Optimizing the Frequency of Quantum Error Correction

A common assumption is that one applies a fault-tolerant quantum error correction (FTQEC) after every gate during quantum computing. However, it is known that this is not always optimal, since the FTQEC procedure itself can introduce errors. Here we vary the number of logical gates between FTQEC operations given that a failure of a postselection condition may cause the FTQEC to be skipped. We derive an expression for the logical error rate as a function of the error-correction frequency and find the optimal frequency for the application of the FTQEC. Furthermore, we show that this is relatively insensitive to the postselection failure probability for a large range of such probabilities. We provide an example of the application of the analytic expression to the $⟦7,1,3⟧$ Steane code and data derived from a Monte Carlo simulation.

Figure 1

“Noisy” QEC for the correction of $X$ errors. The star represents the location of a syndrome error, and the triangle a location of a correction error. The Steane ancilla state is $|{+}_L⟩$. Measurements are performed in the $Z$ basis.

Figure 2

$P_L$ versus $m$ for ${\epsilon }_g=5.0\times {10}^{-5}$, $8.0\times {10}^{-5}$, $1.0\times {10}^{-4}$, and $3.0\times {10}^{-4}$, where ${\epsilon }_a=0.3$. The triangular blue points are the numerical values of $P_L$ as given by Eq. (3). The circular black points are the numerical simulation values of $P_L$.