Conference PaperPDF Available

Quasilinear Time Decoding Algorithm for Topological Codes with High Error Threshold

March 2020

Conference: International Symposium on Computer Architecture (ISCA) 2020

Authors:

Shui Hu

Delft University of Technology

David Elkouss

Okinawa Institute of Science and Technology Graduate University

We propose here a modification of the UF decoder that improves the heuristic for minimum-weight matching. The modified decoder, which we dub the Union-Find Balanced Bloom decoder (UFBB), achieves near MWPM thresholds while retaining a quasilinear time complexity.

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Quasilinear Time Decoding Algorithm for

Topological Codes with High Error Threshold

Mark Shui Hu1, David Elkouss

One of the most promising approaches for fault-tolerant quantum computation is based

on surface quantum error correcting codes [1, 2]. With surface codes, error correction

only requires the measurement of local operators on a 2-dimensional lattice of qubits. The

measurement outcome, called the syndrome, is passed to the decoding algorithm to deduct

the error that has occurred and to supply a correction operator. The resilience against

errors can be improved by increasing the system size whilst the physical error rate is below

a threshold value pth. For this, it is essential that the decoder has low time complexity; if

the clock-rate of the quantum computer becomes limited by the decoder, the advantages

of increasing the system size could be compromised.

Arguably, the most popular decoder for surface codes is the Minimum-Weight Perfect

Matching (MWPM) decoder [1]. The basic principle behind this approach is to identify

the lowest weight error conﬁguration that can produce the syndrome. In general this is

a good approximation to the optimal maximum likelihood decoder [3]. For a toric code

that only suﬀers random Pauli noise, the optimal code threshold is pth = 10.9%, whereas

the MWPM decoder has pth = 10.3%. The MWPM matching are found by constructing

a fully connected graph between nodes of the syndrome, which leads to a cubic worst-case

time complexity of O(n3) [4].

Many other decoding algorithms have been developed [5–12]. Here, we build on top

of a recently proposed decoder called the Union-Find (UF) decoder. It combines a very

low time complexity with a high threshold [13, 14] making it a practical solution for real

devices. The UF decoder maps each syndrome to a vertex in a non-connected graph on the

code lattice, and grows clusters of vertices locally by adding iteratively a layer of edges and

vertices to existing clusters until all clusters have an even number of non-trivial syndrome

vertices. It then trims the clusters until all non-trivial syndrome vertices are paired and

linked by a path, which is the correcting operator. By growing the clusters of vertices in

order of their sizes, the UF-decoder can be regarded as a heuristic for minimum-weight

matching, and has a threshold of pth = 9.9% for the toric code. The complexity of the UF

decoder is driven by the merging between clusters. For this the algorithm uses the Union-

Find or disjoint-set data structure [15], which has worst-cast time complexity O(Nα(N)),

where αis the inverse of Ackermann’s function. For any physical feasible amount of qubits,

this value is α(N)≤3, leading to an “almost-linear” time complexity.

We propose here a modiﬁcation of the UF decoder that improves the heuris-

tic for minimum-weight matching. The modiﬁed decoder, which we dub the

Union-Find Balanced Bloom decoder (UFBB), achieves near MWPM thresh-

olds while retaining a quasilinear time complexity.

In the following we give some intuition into how the UFBB decoder works. Consider

the cluster of vertices V={v0, v1, v2}in Figure 1. Now let us investigate the weights of a

matching if an additional vertex v0is connected to the cluster. If v0is connected to v0or

to v2, then the resulting matching has a total weight of 2: (v0, v0) and (v1, v2), or (v0, v1)

and (v2, v0). However, if v0is connected to vertex v2, then the total weight is 3: (v0, v1)

and (v0, v2). Inspired by this idea, we introduce the concept of potential matching weight

(PMW) of a vertex.

1S.Hu-1@student.tudelft.nl

The calculation of the PMW requires to introduce a new data structure that we call

the node-set of a cluster. The node-set of a cluster is a partition of the cluster such that

each element of the partition consists of a set of adjacent vertices that are either interior

elements in the associated graph or have equal PMW (Figure 2). The calculation of the

PMW is performed by two depth-ﬁrst searches of the node-set (Figure 3), where the PMW

is translated into a node delay that determines the priority for cluster growth, such that

the growth of a cluster moves towards equal PMW in the cluster (Balanced Bloom).

