and its diameter is at mostQn/2.
The term ‘chunk,’ not to be confused with the usage in Section 6.2, is chosen in order to avoid confusion with ‘cluster’, which is used for a set of defects. Note that a level-nchunk contains exactly 2n sites. Given an error E, let En be the union of all level-n chunks of E. If u∈ En+1, then by definitionuis an element of a level-(n+ 1) chunk. Since a level-(n+ 1) chunk is a union of two level-n chunks,uis contained in a level-nchunk. Hence, u∈En, and the sequenceEn form a descending chain
E =E0⊇E1⊇ · · · ⊇Em,
wheremis the smallest integer such thatEm+1=∅. LetFi=Ei\Ei+1, soE=F0∪F1∪ · · · ∪Fm is expressed as a disjoint union, which we call thechunk decompositionofE.
Proposition 7.5.1. LetQ≥6andM be anyQn-connected component ofFn. ThenM has diameter
≤Qn and is separated fromEn\M by distance> 13Qn+1.
Proof. We claim that for any pair of sites u∈Fn =En\En+1 and v∈En we have d(u, v)≤Qn or d(u, v)> 13Qn+1. Suppose on the contrary to the claim, that there is a pairu∈Fn andv∈En
such thatQn< d(u, v)≤Qn+1/3. LetCu3uandCv 3v be level-nchunks that containsuandv, respectively. Since the diameters ofCu,v are≤Qn/2 andd(u, v)> Qn, we deduce thatCu andCv
are disjoint. On the other hand,
d(Cu∪Cv)≤d(u, v) +d(Cu) +d(Cv)≤Qn+1/2
sinceQ≥6. Thus, Cu∪Cv is a level-(n+ 1) chunk that containsuwhich shows thatu∈En+1. It contradicts to our assumption thatu∈Fn=En\En+1.
Note that in the chunk decomposition aQn-connected componentP ofEn may not be separated from the restE\P by distance> Qn.
Lemma 7.5.2. Let Q ≥ 10. If the length m of the chunk decomposition of an error E satisfies Qm+1< Ltqo, then E is corrected by the RG decoder.
Proof. Consider any fixed errorP supported on a set of sitesE. LetE=F0∪F1∪ · · · ∪Fmbe the chunk decomposition of E, and let Fj,α be the Qj-connected components of Fj. Also, let Bj,α be the 1-neighborhood of the smallest box enclosing the syndrome created by the restriction ofP onto Fj,α. Proposition7.5.1implies that
d(Bj,α)≤Qj+ 2 and d(Bj,α, Bk,β)> 1
3Q1+min (j,k)−2. (7.4) LetPec(p) be the accumulated correcting operator returned by the levels 0, . . . , pof the RG decoder.
Let us use induction inpto prove the following statement.
1. The operatorPec(p)has support on the union of the boxesBj,α.
2. The operatorsPec(p) andP have the same restriction onBj,α modulo stabilizers for anyj such that 2p≥Qj+ 2.
The base of induction is p = 0. Using Eq. (7.4) we conclude that any 1-connected component of the syndromeS(P) is fully contained inside some boxBj,α. It proves thatPec(0) has support on the union of the boxesBj,α. The second statement is trivial forp= 0.
Suppose we have proved the above statement for somep. Then the operatorP·Pec(p)has support only inside boxesBj,αsuch that 2p< Qj+1 (modulo stabilizers). It follows that any 2p+1-connected component of the syndrome caused byP·Pec(p)is contained in some boxBj,αwith 2p< Qj+ 1. Note that the RG decoder never adds new defects; we just need to check that 2p+1-connected components do not cross the boundaries between the boxesBj,α with 2p< Qj+ 1. This follows from Eq. (7.4).
