Message passing in fault-tolerant quantum error correction

Inspired by Knill’s scheme for message passing error detection, here we develop a scheme for message passing error correction for the nine-qubit Bacon-Shor code. We show that for two levels of concatenated error correction, where classical information obtained at the first level is used to help interpret the syndrome at the second level, our scheme will correct all cases with four physical errors. This results in a reduction of the logical failure rate relative to conventional error correction by a factor proportional to the reciprocal of the physical error rate.

Figure 1

(a) The nine data qubits of the Bacon-Shor code can be thought of as the vertices of a $3\times 3$ grid. (b) Though there are nine data qubits and nine ancillary qubits, syndrome extraction comprises three separate circuits run in parallel, each involving three data and three ancillary qubits. The $X$ syndrome extraction circuit [18] is shown above for the $i\text{th}$ row of qubits. $Z$ syndrome extraction has a similar circuit (CNOT gates are reversed and measurement and preparation are in the conjugate basis) which operates on the columns of data qubits.

Figure 2

Physical errors during a level-1 two-qubit gate can result in many different combinations of flags and failures on the two logical qubits depending on where the errors occur. In this example there are two physical errors, one of the errors affects both level-1 qubits and the other error only affects the bottom level-1 qubit. Both of the trailing EC boxes will see a nonzero syndrome and so each will raise its flag. In this case the final result will be flagged success on the top qubit and a flagged failure on the bottom qubit. In general, each physical error can affect one or both logical qubits. Any combination of US, FS, FF, and UF is possible as long as the error weight of each logical qubit is not more than the number of physical errors.

Figure 3

Level-2 logical failure rate vs physical error rate with and without message passing. The lines at low error rates are given by Eq. (2), where the probabilities of failure given $i$ errors, $r_i$, are obtained from simulations for $i⩽12$ and set to zero for $i>12$. To show the statistical error in the parameters $r_i$ there are two lines for both standard and message passing error correction. The top line in each pair is $2\sigma$ above the mean and the bottom is $2\sigma$ below the mean. At high error rates, as 12 and more errors become common, the approximation breaks down and so curves drop towards zero. The data points at high error rates come from direct simulations of a level-2 CNOT extended rectangle, where the error bars indicate $\pm 2\sigma$ statistical error. A second set of lines connect these data points.

Figure 4

MPEC failure from a error weight 5 case with two incoming FFs and one FS. In this case the single flag from the FS matches the syndrome and is thus preferred over the pair of flags from the FFs. This will result in an incorrect correction and hence logical failure. In this particular example conventional error correction and MPEC would both fail.

Figure 5

MPEC failure from a error weight 5 case with two incoming FSs and one UF. The two flags match the syndrome and so are assumed to be errors. This causes MPEC to apply the wrong correction and results in logical failure. This case is of particular interest because it is an example of where conventional error correction would succeed but MPEC fails.

Figure 6

Incoming FF with $\mathrm{FF}+\mathrm{FF}$ pair from a CNOT failure is successfully diagnosed and corrected with a three-flag match.

Figure 7

Incoming FF with $\mathrm{FF}+\mathrm{FS}$ pair from a CNOT failure is successfully diagnosed and corrected. The horizontal and vertical flags together match the syndrome.

Figure 8

Incoming FS with $\mathrm{FF}+\mathrm{UF}$ pair from a CNOT failure produces a syndrome which cannot be matched with any combination of the two flags and so the syndrome must be interpreted without the use of the flags. In this case, there is a parity mismatch in the syndrome and so the standard procedure is that no correction should be applied. The error on the third data qubit is let through. The vertical flag does not appear on the ancillary qubits and so it is not removed from that data. Therefore there is an FF leaving this EC box.

Figure 9

Two $\mathrm{FF}+\mathrm{FF}$ pairs from CNOT failures result in a two-flag match which corrects only one of the errors on the data qubits. Since all flags that appear on the ancillary qubits are cleared after error correction there will be a UF leaving this EC box.