Logical Error Rate
The logical error rate is the probability an encoded qubit fails per operation, and below threshold it falls exponentially with code distance.
The figure of merit
The logical error rate p_L is the probability that a logical qubit or logical operation suffers an uncorrectable error, resulting in a logical Pauli that changes the encoded state. It is the ultimate measure of a code's protection, the quantity an algorithm designer must drive below 1 over the total number of logical operations for the computation to succeed.
Exponential suppression
When the physical error rate p is below the threshold p_th, the logical error rate of a distance-d code scales roughly as p_L ~ A (p / p_th)^{(d+1)/2}. Because the exponent grows linearly with distance, p_L falls exponentially as d increases, while the qubit count grows only polynomially (as d^2 for the surface code). This favorable scaling is the entire reason error correction is viable: modest increases in qubits buy dramatic increases in reliability.
- p_L = probability of an uncorrectable logical error per operation.
- Below threshold: p_L ~ A (p/p_th)^{(d+1)/2}, exponential in distance.
- Must satisfy p_L * (number of logical operations) much less than 1.
- Above threshold, increasing distance makes p_L worse, not better.
Measuring and using it
Logical error rate is estimated by simulating the code under a stated error model with a chosen decoder, or measured experimentally by preparing a logical state, running many rounds of correction, and reading out. Experiments demonstrating that p_L decreases as distance increases from, say, 3 to 5 to 7 are milestone evidence that a device is below threshold. The value depends jointly on the code, the noise, and the decoder, so all three must be specified when quoting a number.
Because p_L sets the required distance, it directly determines the overhead: a longer algorithm needs a lower p_L, hence a larger distance and more physical qubits. It is the pivot connecting hardware quality to algorithmic ambition.