Accuracy threshold for concatenated error detection in one dimension

Estimates of the quantum accuracy threshold often tacitly assume that it is possible to interact arbitrary pairs of qubits in a quantum computer with a failure rate that is independent of the distance between them. None of the many physical systems that are candidates for quantum computing possess this property. Here we study the performance of a concatenated error-detection code in a system that permits only nearest-neighbor interactions in one dimension. We make use of a message-passing scheme that maximizes the number of errors that can be reliably corrected by the code. Our numerical results indicate that arbitrarily accurate universal quantum computation is possible if the probability of failure of each elementary physical operation is below approximately ${10}^{-5}$. This threshold is three orders of magnitude lower than the highest known.

Figure 1

Flags are binned according to the effect that an error at the location at which the flag originated would have on the data and ancilla qubits at the point of syndrome extraction. For simplicity, only $X$ errors are considered. An $X$ error in the bin $AG1$ would change the parity of the ancilla while remaining on one of the first pair of data qubits, an $X$ error in $AG2$ would change the parity of the ancilla while remaining on one of the second pair of data qubits, and an $X$ error in $A$ would change the parity of the ancilla. Errors in $G1$ and $G2$ cannot affect parity of the ancilla but the weights of $G1$ and $G2$ are recorded and used to update the weights of $AG1$ and $AG2$ for the next error-correction cycle. Binning of flags corresponding to $Z$ errors is undertaken similarly.

Figure 2

Syndrome-extraction circuit for the [[4,1,2]] subsystem code on a linear nearest-neighbor array. $Q$ denotes the configuration of the encoded qubit after syndrome extraction, bold lines indicate data qubits, and plain lines indicate ancilla qubits.

Figure 3

Encoded CNOT circuit for the [[4,1,2]] subsystem code on a linear nearest-neighbor array, where notation is consistent with that of Fig. 2. Replacing all CNOT gates with SWAP gates gives the encoded SWAP circuit.

Figure 4

Logical failure rate as a function of physical failure rate for up to four levels of error correction (lines labeled 1–4). No error correction (dashed line labeled 0) is shown for reference. Lines for $n=1,2,3$ were calculated using the expansion in Eq. (4). To show the statistical error in the parameters $r_i$ there are two lines for $n=1,2,3$. The top line in each pair is $2\sigma$ above the mean and the bottom is $2\sigma$ below the mean. Data for $n=4$ was obtained directly. The line for $n=4$ joins the means of the three data points, where the error bars indicate $\pm 2\sigma$ statistical error. The threshold is the value of $p$ at which the failure rate of the $n\text{th}$ level of error correction intersects the physical failure rate in the limit $n\to \infty$. Assuming that the lines corresponding to higher-level failure rates continue the trend observed, the threshold will be close to the crossing of lines for $n=3$ and $n=4$ at approximately ${10}^{-5}$.