5.8 Lower bound on accuracy threshold
5.8.2 Correction of higher level errors
5.8. Lower bound on accuracy threshold 129
M U M U
M M
U U
M M
M U M U
U U
M U M U
M U M U
Figure 5.1: Examples of level-0 flip errors being corrected. The leftmost diagram in each row contains a distinct level-0 error. The blue lines are bit flip or phase flip errors. The red squares are sites with a nontrivial syndrome measurement. The green arrows are controlled flips based on the local update rules. Measurement and update steps are denoted by M and U.
5.8. Lower bound on accuracy threshold 130
M U M U
M M
U U
M M
M U M U
U U
M U M U
M U M U
Figure 5.2: Examples of level-0 measurement errors being corrected. The leftmost diagram in each row contains a distinct level-0 error. The blue x’s are measure- ment errors, and the blue lines are qubit errors. The red squares are sites with a nontrivial syndrome measurement. The green arrows are controlled flips based on the local update rules. Measurement and update steps are denoted by M and U.
5.8. Lower bound on accuracy threshold 131
of colonies. Any level-1 error (or higher-level error) that is completely contained within a single colony will be corrected in a straightforward manner by the basic local rules (Section 5.6). The two endpoints of such an error chain would drift toward the center of the colony, where they will meet up and form a trivial loop.
Error chains across colonies, however, require more sophisticated processing.
We will show in the following argument that any actual level-n error will be cor- rected within 3Un time steps, and that it will always be contained inside a space of 2Qn×2Qn. Since actual level-nerrors are (aQn, aQn, bUn)-separated from one another, they will always get corrected before another one occurs (provided a≥2 and b≥3).
Suppose a level-1 error occurs that stretches across neighboring colonies. At the end of the current work period, let xC and xN be the computed syndromes (according to the Count field) at one of the affected colonies of itself and of its neighbor. There are four possible values for xC and xN, depending on at what point during the work period the ends of the error chain migrated to the centers of each colony, and also depending on how the colony centers were affected by level-0 noise. If xC is zero (off), then this colony will do nothing. If both xC and xN are one (on), then the colony will make the proper decision to deal with this error by following the basic rules in Section 5.6 (based on the computed syndromes instead of its local neighborhood). The only dangerous case is when xC is on, but xN is off. The colony may then execute a flip (viaFlipSignal) that moves its syndrome in the wrong direction and thereby increases the size of the level-1 error.
To prevent this last case from ever happening, we want to guarantee that if a level-1 error occurs with enough time left in the work period for the computed syndrome of a colony to be on, then the computed syndromes of its neighbors must be valid. So we will want to set the self-count threshold fC to be as high as possible and the neighbor-count threshold fN to be as low as possible. However, if fC is too high, then level-0 noise could prevent a syndrome from ever turning on in a colony (by temporarily moving a syndrome away from the center), or if fN is too low, then level-0 noise could masquerade as a higher-level syndrome (by
5.8. Lower bound on accuracy threshold 132
temporarily creating a syndrome at the center of a neighboring colony). Since actual level-0 errors are (a, a, b)-separated from one another, and since they are always corrected within two time steps, we could choose
fN = 4
b, fC = b−4
b . (5.4)
Level-0 noise can then never affect the coarse-grained count of higher-level syn- dromes, because the presence or absence of syndromes at a colony center will never be changed by more than two during an interval of b time steps. (See the discus- sion of the Countfield in Section 5.5.2.) Note that we could have chosen fNb= 3 and fCb= (b−3), which would have been sufficient for dealing with level-0 noise, but we will need the choices of equation (5.4) to deal with higher level noise.
It now remains for us to find a bound on b that will guarantee the neighbor counts to be valid whenever a colony’s syndrome is on. Near the end of Section 5.7, we derived another bound on the size of an actual level-1 error. In particular, we found that all of the points in the set of errors differ in time by no more than 7(b+ 2) time steps. If our level-1 error was a long chain of errors connecting the center of one colony to some site in a neighboring colony, the endpoint of the error chain in the neighboring colony should migrate to its center no later than 7(b+ 2) + 2Qtime steps after the appearance of a syndrome in the center of the first colony. It will also take Q time steps for the signal from the neighboring colony to reach the first colony center.
Suppose that the earliest point in the level-1 error occurs with T time steps remaining in the current work period. Then whenever
fCb≤ T
b
(5.5) we require that
fNb≤
T−7(b+ 2)−3Q b
. (5.6)
If we choose a = 2 (which is consistent with all previous bounds on a), then we
5.8. Lower bound on accuracy threshold 133
can choose colony sizeQ= 4(a+ 2) = 16. We can then rearrange equations (5.4), (5.5), and (5.6) to read
T b
≥
11 +62 b
whenever T
b
≥(b−4). (5.7) We can satisfy condition (5.7) for all values of T if we choosebsuch that
(b−5)≥
11 +62 b
. (5.8)
This holds true for b≥ 20. Previously we wantedU =b2 (for the coarse-grained count), and we also required U ≥ 4(b+ 2) (for the error decomposition proof).
We could choose b= 20, U = 400, or if we prefer working with powers of two, let b= 32 andU = 1024. In the first case fN = 1−fC = 1/5 and in the second case fN = 1−fC = 1/8.
Since the same rules are followed for correcting level-1 errors across neighboring colonies as are responsible for correcting level-0 errors across neighboring cells, we know that any actual level-1 error will be corrected within two workperiods, once the syndromes are reported at their colony centers. Depending on the value of T (the number of time steps remaining in the current work period when a level- 1 error occurs), the computed syndromes may not be on until after a full work period. Consequently, the worst case for correcting actual level-1 errors requires no more than 3U time steps.
All of the above reasoning is valid for higher-level errors as well. In particular, the coarse-grained count was defined in such a way that level-(n-1) noise will not disrupt the counting of syndromes in a level-ncolony. The only noteworthy change is that for correcting a level-nerror, the 3Qterm appearing in equation (5.6) gets replaced by U3Qn−1n , which is strictly less than 3Q (if Q < U), so the bound on b becomes more relaxed for higher-level errors (eventually approaching b≥17).
Finally, we have a lower bound on the accuracy threshold of our local error correction scheme. From equation (5.3) and our choices of Q and U, we can
5.8. Lower bound on accuracy threshold 134
maintain quantum memory in our system for arbitrarily long times (by increasing the size of the lattice) provided that
(p+q) < 1
4Q4U2 = 1
4(16)4(400)2 = 2−2210−4 ≈ 2.4×10−11. (5.9) If we instead useU = 1024, the threshold becomes 2−38≈3.6×10−12.
While it may be disturbing to derive such an impractically small number, this is not unusual in these types of proofs, at least in an initial version focusing on ex- istence above other considerations and always assuming the worst case. The proof of stability of Toom’s medium for reliable computation in [34] gave a threshold lower bound of 10−28 (and only then at the request of a referee), which was next improved by [15] to 10−7, although numerical simulations by Bennett suggested a threshold around 0.05. In Section 5.9 we will also show evidence of a much higher value of the threshold from numerical simulations of our model.