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Theory Behind Quantum Error Correcting Codes: An Overview

  • Review Article
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Journal of the Indian Institute of Science Aims and scope

Abstract

Quantum information processing is now a well-evolved field of study with roots to quantum physics that has significantly evolved from pioneering works over almost more than a century. Today, we are at a stage where elementary forms of quantum computers and communication systems are being built and deployed. In this paper, we begin with a historical background into quantum information theory and coding theory for both entanglement-unassisted and assisted quantum communication systems, motivating the need for quantum error correction in such systems. We then begin with the necessary mathematical preliminaries towards understanding the theory behind quantum error correction, central to the discussions within this article, starting from the binary case towards the non-binary generalization, using the rich framework of finite fields. We will introduce the stabilizer framework, build upon the Calderbank-Shor-Steane framework for binary quantum codes and generalize this to the non-binary case, yielding generalized CSS codes that are linear and additive. We will survey important families of quantum codes derived from well-known classical counterparts. Next, we provide an overview of the theory behind entanglement-assisted quantum ECCs along with encoding and syndrome computing architectures. We present a case study on how to construct efficient quantum Reed-Solomon codes that saturate the Singleton bound for the non-degenerate case. We will also show how positive coding rates can be realized using tensor product codes from two zero-rate entanglement-assisted CSS codes, an effect termed as the coding analog of superadditivity, useful for entanglement-assisted quantum communications. We discuss how quantum coded networks can be realized using cluster states and modified graph state codes. Last, we will motivate fault-tolerant quantum computation from the perspective of coding theory. We end the article with our perspectives on interesting open directions in this exciting field.

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Notes

  1. The reader must note that this tunneling probability depends on both the potential barrier and the scale i.e., narrow vs. wide barriers. For example, in scanning tunneling microscopy, a probe tip is very close to a sample (typically, a fraction of a nm.). An electron with an energy above the work function of the sample is able to tunnel across this gap, creating a current in the probe as a function of the gap length. As the probe scans the surface of the sample, changes in the tunneling current can provide the needed information on the topography of the surface.

  2. In fact, in the absence of a perfect single photon source, the state-of-the-art practical QKD systems employ a weak coherent source with probabilistic emission of single photons, driven by Poissonian statistics.

  3. Quantum no-cloning theorem has an important bearing in quantum coding and communications since we cannot clone a quantum bit, unlike a classical bit that can be copied using registers and stored in flip-flops (memory elements). Quantum no-broadcasting theorem has an important bearing while dealing with relay and multicasting equivalents within a quantum setting, useful for quantum network information theory and coding.

  4. The reader must note that the failed node can be identified using quantum non-demolition experiments.

  5. Since the original work of Kitaev57, there has been several research works in the field of topological QECCs. We will not be listing all the related works here, and instead focus on major works on algebraic quantum ECCs.

  6. Construction of practical encoding and error correction circuits requires the knowledge of quantum gates, which could be transversal gates or controlled gates.

  7. RM codes are constructed using the concept of multivariate polynomials, and they are based on the idea of evaluating a polynomial at different points in a finite field to generate code words.

  8. The reader must note that quantum states are in fact rays in a Hilbert space since there is a global phase. In the projective Hilbert space when the phase is quotiented out, what remains is a normalized vector. For more details, the reader is referred to128.

  9. The Werner state \(\rho ^{AB}=\lambda \vert {\psi ^{-}}\rangle \langle {\psi ^{-}}\vert +(1-\lambda )\frac{\mathbb{I}}{4}\), where \(\vert {\psi ^{-}}\rangle =\frac{\vert {01}\rangle -\vert {10}\rangle }{\sqrt{(}2)}\) is a noisy version of the Bell state. By varying \(\lambda \), one can go from separable states to a pure Bell state.

  10. The reader must note that, in general, one could share a entangled qudit i.e., generalized d-level quantum states.

  11. The reader must note that the CSS code over qubits is a special case of this generalization.

  12. Minimal generators of a group are a set of generators that generate the group and none of the generators can be generated by other generators of the set. This is analogous to a basis for a vector space.

  13. The reader must note that unlike classical binary codes where the code coordinates can be flipped due to noise, we have phase flips in the quantum case, requiring stabilizers for correcting both these types and their combinations.

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Acknowledgements

The authors would like to thank Abhi K. Sharma, Sudhir K. Sahoo and Arun M. (Ph. D students from the Physical Nanomemories Signal and Information Processing Laboratory (PNSIL), IISc) for their help in contributing to reviewing fault-tolerant literature, compiling the list of references, and typesetting a few figures.

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  1. Department of Electronic Systems Engineering, Division of EECS, Indian Institute of Science, C. V. Raman Road, Bengaluru, KA, 560012, India

    Shayan Srinivasa Garani & Priya J. Nadkarni

  2. Department of EECS, Indian Institute of Science Education and Research, Bhopal Bypass Road, Bhopal, MP, 462066, India

    Ankur Raina

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Garani, S.S., Nadkarni, P.J. & Raina, A. Theory Behind Quantum Error Correcting Codes: An Overview. J Indian Inst Sci 103, 449–495 (2023). https://doi.org/10.1007/s41745-023-00392-7

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