valid for any X ∈ D. Indeed, any quantum jump operatorAα,ω changes the energy at most byf, so thatL+(Q⊥mX) =L+(XQ⊥m) = 0 for anyX ∈ D(note that anyX ∈ Dcommutes withQm). We arrive at
(t)≤t· kL+(QmeLs(ρ0)Qm)k1≤t· kL+k1· kQmeLs(ρ0)Qmk1≤O(tN) TrQmeLs(ρ0), (8.15)
where the maximization over s is implicit. We used Proposition8.3.1 and the positivity ofL(ρ0).
in the last inequality. Since the ground-state energy ofH is zero, one has
ρ0=Zβρβρ0, (8.16)
whereZβ is the partition function. It yields
TrQmeLs(ρ0) = Trρ0eL∗s(Qm) =ZβTrρβρ0eL∗s(Qm) =Zβhρ0, eL∗s(Qm)iβ, (8.17) Proposition 8.3.4 implies that the map eL∗s is also self-adjoint with respect to the Liouville inner product. Hence, we have
TrQmeLs(ρ0) =Zβhρ0, eL∗s(Qm)iβ =ZβheL∗s(ρ0), Qmiβ= TreL∗s(ρ0)Qme−βH ≤TrQme−βH, (8.18) where the last inequality is becauseeL∗s is a unital completely positive map andρ0≤I.
Let Γ = (P0, P1, . . . , Pt) be an error path implementingP with the energy costm(Γ) =m. Here P0=I, Pt=P, whileEj ≡PjPj−1 are single-qubit Pauli operators for all j. Lemma6.1.1implies that there is a levelpmax< m such that in the level-pmax syndrome history only the initial empty syndromeS(0) = 0 and the final syndromeS(t) =S are possibly non-sparse.
Lemma 8.5.1. Let P be any Pauli error, S=S(P)be its syndrome, and m= ∆(P) be its energy barrier. Suppose
16m(10α)m< Ltqo. (8.19)
Then, there exists a stabilizer G ∈ G such that P ·G has support on the ξ(m)-neighborhood of S.
Any R-connected component ofS is neutral for2ξ(m)< R≤4ξ(m).
Proof. Let us apply Lemma 6.1.3 to the level pmax, the smallest integer such that the level-pmax
syndrome history has the initial and final syndrome, and level-pmax-sparse syndromes. Sincepmax<
m, the condition 16mξ(pmax) < Ltqo in Lemma6.1.3 is satisfied. We have S0 = 0, S00 =S, and E = P. Hence, there exists a stabilizer G ∈ G such that P ·G is supported on the ξ(pmax)- neighborhood ofS. It proves the first statement of the lemma.
Let r=ξ(m) = (10α)m. Choose any R such that 2r < R≤4r and letCa be anyR-connected component ofS. SinceCacontains at mostmdefects, the diameter ofCais at mostmR. Restricting P·Gon ther-neighborhood ofCa, we obtain a Pauli operatorPa supported on a cube of linear size at most mR+ 2r≤4rm+ 2r < Ltqo by assumption. Furthermore, the support ofPa is separated from (Pa)−1(P ·G) by distance at least R−2r > 0. Hence, Pa creates the cluster Ca from the vacuum. Therefore,Ca is neutral.
We wish to have a well-separated cluster decomposition.
Lemma 8.5.2. Let S be any cluster of m > 0 defects. For any integer µ ≥ 1, there exists a nonnegative integer p < mand a decomposition
S =C1∪ · · · ∪Cn such that d(Ca)≤4pµ and d(Ca, Cb)> 1
2·4p+1µ fora6=b. (∗) Proof. The only nontrivial part is that pcan be chosen as p < m. Let us say that a partition of S into clusters isp-good if it satisfies (∗). By grouping all defects occupying the same elementary cube into a cluster, one obtains a partitionS =C1∪. . .∪Cg. Obviously,g ≤m, andd(Ca)≤µ.
If this partition is not 0-good, theng≥2 and there is a pair, say,C1, C2such that d(C1, C2)≤2µ.
MergingC1 andC2 into a single clusterC20, one obtains a partition S =C20 ∪C3∪. . .∪Cg where d(C20)≤4µ. If this partition is not 1-good, theng≥3 and one can repeat the merging again. After at mostg−1 iterations, one arrives at a good partition.
Note that the minimal enclosing boxes of distinct cluster do not overlap, since
d(b(Ca), b(Cb))>2·4pµ−4pµ−4pµ= 0.
The following is the desired property of the RG decoder.
Lemma 8.5.3. LetP be any Pauli error with energy barrier m= ∆(P). Suppose (160α)m< Ltqo.
Then calling the RG decoder on the syndromeS(P)returns a correcting operatorPecsuch thatP Pec is a stabilizer. Thus, the RG decoder correctsP if∆(P)< log(160α)γ logL.
Proof. Let S = S(P) be the syndrome. Let p be the integer such that 2ξ(m) < 2p ≤ 4ξ(m).
Setting µ = 2p in Lemma 8.5.2, S is decomposed into S = C1∪. . .∪Cn such that d(Ca) ≤ 2p0 and d(Ca, Cb) > 2p0+1 for all a 6= b, where p0 is an integer such that p ≤ p0 < 2m+p. Since 16m(10α)m ≤(160α)m, Lemma8.5.1 implies that eachCa is neutral for being a disjoint union of neutral 2p-connected components. The RG subroutines EC(s) with s = 0,1, . . . , p−1, can only annihilate some neutral 2s-connected components of Ca, which does not alter the neutrality ofCa. Therefore, the RG decoder from level-0 topwill annihilate each clusterCa, and henceS at last.
We need to show thatP·Pecis a stabilizer, wherePecis the returned correcting operator. LetBa be the (10α)m-neighborhood of b(Ca). Our assumptions imply thatBa has diameter smaller than Ltqo and distinctBa’s do not intersect. By construction, the operatorsPec andP·Ghave support in the unionB1∪ · · · ∪Bn. Therefore,P·G·Pec=Q1· · ·Qn, whereQahas support onBa and has trivial syndrome. Topological order condition implies thatQa are stabilizers, so is the product.
The full hierarchy of the RG decoder is not necessary to correct the error with the low energy barrier. A single level-p error correction with p proportional to logLtqo, will be sufficient. We nevertheless include the hierarchy since in practice it corrects errors with slightly higher (although only by a constant factor) energy barrier at a marginal slowdown of the decoder. If one wishes to apply the decoder against random errors, the hierarchy becomes necessary, as we have discussed in Section7.5.
Remark 8.1. A closer analysis reveals a simplification of TestNeutral defined in Section 7.2 for the 3D Cubic Code. We defined TestNeutral to return the identity operator if a cluster turns out to be charged. The modified TestNeutral0 just applies the broom algorithm and returns recorded operator in any case. It gives thesame characteristic as stated in the Lemma8.5.3. EC0(p) using TestNeutral0 will transform a charged clusterCa to adifferentcluster Ca0, butCa0 is still contained in b(Ca). Due to Lemma 8.5.2, b(Ca) do not overlap at a high level p, and EC0(p) will eliminate
neutral clusters at last. This specialized version of RG decoder is used in our numerical simulation in Section8.7.