Hence Pec(p+1) has support in the union ofBj,α. Furthermore, if 2p < Qj+ 1≤2p+1, the cluster of defects created by P ·Pec(p) inside Bj,α forms a single 2p+1-connected component of the syndrome examined by EC(p+ 1). This cluster is neutral since we assumedQm+1< Ltqo. HencePec(p+1) will annihilate this cluster. The annihilation operator is equivalent to the restriction of P ·Pec(p) onto Bj,α modulo stabilizers, since the linear size of Bj,α is smaller than Ltqo. It proves the induction hypothesis for the levelp+ 1.
The preceding lemma says that errors by which the RG decoder could be confused are those from very high level chunks. What is the probability of the occurrence of such a high level chunk if the error is random according to Eq.(7.3)? Since our probability distribution of errors depend only on the number of sites inE, this question is completely percolation-theoretic.
Let us review some terminology from the percolation theory[121]. An event is a collection of configurations. In our setting, a configuration is a subset of the lattice. Hence, we have a partial order in the configuration space by the set-theoretic inclusion. An eventE is said to beincreasing ifE ∈ E, E⊆E0 impliesE0 ∈ E. For example, the event defined by the criterion that there exists an error at (0,0), is increasing. The disjoint occurrence A ◦ B of the events AandB is defined as the collection of configurationsE such thatE=Ea∪Eb is a disjoint union ofEa∈ AandEb∈ B.
To illustrate the distinction between A ◦ B andA ∩ B, consider two events defined as A= “there are errors at (0,0) and at (1,0)”, andB= “there are errors at (0,0) and at (0,1)”. The intersection A ∩ B contains a configuration{(0,0),(1,0),(0,1)}, but the disjoint occurrence A ◦ B does not. A useful inequality by van den Berg and Kesten (BK) reads [122,121]
Pr[A ◦ B]≤Pr[A]·Pr[B] (7.5)
provided the eventsAand Bare increasing.
Proof of Theorem 7.1. Consider aD-dimensional lattice and a random errorEdefined by Eq. (7.3).
LetBn be a fixed cubic box of linear sizeQn andBn+ be the box of linear size 3Qn centered atBn. Define the following probabilities:
pn = Pr [Bn has a nonzero overlap with a level-nchunk ofE]
˜
pn = Pr [Bn+ contains a level-nchunk ofE]
qn = Pr [Bn+ contains 2 disjoint level-(n−1) chunks ofE]
rn = Pr [Bn+ contains a level-(n−1) chunk ofE]
Note that all these probabilities do not depend on the choice of the box Bn due to translation invariance. Since a level-0 chunk is just a single site ofE, we havep0=. We begin by noting that
pn ≤p˜n≤qn.
Here we used the fact that any level-nchunk has diameter at mostQn/2 and that any level-nchunk consists of a disjoint pair of level-(n−1) chunks. Let us fix the box Bn+ and let Qn be the event thatBn+ contains a disjoint pair of level-(n−1) chunks ofE. LetRn be the event thatBn+contains a level-(n−1) chunk ofE. ThenQn =Rn◦ Rn. It is clear thatQn and Rn are increasing events.
Applying the van den Berg and Kesten inequality we arrive at qn≤r2n.
Finally, sinceB+n is a disjoint union of (3Q)D boxes of linear sizeQn−1, the union bound yields rn≤(3Q)Dpn−1.
Combining the above inequalities we getpn≤(3Q)2Dp2n−1, and hence
pn ≤(3Q)−2D((3Q)2D)2n.
The probabilitypn is doubly exponentially small innwhenever <(3Q)−2D. If there exists at least one level-nchunk, there is always a box of linear sizeQn that overlaps with it. Hence, on the finite system of linear size L, the probability of the occurrence of a level-m chunk is bounded above by LDpm. Employing Lemma 7.5.2, we conclude that the RG decoder fails with probability at most pf ail=LDpmfor anymsuch thatQm+1< Ltqo. Since we assumed thatLtqo≥Lδ, one can choose m≈δlogL/logQ. In this casepf ail = exp (−Ω(Lη)) forη≈δ/logQ. We have proved our theorem with0= (3Q)−2D whereQ= 10.