The UFBB decoder has a higher complexity than the UF decoder. Due to the merges

of clusters, the PMW values in a cluster change and need to be recalculated, which in turn

requires to join the associated node-sets. These join operations are more complex than

those of the UF (union of vertex sets). However, we can show that the worst-case time

complexity of UFBB is O(nlog n), which is still quasilinear.

Our numerical results show that the error threshold of the UFBB decoder (Figure 4)

is pth = 10.25% compared to 10.02% for our implementation of the UF decoder and near

the threshold of MWPM 10.35% (thresholds diﬀer from reported value due to changes

in cluster sorting, ﬁtting parameters, etc.). Including measurement errors, the UFBB

decoder achieves a threshold of 2.86%, compared to 2.70% (UF) and 2.98% (MWPM).

We also ﬁnd that the numeric average-case time complexity (for a range of error rates in

Figure 5) matches the complexity of the UF decoder up to a constant factor.

[1] Eric Dennis et al. “Topological quantum memory”. In: Journal of Mathematical Physics 43.9 (2002),

pp. 4452–4505.

[2] A Yu Kitaev. “Fault-tolerant quantum computation by anyons”. In: Annals of Physics 303.1 (2003),

pp. 2–30.

[3] Sergey Bravyi, Martin Suchara, and Alexander Vargo. “Eﬃcient algorithms for maximum likelihood

decoding in the surface code”. In: Physical Review A 90.3 (2014), p. 032326.

[4] Vladimir Kolmogorov. “Blossom V: a new implementation of a minimum cost perfect matching

algorithm”. In: Mathematical Programming Computation 1.1 (2009), pp. 43–67.

[5] Austin G Fowler. “Minimum weight perfect matching of fault-tolerant topological quantum error

correction in average O(1) parallel time”. In: arXiv preprint arXiv:1307.1740 (2013).

[6] Guillaume Duclos-Cianci and David Poulin. “Fault-tolerant renormalization group decoder for Abelian

topological codes”. In: arXiv preprint arXiv:1304.6100 (2013).

[7] Adrian Hutter, Daniel Loss, and James R Wootton. “Improved HDRG decoders for qudit and non-

abelian quantum error correction”. In: New Journal of Physics 17.3 (2015), p. 035017.

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qudit surface codes”. In: Physical Review A 92.3 (2015), p. 032309.

[9] David K Tuckett, Stephen D Bartlett, and Steven T Flammia. “Ultrahigh error threshold for surface

codes with biased noise”. In: Physical review letters 120.5 (2018), p. 050505.

[10] Aleksander Kubica and John Preskill. “Cellular-automaton decoders with provable thresholds for

topological codes”. In: Physical review letters 123.2 (2019), p. 020501.

[11] Giacomo Torlai and Roger G Melko. “Neural decoder for topological codes”. In: Physical review

letters 119.3 (2017), p. 030501.

[12] Savvas Varsamopoulos, Ben Criger, and Koen Bertels. “Decoding small surface codes with feedfor-

ward neural networks”. In: Quantum Science and Technology 3.1 (2017), p. 015004.

[13] Nicolas Delfosse and Gilles Z´emor. “Linear-time maximum likelihood decoding of surface codes over

the quantum erasure channel”. In: arXiv preprint arXiv:1703.01517 (2017).

[14] Nicolas Delfosse and Naomi H Nickerson. “Almost-linear time decoding algorithm for topological

codes”. In: arXiv preprint arXiv:1709.06218 (2017).

[15] Robert Endre Tarjan. “Eﬃciency of a good but not linear set union algorithm”. In: Journal of the

ACM (JACM) 22.2 (1975), pp. 215–225.

Figure 1: Unbalanced matching

weight in cluster vertex set V.

The matching edges (dashed)

correspond to the position of v0.

σ0σ1σ2

Figure 2: A node-set Nvs. a

vertex set V, both representing

the same cluster. Each shaded

area covers the vertices of a dif-

ferent node.

Parity

Delay

Figure 3: Two depth-ﬁrst

searches on Nto compute node

parities (head recursively) and

delays (tail recursively).

Figure 4: Decoder success rate and threshold. Figure 5: Decoder performance in computation time.

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Quasilinear Time Decoding Algorithm for Topological Codes with High Error Threshold

Abstract

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