Decoder Construction

The central concept required for the description of the HDRG decoder is that of a metric – a geometric distance function between any pair of elements from a set. Specifically, we wish to associate a metric between pairs of syndromes in the set S0. The metric we use for the decoder presented in this chapter is the Manhattan distance, denoted here by δ, which maps two syndromes as follows:

4.4. HDRG Decoder with Noisy Syndromes

The set of syndromes at a fixed distance δ from a chosen syndrome form an approxi- mately octahedral structure around it in 3D – see figure 4.2for an illustration for how this grows as δ increases.

We say that two syndromes are δ-connected if the Manhattan distance between them is less than or equal to δ. For a given metric value δ, we define a cluster C to be the set of non-trivial syndromes such that every element of the set is δ-connected to at least one other syndrome within that cluster. For a fixed δ, the syndrome changes history S0 can always be partitioned into a set of disjoint clusters such that S0 =C1∪ C2∪ · · · ∪ Cn, for some integer n.

t_{− 1}

t_{− 2}

t

Figure 4.2: An illustration of the Manhattan distance metric. The figure shows the same portion of the lattice at each of five time steps from the syndrome changes history. The yellow plaquettes are 1-connected to the central red plaquette. Blue and yellow plaquettes are 2-connected to the central plaquette.

For each partitioning, or clustering, of S0 corresponding to a fixed δ we classify clusters in one of three ways. This classification depends on the cluster charge, the sum over all the charges of the syndromes contained in the cluster _⊕Cst,x,y, where

the summation is performed modulo d.

• Neutral clusters. The total charge is zero. Such a cluster can be annihilated by fusing all of the syndromes contained within the cluster locally, meaning that the Pauli correction operator e0 will have support only within a box enclosing the cluster.

• Boundary-neutral clusters. The charge is non-zero but the cluster is δ-connected to any of the three smooth boundaries (two spatial and one time-like). Clusters of this type can be annihilated by fusing the syndromes locally and then connecting the remaining charge to the nearest smooth boundary.

4.4. HDRG Decoder with Noisy Syndromes

• Charged clusters. The charge is non-zero and the cluster is more than a distance δ from a smooth edge. No annihilation is possible and consequently the cluster is assumed to be part of a larger neutral cluster.

We note that the t = L boundary is both rough and smooth. This simply means that we allow both vertex and plaquette syndromes to form boundary-neutral clusters if they are δ-connected to this boundary. Therefore boundary-neutral clusters can be defined in relation to two physical boundaries, either rough or smooth depending on the type of the syndrome, and one time-like boundary.

Here we explicitly explain how the decoder calculates e0 for a given cluster by introducing the concept of syndrome transportation: a syndrome can be transported in any direction by applying the appropriate operator as illustrated by the example in figure 2.9(c). In doing this the charge at the initial syndrome location becomes zero, while the original charge a appears at one of the four neighbouring plaquettes. This is where the term ‘transportation’ arises – the syndrome value appears to move. Moreover, by transporting one syndrome to the location of a second, the two become fused into a single syndrome such that their charges are added (modulo d).

The HDRG decoder involves multiple levels of decoding to eliminate the non-trivial syndromes by fusing all the elements in S0 and returning the resultant correction operator. Every decoding level ` is associated with a distance determining the connectivity of the disjoint clusters at that level. For the metric we have defined we will use δ = 2` _{starting with ` = 0 and running to a maximum length scale}

of ` = <sub>dlog2</sub>L<sub>e. The algorithm terminates if all of the non-trivial syndromes</sub> are removed before the maximum level is reached. This means that the cluster connectivity increases exponentially as we increase the decoding levels linearly1. At each level the neutral and boundary-neutral clusters are removed using the techniques listed above, leaving any charged clusters to be combined to form neutral clusters at subsequent levels. The correction operator identified at each levelE0<sub>`</sub> is tracked and the total correction calculated as E0 = E0₁E0₂. . . E0<sub>`max</sub>, where `max is the level where

the decoder terminated.

1_{The exponential scaling of the metric improves the run time of the decoder at the expense}

of the threshold. However, the first two levels are identical when the metric increases linearly. This means that the threshold is not affected significantly by the choice of exponential scaling. Indeed, numerical tests indicate that the threshold values using exponentially and linearly increasing metrics, p(d)_th,e and p(d)_th,l respectively, satisfy 0.95p(d)_th,e< p(d)_th,l< 1.05p(d)_th,e for all d tested.

4.4. HDRG Decoder with Noisy Syndromes

The decoding procedure can now be summarised as follows, starting with ` = 0.

1. Clustering: Identify all the disjoint δ-connected clusters at level `.

2. Neutral annihilation: Fuse each neutral and boundary-neutral cluster locally and return a correction operator.

3. Renormalise : If there are clusters that have not been annihilated, then incre- ment ` by 1 and return to step 1.

The decoder stops when there are no non-trivial syndromes remaining. The crucial feature of this decoder is that part of the total correction operator is fixed after each level of decoding. In classical coding theory, decoding algorithms exhibiting such a feature are referred to as a hard-decision decoders. An explicit example for a small lattice simulation is illustrated in figure 4.3.