We introduce a decoder for the 3D color code with boundaries, which is a variation of the restriction decoder introduced by Kubicka and Delfosse. Specifically, we adapt the lift procedure to efficiently find a correction on qubits adjacent to a boundary. We numerically estimate a threshold of $4\% - 8\%$ for Pauli $X$ errors and a threshold of $0.7\% - 0.8\%$ for Pauli $Z$ errors. Our work is a first step towards characterizing the performance of Bombín's recently proposed "colorful quantum computing."
We introduce an efficient decoder of the color code in $d\geq 2$ dimensions, the Restriction Decoder, which uses any $d$-dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-flip and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold $…
Fault-tolerant quantum computation relies on scaling up quantum error correcting codes in order to suppress the error rate on the encoded quantum states. Topological codes, such as the surface code or color codes are leading candidates for practical scalable quantum error correction and require efficient and scalable decoders. In this work, we propose and study the efficiency of a decoder for 2D topological color codes on the 4.8.8 lattice, by building on the work of [1] for color codes on hexagonal lattices. The decoder is based on a rescaling approach, in which syndrome information on a part…
The color code is remarkable for its ability to perform fault-tolerant logic gates. This motivates the design of practical decoders that minimise the resource cost of color-code quantum computation. Here we propose a decoder for the planar color code with a triangular boundary where we match syndrome defects on a nontrivial manifold that has the topology of a Möbius strip. A basic implementation of our decoder used on the color code with hexagonal lattice geometry demonstrates a logical failure rate that is competitive with the optimal performance of the surface code, $\sim p^{α\sqrt{n}}$, wit…
Topological subsystem color codes (TSCCs) are an important class of topological subsystem codes that allow for syndrome measurement with only 2-body measurements. It is expected that such low complexity measurements can help in fault tolerance. While TSCCs have been studied over depolarizing noise model, their performance over the erasure channel has not been studied as much. Recently, we proposed erasure decoders for TSCCs and reported a threshold of 9.7%. In this paper, we continue our study of TSCCS over the erasure channel. We propose two erasure decoders for topological subsystem color co…
Color codes are a class of topological quantum codes with a high error threshold and large set of transversal encoded gates, and are thus suitable for fault tolerant quantum computation in two-dimensional architectures. Recently, computationally efficient decoders for the color codes were proposed. We describe an alternate efficient iterative decoder for topological color codes, and apply it to the color code on hexagonal lattice embedded on a torus. In numerical simulations, we find an error threshold of 7.8% for independent dephasing and spin flip errors.
Self-correcting quantum memories demonstrate robust properties that can be exploited to improve active quantum error-correction protocols. Here we propose a cellular automaton decoder for a variation of the color code where the bases of the physical qubits are locally rotated, which we call the XYZ color code. The local transformation means our decoder demonstrates key properties of a two-dimensional fractal code if the noise acting on the system is infinitely biased towards dephasing, namely, no string-like logical operators. As such, in the high-bias limit, our local decoder reproduces the b…
In previous work, we have shown that pseudocodewords can be used to characterize the behavior of decoders not only for classical codes but also for quantum stabilizer codes. With the insights obtained from this pseudocodewords-based analysis, we have also introduced a two-stage decoder based on pseudocodewords for quantum cycle codes that leads to improved decoding performance. In this paper, we consider quantum (stabilizer) color codes and propose a two-stage decoder that is a generalization of the pseudocodeword-based decoder for quantum cycle codes. Our decoder has only local or error-weigh…
The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a $d$-dimensional closed manifold is equivalent to multiple decoupled copies of the $d$-dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for $d=2$, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for …
We introduce and analyze a new type of decoding algorithm called General Color Clustering (GCC), based on renormalization group methods, to be used in qudit color codes. The performance of this decoder is analyzed under code capacity depolarizing noise, and is used to obtain the first fault-tolerant threshold estimates for qudit 6-6-6 color codes. The proposed decoder is compared with similar decoding schemes for qudit surface codes as well as the current leading qubit decoders for both sets of codes. We find that, as with surface codes, clustering performs sub-optimally for qubit color codes,…
The color code is both an interesting example of an exactly solved topologically ordered phase of matter and also among the most promising candidate models to realize fault-tolerant quantum computation with minimal resource overhead. The contributions of this work are threefold. First of all, we build upon the abstract theory of boundaries and domain walls of topological phases of matter to comprehensively catalog the objects realizable in color codes. Together with our classification we also provide lattice representations of these objects which include three new types of boundaries as well a…
The color code is a topological quantum error-correcting code supporting a variety of valuable fault-tolerant logical gates. Its two-dimensional version, the triangular color code, may soon be realized with currently available superconducting hardware despite constrained qubit connectivity. To guide this experimental effort, we study the storage threshold of the triangular color code against circuit-level depolarizing noise. First, we adapt the Restriction Decoder to the setting of the triangular color code and to phenomenological noise. Then, we propose a fault-tolerant implementation of the …
A quantum computer needs the assistance of a classical algorithm to detect and identify errors that affect encoded quantum information. At this interface of classical and quantum computing the technique of machine learning has appeared as a way to tailor such an algorithm to the specific error processes of an experiment --- without the need for a priori knowledge of the error model. Here, we apply this technique to topological color codes. We demonstrate that a recurrent neural network with long short-term memory cells can be trained to reduce the error rate $ε_{\rm L}$ of the encoded logical …
