Quantum error correction protocols will play a central role in the realisation of quantum computing; the choice of error correction code will influence the full quantum computing stack, from the layout of qubits at the physical level to gate compilation strategies at the software level. As such, familiarity with quantum coding is an essential prerequisite for the understanding of current and future quantum computing architectures. In this review, we provide an introductory guide to the theory and implementation of quantum error correction codes. Where possible, fundamental concepts are described using the simplest examples of detection and correction codes, the working of which can be verified by hand. We outline the construction and operation of the surface code, the most widely pursued error correction protocol for experiment. Finally, we discuss issues that arise in the practical implementation of the surface code and other quantum error correction codes.

No abstract

Coherent parity check (CPC) codes are a new framework for the construction of quantum error correction codes that encode multiple qubits per logical block. CPC codes have a canonical structure involving successive rounds of bit and phase parity checks, supplemented by cross-checks to fix the code distance. In this paper, we provide a detailed introduction to CPC codes using conventional quantum circuit notation. We demonstrate the implementation of a CPC code on real hardware, by designing a [[4, 2, 2]] detection code for the IBM 5Q superconducting qubit device. Whilst the individual gate-error rates on the IBM device are too high to realise a fault tolerant quantum detection code, our results show that the syndrome information from a full encode-decode cycle of the [[4, 2, 2]] CPC code can be used to increase the output state fidelity by post-selection. Following this, we generalise CPC codes to other quantum technologies by showing that their structure allows them to be efficiently compiled using any experimentally realistic native two-qubit gate. We introduce a threestage CPC design process for the construction of hardware-optimised quantum memories. As a proofof-concept example, we apply our design process to an idealised linear seven-qubit ion trap. In the first stage of the process, we use exhaustive search methods to find a large set of [[7, 3, 3]] codes that saturate the quantum Hamming bound for seven qubits. We then optimise over the discovered set of codes to meet the hardware and layout demands of the ion trap device. We also discuss how the CPC design process will generalise to larger-scale codes and other qubit technologies.

We introduce a high-level graphical framework for the design, analysis, and verification of quantum error correcting codes. The coherent parity check construction for stabilizer codes allows us to construct a broad range of quantum codes based on classical codes, and gives a framework in which large classes of such codes can be both analytically and numerically discovered. Using graphical tools based on the ZX calculus, we explicitly construct small distance 3 and 5 codes with high code rates using this framework. We also show how this framework can be used to represent CSS codes and conversely how to compute stabilisers for a CPC code. We give a construction turns (almost) any pair of classical [n, k, 3] codes into a [[2n − k + 2, k, 3]] CPC code, and give a straightforward technique for machine search which yields thousands of potential codes, and demonstrate its operation for distance 3 and 5 codes. Finally we use the graphical tools we introduce to demonstrate how Clifford computation can be performed within CPC codes.

We introduce a new graphical framework for designing quantum error correction codes based on classical principles. A key feature of this graphical language, over previous approaches, is that it is closely related to that of factor graphs or graphical models in classical information theory and machine learning. It enables us to formulate the description of the recentlyintroduced 'coherent parity check' quantum error correction codes entirely within the language of classical information theory. This makes our construction accessible without requiring background in quantum error correction or even quantum mechanics in general. More importantly, this allows for a collaborative interplay where one can design new quantum error correction codes derived from classical codes.