Quantum information is fragile and needs to be protected from errors introduced by faulty gates, unwanted interactions with the environment, and other sources of noise. This is typically done with a quantum error-correcting code, which encodes a logical qubit into many physical qubits. Thanks to this redundancy, the logical information can be preserved even if some physical qubits are corrupted.
To define the concepts of fault-tolerant threshold and breakeven point, one compares the physical error rate (for the errors relevant to the hardware) to the logical error rate, which is the probability that the logical qubit is not the correct one after an error-correction cycle. The extra qubits and structure introduced by error-correcting codes only suppress errors when the physical error rate is sufficiently low: if the noise is too strong, redundancy doesn’t help. The critical value such that the logical error rate is equal to the physical error rate is called the breakeven point. Above that point, the encoding makes things worse. Below that point, using the error-correcting code reduces the error rate.
When considering a family of codes parametrised by the number of physical qubits in the code, if the physical error rate is sufficiently low, one can reduce the logical error rate to an arbitrarily low level by increasing the number of qubits in the code, i.e. by using a bigger code in the code family. The value of the physical error rate at which this happens is referred to as the fault-tolerant threshold of that code family. It depends on several factors, such as the noise model and the way the logical operations are performed.
At Quandela, we have simulated a quantum memory experiment for our latest fault-tolerant architecture, showing that it achieves a loss threshold of 6.3% with the honeycomb Floquet code family. This is almost twice as high as the loss threshold we previously estimated, when using the surface code on a former version of the architecture, all the while requiring less physical qubits. An important effort of our R&D teams is to bring this threshold (and the thresholds for the other types of noise we encounter) higher and higher.
Frequently asked questions
- How is the threshold of a code family determined? The threshold is usually estimated by numerical simulations. One simulates how the logical error rate varies with the relevant physical error rate and the code size. The simulations can be carried out in the simple code-capacity model, where measurements are assumed to be perfect, or in the more advanced circuit-level noise model, where any operation can be faulty.
- Do all quantum error-correcting codes have a fault-tolerant threshold? No, the notion of fault-tolerant threshold is only defined for families of codes that can be extended systematically (e.g., surface codes, color codes…), while for fixed-size codes such as Steane 7-qubit code or Shor’s 9-qubit code, the notion of fault-tolerant threshold doesn’t make sense. Moreover, only certain code families have been proven to exhibit a threshold.
- Is the threshold the only useful metric for evaluating a quantum error-correcting scheme? No, the threshold indicates whether fault tolerance is in principle achievable, but it does not capture the resource overhead, the decoding complexity, or how to perform computation on the encoded information. Notions such as the teraquop footprint, which measures the qubit overhead in the low-error regime, and the existence of easy fault-tolerant schemes to perform logical gates are essential to build a practical fault-tolerant quantum computer.
