Tutorial on Quantum Error Correction for 2024 Quantum Information Knowledge (QuIK) Workshop

TL;DR

This tutorial addresses the problem of achieving scalable, fault-tolerant quantum computing by surveying the key quantum error correction formalisms and code families used today. It outlines stabilizer and CSS constructions, quantum LDPC codes, and bosonic encodings, along with practical issues in decoding and fault-tolerant gate implementation. The discussion covers major code families (surface/toric, hypergraph and lifted product, concatenated), decoding strategies, and the role of magic-state distillation in enabling universal computation. By connecting classical coding theory with quantum error correction, the paper highlights actionable paths toward reducing overhead and improving thresholds for practical quantum computers.

Abstract

We provide a brief review of the fundamentals of quantum computation and quantum error correction for the participants of the first Quantum Information Knowledge (QuIK) workshop at the 2024 IEEE International Symposium on Information Theory (ISIT 2024). While this is not a comprehensive review, we provide many references for the reader to delve deeper into the concepts and research directions.

Figures (2)

• Figure 1: The standard $[\![ 41,1,5 ]\!]$ surface code with side length $5$. Qubits are represented by circles. The orange boxes highlight weight-$4$$\mathrm{X}$-type and $\mathrm{Z}$-type stabilizers represented by red and blue squares, respectively. Note that there are weight-$3$ stabilizers in the boundaries of the lattice. All connecting lines represent local connectivity natively made available in the hardware; solid lines show the faces (or plaquettes) of the lattice and dashed lines connect the blue $\mathrm{Z}$-checks to qubits incident to a face. A possible choice for the logical $\mathrm{Z}$ and logical $\mathrm{X}$ operators is shown, both of minimum weight $5$.
• Figure 2: The surface code constructed as the hypergraph product of classical repetition codes. The intersection of bits and checks of the repetition codes determine the qubits and stabilizers of the hypergraph product code.