qubit Pauli operations that violate either three or six stabilisers as opposed to the toric code where single qubit Pauli operations violate two or four stabilisers. Exploiting this fact we construct several non-CSS versions of the two-dimensional color code falling into two families - those where either three, four or five stabilisers are violated and those which violate exactly four stabilisers for all the three types of single qubit Pauli operations. These models are not equivalent to the original color code by a local unitary transformation. Nevertheless the code spaces of the CSS and non-CSS versions are related by local unitaries which reflects the fact that their long range physics coincide. This also implies that the non-CSS versions host transversal Clifford gates and hence support fault-tolerant computations. As a consequence of the non-CSS structure, the logical operators are of a mixed type which in some cases include all the three Pauli operators making them potentially useful for protection against biased Pauli 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.
The color code model is a crucial instance of a Calderbank-Shor-Steane (CSS)-type topological quantum error-correcting code, which notably supports transversal implementation of the full Clifford group. Its robustness against local noise is rooted in the structure of its topological excitations. From the perspective of quantum phases of matter, it is essential to understand these excitations in the thermodynamic limit. In this work, we analyze the color code model on an infinite lattice within the quasi-local C∗-algebra framework, using a cone-localized Doplicher-Haag-Roberts (DHR) analysis. We classify its irreducible anyon superselection sectors and construct explicit string operators that generate anyonic excitations from the ground state. We further examine the fusion and braiding properties of these excitations. Our results show that the topological order of the color code is described by Rep(D(Z2×Z2))≃Rep(D(Z2))⊠Rep(D(Z2)), which is equivalent to a double layer of the toric code and consistent with established analyses on finite lattices. We also establish the existence and uniqueness of the KMS state at every finite inverse temperature and derive an explicit description of the corresponding thermal distribution of stabilizer excitations.
Topological Calderbank-Shor-Steane (CSS) codes in more than two spatial dimensions offer features useful for fault-tolerant quantum computation, such as transversal non-Clifford gates and single-shot decoding, but a systematic understanding of them is still lacking. One approach is to decouple a code: map it to a stack of higher-form toric codes by a finite-depth local CSS-preserving Clifford circuit, possibly after adding ancillas. For translation-invariant p-ary CSS codes, where p is prime (p=2 for qubits), we require the circuit to commute with the translations of a superlattice. In two dimensions, such a decoupling exists once these translations act trivially on every anyon type. In higher dimensions, excitations are sorted by their dimension into superselection layers, which can remain glued together even when translations act trivially on each layer. We show that this gluing is the only remaining obstruction: a code can be decoupled on a superlattice exactly when, after coarse-graining to it, translations act trivially on every layer and the gluing vanishes. Moreover, for a code with finitely many excitation types in each layer and m nonzero layers, once translations act trivially on every layer, further coarse-graining by a factor of pm−1 in every direction removes the aforementioned gluing. Physically, the same criterion determines when the code encodes the maximal number of logical qudits on a finite torus. Finally, we give an explicit algorithm that constructs the decoupling as a circuit of SUM (qudit CNOT) and SWAP gates.
Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit X corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of ≈2.5%, close to the ≈2.9% of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for Z errors is required and the non-Clifford gates in the circuit reduce the threshold from ≈2.9% to ≈1.8% with a naive decoder. We show how Z decoding can be improved using knowledge of the X corrections, pushing the threshold to ≈2.2%. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose X stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by CCZ, CS, and T gates.
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.
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.