We propose an error correction procedure based on a cellular automaton, the sweep rule, which is applicable to a broad range of codes beyond topological quantum codes. For simplicity, however, we focus on the three-dimensional (3D) toric code on the rhombic dodecahedral lattice with boundaries and prove that the resulting local decoder has a non-zero error threshold. We also numerically benchmark the performance of the decoder in the setting with measurement errors using various noise models. We find that this error correction procedure is remarkably robust against measurement errors and is al…
The Gottesman-Kitaev-Preskill (GKP) code is an important type of bosonic quantum error-correcting code. Since the GKP code only protects against small shift errors in $\hat{p}$ and $\hat{q}$ quadratures, it is necessary to concatenate the GKP code with a stabilizer code for the larger error correction. In this paper, we consider the concatenation of the single-mode GKP code with the two-dimension (2D) color code (color-GKP code) on the square-octagon lattice. We use the Steane type scheme with a maximum-likelihood estimation (ME-Steane scheme) for GKP error correction and show its advantage fo…
We describe a computationally-efficient heuristic algorithm based on a renormalization-group procedure which aims at solving the problem of finding minimal surface given its boundary (curve) in any hypercubic lattice of dimension $D>2$. We use this algorithm to correct errors occurring in a four-dimensional variant of the toric code, having open as opposed to periodic boundaries. For a phenomenological error model which includes measurement errors we use a five-dimensional version of our algorithm, achieving a threshold of $4.35\pm0.1\%$. For this error model, this is the highest known thresho…
We propose a new cellular automaton (CA), the Sweep Rule, which generalizes Toom's rule to any locally Euclidean lattice. We use the Sweep Rule to design a local decoder for the toric code in $d\geq 3$ dimensions, the Sweep Decoder, and rigorously establish a lower bound on its performance. We also numerically estimate the Sweep Decoder threshold for the three-dimensional toric code on the cubic and body-centered cubic lattices for phenomenological phase-flip noise. Our results lead to new CA decoders with provable error-correction thresholds for other topological quantum codes including the c…
Hard-decision renormalization group (HDRG) decoders are an important class of decoding algorithms for topological quantum error correction. Due to their versatility, they have been used to decode systems with fractal logical operators, color codes, qudit topological codes, and non-Abelian systems. In this work, we develop a method of performing HDRG decoding which combines strenghts of existing decoders and further improves upon them. In particular, we increase the minimal number of errors necessary for a logical error in a system of linear size $L$ from $Θ(L^{2/3})$ to $Ω(L^{1-ε})$ for any $ε…
Recent developments in the field of deep learning have motivated many researchers to apply these methods to problems in quantum information. Torlai and Melko first proposed a decoder for surface codes based on neural networks. Since then, many other researchers have applied neural networks to study a variety of problems in the context of decoding. An important development in this regard was due to Varsamopoulos et al. who proposed a two-step decoder using neural networks. Subsequent work of Maskara et al. used the same concept for decoding for various noise models. We propose a similar two-ste…
We study $(d-1)$-dimensional excitations in the $d$-dimensional color code that are created by transversal application of the $R_{d}$ phase operators on connected subregions of qubits. We find that such excitations are superpositions of electric charges and can be characterized by fixed-point wavefunctions of $(d-1)$-dimensional bosonic SPT phases with $(\mathbb{Z}_{2})^{\otimes d}$ symmetry. While these SPT excitations are localized on $(d-1)$-dimensional boundaries, their creation requires operations acting on all qubits inside the boundaries, reflecting the non-triviality of emerging SPT wa…
Finding optimal correction of errors in generic stabilizer codes is a computationally hard problem, even for simple noise models. While this task can be simplified for codes with some structure, such as topological stabilizer codes, developing good and efficient decoders still remains a challenge. In our work, we systematically study a very versatile class of decoders based on feedforward neural networks. To demonstrate adaptability, we apply neural decoders to the triangular color and toric codes under various noise models with realistic features, such as spatially-correlated errors. We repor…
In recent years, there have been many studies on local stabilizer codes. Under the assumption of translation and scale invariance Yoshida classified such codes. His result implies that translation invariant 2D color codes are equivalent to copies of toric codes. Independently, Bombin, Duclos-Cianci, and Poulin showed that a local translation invariant 2D topological stabilizer code is locally equivalent to a finite number of copies of Kitaev's toric code. In this paper we focus on 2D topological color codes and relax the assumption of translation invariance. Using a linear algebraic framework …
Trellis decoders are a general decoding technique first applied to qubit-based quantum error correction codes by Ollivier and Tillich in 2006. Here we improve the scalability and practicality of their theory, show that it has strong structure, extend the results using classical coding theory as a guide, and demonstrate a canonical form from which the structural properties of the decoding graph may be computed. The resulting formalism is valid for any prime-dimensional quantum system. The modified decoder works for any stabilizer code $S$ and separates into two parts: a one-time, offline comput…
USDs: A universal stabilizer decoder framework using symmetry
Computer Science Program, Graduate School of Science and Technology, Meiji University（Meiji大学计算机科学系）；Department of Computer Science, School of Science and Technology, Meiji University（Meiji大学计算机科学系）
Generalized matching decoders for 2D topological translationally-invariant codes
Joint Center for Quantum Information and Computer Science, University of Maryland（马里兰大学量子信息与计算机科学联合中心）；QuEra Computing Inc.（QuEra 计算公司）；Yale Quantum Institute & Department of Applied Physics, Yale University（耶鲁大学应用物理系与耶鲁量子研究所）；Department of Computer Science, Carnegie Mellon University（卡内基梅隆大学计算机科学系）；Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology（麻省理工学院电气工程和计算机科学系）
