Tailoring surface codes for highly biased noise

The surface code, with a simple modification, exhibits ultra-high error correction thresholds when the noise is biased towards dephasing. Here, we identify features of the surface code responsible for these ultra-high thresholds. We provide strong evidence that the threshold error rate of the surface code tracks the hashing bound exactly for all biases, and show how to exploit these features to achieve significant improvement in logical failure rate. First, we consider the infinite bias limit, meaning pure dephasing. We prove that the error threshold of the modified surface code for pure dephasing noise is $50\%$, i.e., that all qubits are fully dephased, and this threshold can be achieved by a polynomial time decoding algorithm. We demonstrate that the sub-threshold behavior of the code depends critically on the precise shape and boundary conditions of the code. That is, for rectangular surface codes with standard rough/smooth open boundaries, it is controlled by the parameter $g=\gcd(j,k)$, where $j$ and $k$ are dimensions of the surface code lattice. We demonstrate a significant improvement in logical failure rate with pure dephasing for co-prime codes that have $g=1$, and closely-related rotated codes, which have a modified boundary. The effect is dramatic: the same logical failure rate achievable with a square surface code and $n$ physical qubits can be obtained with a co-prime or rotated surface code using only $O(\sqrt{n})$ physical qubits. Finally, we use approximate maximum likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased towards dephasing. In particular, comparing with a square surface code, we observe a significant improvement in logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.Comment: 18+4 pages, 24 figures; v2 includes additional coauthor (ASD) and new results on the performance of surface codes in the finite-bias regime, obtained with beveled surface codes and an improved tensor network decoder; v3 published versio

Neural Decoder for Topological Codes using Pseudo-Inverse of Parity Check Matrix

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-step neural decoder using inverse parity-check matrix for topological color codes. We show that it outperforms the state-of-the-art performance of non-neural decoders for independent Pauli errors noise model on a 2D hexagonal color code. Our final decoder is independent of the noise model and achieves a threshold of $10 \%$. Our result is comparable to the recent work on neural decoder for quantum error correction by Maskara et al.. It appears that our decoder has significant advantages with respect to training cost and complexity of the network for higher lengths when compared to that of Maskara et al.. Our proposed method can also be extended to arbitrary dimension and other stabilizer codes.Comment: 12 pages, 12 figures, 2 tables, submitted to the 2019 IEEE International Symposium on Information Theor

CoBe -- Coded Beacons for Localization, Object Tracking, and SLAM Augmentation

This paper presents a novel beacon light coding protocol, which enables fast and accurate identification of the beacons in an image. The protocol is provably robust to a predefined set of detection and decoding errors, and does not require any synchronization between the beacons themselves and the optical sensor. A detailed guide is then given for developing an optical tracking and localization system, which is based on the suggested protocol and readily available hardware. Such a system operates either as a standalone system for recovering the six degrees of freedom of fast moving objects, or integrated with existing SLAM pipelines providing them with error-free and easily identifiable landmarks. Based on this guide, we implemented a low-cost positional tracking system which can run in real-time on an IoT board. We evaluate our system's accuracy and compare it to other popular methods which utilize the same optical hardware, in experiments where the ground truth is known. A companion video containing multiple real-world experiments demonstrates the accuracy, speed, and applicability of the proposed system in a wide range of environments and real-world tasks. Open source code is provided to encourage further development of low-cost localization systems integrating the suggested technology at its navigation core