We consider a topological stabilizer code on a honeycomb grid, the "XYZ 2 " code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a "matching code" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six ( XYZXYZ ) plaquette stabilizers and weight-two ( XX ) link stabilizers on a planar hexagonal grid composed of 2d2 qubits for code distance d , with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure X , Y , or Z noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance d2 only for pure Y noise, here the code distance is 2d2 for both pure Z and pure Y noise. Thresholds for noise with finite Z bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding.
We consider a topological stabilizer code on a honeycomb grid, the "XYZ 2 " code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a "matching code" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six ( XYZXYZ ) plaquette stabilizers and weight-two ( XX ) link stabilizers on a planar hexagonal grid composed of 2d2 qubits for code distance d , with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure X , Y , or Z noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance d2 only for pure Y noise, here the code distance is 2d2 for both pure Z and pure Y noise. Thresholds for noise with finite Z bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding.
Sequential decoding of the XYZ2 hexagonal stabilizer code
Quantum error correction requires accurate and efficient decoding to optimally suppress errors in the encoded information. For concatenated codes, where one code is embedded within another, optimal decoding can be achieved using a message-passing algorithm that sends conditional error probabilities from the lower-level code to a higher-level decoder. In this work, we study the XYZ2 topological stabilizer code, defined on a honeycomb lattice, and use the fact that it can be viewed as a concatenation of a [[2, 1, 1]] phase-flip parity check code and the surface code with YZZY stabilizers, to decode the syndrome information in two steps. We use this sequential decoding scheme to correct errors on data qubits, as well as measurement errors, under various biased error models using both a maximum-likelihood decoder (MLD) and more efficient matching-based decoders. For depolarizing noise we find that the sequential matching decoder gives a threshold of 18.3%, close to optimal, as a consequence of a favorable, effectively biased, error model on the upper-level YZZY code. For phase-biased noise on data qubits, at a bias η=px+pypz=10, we find that a belief-matching-based decoder reaches thresholds of 24.1%, compared to 28.6% for the MLD. With measurement errors the thresholds are reduced to 3.4% and 4.3%, for depolarizing and biased noise respectively, using the belief-matching decoder. This demonstrates that the XYZ2 code has thresholds that are competitive with other codes tailored to biased noise. The results also showcase two approaches to taking advantage of concatenated codes: 1) tailoring the upper-level code to the effective noise profile of the decoded lower-level code, and 2) making use of an upper-level decoder that can utilize the local information from the lower-level code.
Efficient high-performance decoding of topological stabilizer codes has the potential to crucially improve the balance between logical failure rates and the number and individual error rates of the constituent qubits. High-threshold maximum-likelihood decoders require an explicit error model for Pauli errors to decode a specific syndrome, whereas lower-threshold heuristic approaches such as minimum-weight matching are error agnostic. Here we consider an intermediate approach, formulating a decoder that depends on the bias, i.e., the relative probability of phase-flip to bit-flip errors, but is agnostic to error rate. Our decoder is based on counting the number and effective weight of the most likely error chains in each equivalence class of a given syndrome. We use Metropolis-based Monte Carlo sampling to explore the space of error chains and find unique chains that are efficiently identified using a hash table. Using the error-rate invariance, the decoder can sample chains effectively at an error rate which is higher than the physical error rate and without the need for thermalization between chains in different equivalence classes. Applied to the surface code and the XZZX code, the decoder matches maximum-likelihood decoders for moderate code sizes or low error rates. We anticipate that, because of the compressed information content per syndrome, it can be taken full advantage of in combination with machine-learning methods to extrapolate Monte Carlo--generated data.
