Quantum Fault Tolerance Concatenated Codes

Short answer: Concatenated codes build fault tolerance by nesting one error-correcting code inside another, and recent papers use them to avoid magic state distillation or to cut qubit overhead. A 2016 pair of papers analyzes a 105-qubit scheme built from the 7-qubit Steane code and the 15-qubit Reed-Muller code. A 2025 npj Quantum Information paper proposes a concatenated protocol with constant space overhead. These are theoretical and numerical analyses, not hardware demonstrations.

The idea

A 2016 arXiv paper says concatenated codes have played a key role in determining fault-tolerance threshold rates because they can iteratively suppress errors as the levels of concatenation increase. It notes that a rigorous lower bound for the threshold of concatenated codes was established by Aliferis, Gottesman and Preskill using a technique called malignant set counting.

The 105-qubit scheme

The 2016 paper says that by concatenating the 7-qubit Steane and 15-qubit Reed-Muller codes, the resulting 105-qubit code enables a universal set of fault-tolerant gates even though not all of them are transversal. The CNOT gate remains transversal in both codes, so it has more error protection than the single-qubit logical gates. The authors show that the level-1 pseudo-threshold is limited by the logical Hadamard, but that the error suppression of logical CNOTs lets the asymptotic threshold rise by orders of magnitude at higher levels. They establish a lower bound of 1.28 x 10^-3 for the asymptotic threshold, which they call competitive with known concatenated models, and say it does not rely on magic state preparation for universal computation.

Resource overhead of the scheme

A second 2016 paper analyzes the resource overhead of the concatenated approach, examining the Steane code concatenated with the Reed-Muller code to build 49-qubit and 105-qubit codes that do not need magic state distillation for universality. It computes a lower bound of 8.33 x 10^-6 for the adversarial noise threshold of the 105-qubit code, and a depolarizing noise threshold of 9.69 x 10^-4 for the 49-qubit code, which it calls competitive with the 105-qubit result of 1.28 x 10^-3. It then compares resource requirements with a surface code implementation of a logical T gate using magic state distillation. For the sampled input error rates and noise model, it finds that the surface code achieves a smaller overhead than these concatenated schemes. So in that comparison, the concatenated approach did not win.

A 2025 protocol

A 2025 paper in npj Quantum Information says the essential requirement for fault-tolerant quantum computation is a protocol design that balances space overhead, threshold and modularity. It says a major obstacle for conventional protocols, such as those based on the surface code and the concatenated Steane code, has been space overhead, and that high-rate quantum LDPC codes reduce overhead but existing fault-tolerant protocols for them sacrifice other factors. The authors construct a protocol based on recent techniques for concatenated codes, achieving constant space overhead, a high threshold and flexible modular design. Under a physical error rate of 0.1 percent, they report reducing the space overhead for logical CNOT error rates of 10^-10 and 10^-24 by more than 90 percent and 96 percent respectively, compared with a surface code protocol, and a threshold of 2.5 percent under a conventional circuit-level error model. These figures are the authors' results under their stated models.

What this does not tell you

The papers give different, model-dependent answers on overhead: the 2016 analysis favours the surface code, while the 2025 protocol reports large savings. None reports a hardware demonstration, and each depends on its noise model.

FAQ

What is a concatenated code?

A code nested inside another to suppress errors at each level, per the 2016 paper.

Why concatenate Steane and Reed-Muller codes?

The 105-qubit result gives a universal fault-tolerant gate set without magic state preparation, according to the authors.

Is it cheaper than the surface code?

A 2016 analysis found the surface code had smaller overhead for its sampled rates. A 2025 paper reports large savings for a different concatenated protocol.

What threshold was found?

A lower bound of 1.28 x 10^-3 for the asymptotic threshold of the 105-qubit code, per the 2016 paper.

Is this on hardware?

These sources are theoretical and numerical.