Graphical Structures for Design and Verification of Quantum Error Correction

We introduce a high-level graphical framework for designing and analysing quantum error correcting codes, centred on what we term the coherent parity check (CPC). The graphical formulation is based on the diagrammatic tools of the zx-calculus of quantum observables. The resulting framework leads to a construction for stabilizer codes that allows us to design and verify a broad range of quantum codes based on classical ones, and that gives a means of discovering large classes of codes using both analytical and numerical methods. We focus in particular on the smaller codes that will be the first used by near-term devices. We show how CSS codes form a subset of CPC codes and, more generally, how to compute stabilizers for a CPC code. As an explicit example of this framework, we give a method for turning almost any pair of classical [n,k,3] codes into a [[2n - k + 2, k, 3]] CPC code. Further, we give a simple 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 to demonstrate how Clifford computation can be performed within CPC codes. As our framework gives a new tool for constructing small- to medium-sized codes with relatively high code rates, it provides a new source for codes that could be suitable for emerging devices, while its zx-calculus foundations enable natural integration of error correction with graphical compiler toolchains. It also provides a powerful framework for reasoning about all stabilizer quantum error correction codes of any size.Comment: Computer code associated with this paper may be found at https://doi.org/10.15128/r1bn999672

Quantum Codes from Classical Graphical Models

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 recently-introduced ‘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

Analog information decoding of bosonic quantum LDPC codes

Quantum error correction is crucial for scalable quantum information processing applications. Traditional discrete-variable quantum codes that use multiple two-level systems to encode logical information can be hardware-intensive. An alternative approach is provided by bosonic codes, which use the infinite-dimensional Hilbert space of harmonic oscillators to encode quantum information. Two promising features of bosonic codes are that syndrome measurements are natively analog and that they can be concatenated with discrete-variable codes. In this work, we propose novel decoding methods that explicitly exploit the analog syndrome information obtained from the bosonic qubit readout in a concatenated architecture. Our methods are versatile and can be generally applied to any bosonic code concatenated with a quantum low-density parity-check (QLDPC) code. Furthermore, we introduce the concept of quasi-single-shot protocols as a novel approach that significantly reduces the number of repeated syndrome measurements required when decoding under phenomenological noise. To realize the protocol, we present a first implementation of time-domain decoding with the overlapping window method for general QLDPC codes, and a novel analog single-shot decoding method. Our results lay the foundation for general decoding algorithms using analog information and demonstrate promising results in the direction of fault-tolerant quantum computation with concatenated bosonic-QLDPC codes.Comment: 30 pages, 15 figure

Bias-tailored quantum LDPC codes

Bias-tailoring allows quantum error correction codes to exploit qubit noise asymmetry. Recently, it was shown that a modified form of the surface code, the XZZX code, exhibits considerably improved performance under biased noise. In this work, we demonstrate that quantum low density parity check codes can be similarly bias-tailored. We introduce a bias-tailored lifted product code construction that provides the framework to expand bias-tailoring methods beyond the family of 2D topological codes. We present examples of bias-tailored lifted product codes based on classical quasi-cyclic codes and numerically assess their performance using a belief propagation plus ordered statistics decoder. Our Monte Carlo simulations, performed under asymmetric noise, show that bias-tailored codes achieve several orders of magnitude improvement in their error suppression relative to depolarising noise.Comment: 21 Pages, 13 Figures. Comments welcome

Correcting non-independent and non-identically distributed errors with surface codes

A common approach to studying the performance of quantum error correcting codes is to assume independent and identically distributed single-qubit errors. However, the available experimental data shows that realistic errors in modern multi-qubit devices are typically neither independent nor identical across qubits. In this work, we develop and investigate the properties of topological surface codes adapted to a known noise structure by Clifford conjugations. We show that the surface code locally tailored to non-uniform single-qubit noise in conjunction with a scalable matching decoder yields an increase in error thresholds and exponential suppression of sub-threshold failure rates when compared to the standard surface code. Furthermore, we study the behaviour of the tailored surface code under local two-qubit noise and show the role that code degeneracy plays in correcting such noise. The proposed methods do not require additional overhead in terms of the number of qubits or gates and use a standard matching decoder, hence come at no extra cost compared to the standard surface-code error correction

Quantum error correction : an introductory guide

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

Protecting quantum memories using coherent parity check codes

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 three-stage CPC design process for the construction of hardware-optimised quantum memories. As a proof-of-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