The code-capacity threshold of a scalable quantum error correcting stabilizer code can be expressed as a thermodynamic phase transition of a corresponding random-bond Ising model. Here we study the XY and XZZX surface codes under phase-biased noise, px=py=pz/(2η) , with η≥1/2 , and total error rate p=px+py+pz . By appropriately formulating the boundary conditions, in the rotated code geometry, we find exact solutions at a special disordered point, p=1+η−12+η−1≳0.5 , for arbitrary odd code distance d , where the codes reduce to one-dimensional Ising models. The total logical failure rate is given by Pf=34−14e−2dZartanh(1/2η) , where dZ=d2 and d for the two codes respectively, is the effective code distance for pure phase-flip noise. As a consequence, for code distances d≪η , and error rates near the threshold, the XZZX code is effectively equivalent to the phase-flip correcting repetition code over d qubits. The large finite size corrections for dZ<η also make threshold extractions, from numerical calculations at moderate code distances, unreliable. We show that calculating thresholds based not only on the total logical failure rate, but also independently on the phase- and bit-flip logical failure rates, gives a more confident estimate. Using this method for the XZZX code with a tensor-network based decoder and code distances up to d≈100 , we find that the thresholds converge to a single value at moderate bias ( η=30,100 ), at an error rate above the hashing bound.
Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topological codes that we call generalized toric codes (GTCs). We show that these XZZX codes are highly qubit efficient if tailored to biased noise. To characterize the code performance, we use the notion of effective distance, which generalizes code distance to the case of biased noise and constitutes a proxy for the logical failure rate. We find that the XZZX codes can achieve a favorable resource scaling by this metric under biased noise. We also show that the XZZX codes have remarkably high thresholds that reach what is achievable by random codes, and furthermore they can be efficiently decoded using matching decoders. Finally, by adding only one flag qubit, the XZZX codes can realize fault-tolerant QEC while preserving their large effective distance. In combination, our results show that tailored XZZX codes give a resource-efficient scheme for fault-tolerant QEC against biased noise.
Tailored topological stabilizer codes in two dimensions have been shown to exhibit high storage threshold error rates and improved subthreshold performance under biased Pauli noise. Three-dimensional (3D) topological codes can allow for several advantages including a transversal implementation of non-Clifford logical gates, single-shot decoding strategies, parallelized decoding in the case of fracton codes as well as construction of fractal lattice codes. Motivated by this, we tailor 3D topological codes for enhanced storage performance under biased Pauli noise. We present Clifford deformations of various 3D topological codes, such that they exhibit a threshold error rate of 50% under infinitely biased Pauli noise. Our examples include the 3D surface code on the cubic lattice, the 3D surface code on a checkerboard lattice that lends itself to a subsystem code with a single-shot decoder, the 3D color code, as well as fracton models such as the X-cube model, the Sierpinski model and the Haah code. We use the belief propagation with ordered statistics decoder (BP-OSD) to study threshold error rates at finite bias. We also present a rotated layout for the 3D surface code, which uses roughly half the number of physical qubits for the same code distance under appropriate boundary conditions. Imposing coprime periodic dimensions on this rotated layout leads to logical operators of weight O(n) at infinite bias and a corresponding exp[−O(n)] subthreshold scaling of the logical failure rate, where n is the number of physical qubits in the code. Even though this scaling is unstable due to the existence of logical representations with O(1) low-rate Pauli errors, the number of such representations scales only polynomially for the Clifford-deformed code, leading to an enhanced effective distance.
Quantum error correction, a technique that relies on the principle of redundancy to encode logical information into additional qubits to better protect the system from noise, is necessary to design a viable quantum computer. For this new topological stabilizer code-XYZ 2 code defined on the cellular lattice, it is implemented on a hexagonal lattice of qubits and it encodes the logical qubits with the help of stabilizer measurements of weight six and weight two. However topological stabilizer codes in cellular lattice quantum systems suffer from the detrimental effects of noise due to interaction with the environment. Several decoding approaches have been proposed to address this problem. Here, we propose the use of a state-attention based reinforcement learning decoder to decode XYZ 2 codes, which enables the decoder to more accurately focus on the information related to the current decoding position, and the error correction accuracy of our reinforcement learning decoder model under the optimisation conditions can reach 83.27% under the depolarizing noise model, and we have measured thresholds of 0.18856 and 0.19043 for XYZ 2 codes at code spacing of 3–7 and 7–11, respectively. our study provides directions and ideas for applications of decoding schemes combining reinforcement learning attention mechanisms to other topological quantum error-correcting codes.