In this section we prove that any topological stabilizer code can tolerate stochastic local errors with a small constant rate assuming that the error correction is performed using the RG decoder. We assume without loss of generality that each stabilizer generator is supported on a unit cube. Each site of the lattice may contain finitely many qubits. A generator at a cubecmay act only on qubits ofc. We shall assume that errors at different sites are independent and identically distributed. More precisely, letE(P) be the set of sites at which a Pauli error P acts nontrivially. We shall assume that
Pr[E(P) =E] = (1−)V−|E||E| (7.3) where 0 ≤ ≤ 1 is the error rate and V = LD is the total number of sites (the volume of the lattice). For example, the depolarizing noise in which every qubit experiences X, Y, Z errors with the probabilityp/3 each, satisfies Eq. (7.3) with the error rate= 1−(1−p)q, whereqis the number
of qubits per site.
Theorem 7.1. Suppose a family of stabilizer codes has topological order satisfying TQO1,2. (In particular, every translationally invariant exact codes do.) Then, there exists a constant threshold
0>0such that for any < 0 the RG decoder corrects random independent errors with rate with
the failure probability at moste−Ω(Lη)for some constant η >0.
In the rest of this section we prove the theorem. Our proof borrows some techniques from [119,120, 114], specifically Section 5.1 of Gray’s review [119] on G´acs’ 1D cellular automata [120].
Recall that we use `∞-metric, so a cube of linear size r thus has diameter r. We keep the terminologies and conventions from Section 7.1, and our decoder is what we have explained in the previous sections: The level-perror correction EC(p) on a syndromeS is the following subroutine. (i) find all neutral 2p-connected componentsM ofS, (ii) for eachM found at step 1, calculate and
apply a Pauli operatorP supported on the 1-neighborhood ofb(M) that annihilatesM, and update the syndrome accordingly. Calling the full RG decoder on a syndrome S involves the following steps: (i) run EC(0), EC(1), ..., EC(blog2Ltqoc), (ii) if the resulting syndromeS is empty, return
the accumulated Pauli operator applied by the subroutines EC(p). Otherwise, declare a failure. Below we shall use the term ‘error’ both for the error operator P and for the subset of sites E acted on by P, whenever the meaning is clear from the context. Let us choose an integer Q 1 and find a class of errors which are properly corrected by the RG decoder, see Lemma7.5.2below. We will see later that this class of errors actually includes all errors which are likely to appear for small enough error rate.
Definition 7.1. Let E be a fixed error. A site u ∈ E is called a level-0 chunk. A non-empty subset ofE is called a level-n chunk (n≥1) if it is a disjoint union of two level-(n−1) chunks
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 E
n 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 ofF
n. ThenM has diameter
≤Qn and is separated fromE
n\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∈F
n andv∈En
such thatQn< d(u, v)≤Qn+1/3. LetC
u3uandCv 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 3Q
1+min (j,k)−2. (7.4)
LetPec(p) be the accumulated correcting operator returned by the levels 0, . . . , pof the RG decoder.
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 thatP
(0)
ec 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. HenceP
(p+1)
ec 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)
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 B+
n and let Qn be the event
thatB+
n 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)2
n .
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