Trapping Sets of Quantum LDPC Codes - Nithin Raveendran and Bane Vasić
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Trapping Sets of Quantum LDPC Codes
Nithin Raveendran and Bane Vasić
Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721, USA
Iterative decoders for finite length quan- The existing QLDPC code literature primarily fo-
tum low-density parity-check (QLDPC) cuses on constructing asymptotically good code
codes are attractive because their hard- families with improved minimum distance scal-
ware complexity scales only linearly with ing with the block length and higher code rates,
the number of physical qubits. However, as well as on designing better iterative decoding
they are impacted by short cycles, detri- algorithms [6–12]. However, QLDPC codes im-
mental graphical configurations known as plemented in practical QEC systems will be of
trapping sets (TSs) present in a code graph finite length and will exhibit performance degra-
arXiv:2012.15297v2 [cs.IT] 7 Oct 2021
as well as symmetric degeneracy of errors. dation due to the failure of iterative decoders to
These factors significantly degrade the de- converge to a correct error pattern. This phe-
coder decoding probability performance nomenon specific to finite-length codes is well un-
and cause so-called error floor. In this pa- derstood in classical literature, but similar anal-
per, we establish a systematic methodol- ysis for QLDPC codes and precise mathematical
ogy by which one can identify and clas- characterization requires further attention in the
sify quantum trapping sets (QTSs) accord- QEC literature [7, 13]. The convergence failure
ing to their topological structure and de- manifests itself as an error floor of the decod-
coder used. The conventional definition of ing probability of error [14] at low physical error
a TS from classical error correction is gen- rate levels – an operating regime for large-scale
eralized to address the syndrome decod- FTQCs – and is observed in all state-of-the-art
ing scenario for QLDPC codes. We show iterative message-passing decoders for QLDPC
that the knowledge of QTSs can be used codes such as belief propagation (BP), min-sum
to design better QLDPC codes and de- algorithm (MSA) and their variants [15–17].
coders. Frame error rate improvements of A typical approach in QEC literature to reduce
two orders of magnitude in the error floor the error floor of the above decoding algorithms
regime are demonstrated for some practi- is to couple them with ordered statistics decoding
cal finite-length QLDPC codes without re- (OSD) and post-processing [15, 17]. However, al-
quiring any post-processing. though exhibiting good performance, this tech-
nique is too complex to implement in hardware
due to the high complexity of the OSD algorithm
1 Introduction [18] which scales cubically with the code dimen-
sion (see Eq. 4 in [19]). In contrast, the philos-
Quantum low-density parity check (QLDPC) ophy of our approach and our ultimate goal is
codes are an important class of quantum er- to develop message-passing decoders for QLDPC
ror correction (QEC) [2, 3] codes that can re- codes that do not require a post-processing step
alize scalable fault-tolerant quantum computers to achieve strong error correction.
(FTQCs) with a finite multiplicative overhead
Iterative message-passing decoder operates on
[4]. In addition, they have finite asymptotic
a Tanner graph which is the graphical represen-
rates with non-zero fault-tolerant thresholds [5],
tation of a parity check matrix of the underlying
and support low-complexity iterative decoding.
code. Error floor is attributed to the presence
Nithin Raveendran: nithin@email.arizona.edu of specific topologies of sub-graphs in the Tan-
Bane Vasić: vasic@ece.arizona.edu, ner graph, generically referred to as trapping sets
A part of this paper was presented at the 2021 IEEE Inter-
(TSs) that are detrimental to iterative decoders.
national Symposium on Information Theory (ISIT) [1].
Since a trapping set depends both on the topol-
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 1
ogy of the sub-graph and on the decoder, one ular iterative decoder used along with their loca-
must understand key differences of QEC, specifi- tion in the Tanner graph [22]. In the same flavor
cally QLDPC codes and decoding with respect to as in classical LDPC codes where the knowledge
classical error correction. of trapping sets has resulted in low-complexity
The first difference comes as the fact that the decoders surpassing the traditional BP decoder
stabilizer commutativity/symplectic inner prod- [23], in this paper, we demonstrate two prac-
uct (SIP) requirement for the parity check matri- tical advantages of our quantum trapping set
ces introduces additional code construction con- study: (i) ability to construct QLDPC codes de-
straints resulting in unavoidable cycles in the void of such graphical configurations, and (ii)
Tanner graph. Furthermore, QLDPC codes are ability to devise new decoding algorithms that
known to be highly degenerate, i.e., their mini- escape from such graphical configurations with-
mum distance is higher than the weight of their out post-processing. The message update rules
stabilizers. From the decoder perspective, this and scheduling strategies thus identified help to
implies that the decoders need to account for de- improve the error floor performance.
generate errors, which have no equivalent in clas- The rest of this paper is organized as follows.
sical error correction. However, iterative algo- In Section 2, we introduce QLDPC codes using
rithms based on BP are sub-optimal in the pres- the stabilizer formalism reviewing some basic no-
ence of cycles and, also, are not capable of cor- tations, and then discuss the syndrome decod-
recting all degenerate errors [13, 20]. Another key ing problem and classical trapping sets. In Sec-
difference from classical LDPC decoding stems tion 3, we analyze the different failure configura-
from the inability to directly measure qubits for tions and the relation between trapping sets and
error correction. Hence, the iterative message- decoder-error correction properties. We also for-
passing algorithms used for decoding of QLDPC mally define quantum trapping sets and describe
codes are modified to only make use of the syn- the methodology used to identify those specifi-
drome information to infer the error introduced cally for Calderbank, Shor, Steane (CSS) codes
by the channel. How the classical trapping sets [24]. Trapping sets of some classes of CSS codes
definition accommodates a syndrome-based de- are analyzed in Section 4. Based on these anal-
coder is not clearly understood. As we show, yses, we present simulation results that briefly
degenerate errors having no classical analogy in- explore two strategies of code and decoder im-
troduces new failure configurations unique to the provement. We explore CSS code constructions
QLDPC codes. The approach presented in this without some of the harmful configurations and
paper accounts for these key differences and their compare the performance of trapping set-aware
implications. decoding strategies in Section 5 followed by con-
cluding remarks and future research directions in
Failure configurations of QLDPC codes are rel-
Section 6.
atively unknown when compared to the classical
trapping set research. One major drawback of BP
as pointed out in [21] is that the decoding abil- 2 Preliminaries
ity of BP is typically limited by the row weight
of the parity-check matrix due to the SIP con- 2.1 Stabilizer Formalism
straint and identifies pseudo-codeword structures Stabilizer codes, the quantum analog of classi-
for cycle codes. However, generalization from cy- cal linear codes, are the most common type of
cle codes to QLDPC codes is non-trivial. QEC codes considered in both theory and prac-
In this paper, we define quantum trapping sets tice [25, 26]. An Jn, k, dK quantum stabilizer code
(QTSs) by investigating failure configurations for maps k qubit quantum state |φi to an entan-
syndrome based iterative message passing algo- gled n-qubit codeword |ψi (a unit vector in the
rithms. The quantum trapping set formulation 2n -dimensional Hilbert space) and is defined as
is modified to the syndrome decoding scenario a 2k -dimensional subspace of the Hilbert space
for QLDPC codes considering Pauli X and Z er- which is a common +1 eigenspace of the sta-
rors separately. We identify QTSs of prominent bilizer group S. The n-qubit codeword |ψi is
QLDPC code families and show that the QTSs stabilized by all stabilizer elements in S. i.e.,
must be analyzed in conjunction with the partic- sj |ψi = + |ψi for any sj ∈ S. We denote a gen-
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 2erator set of a given stabilizer group S by the set stabilizer matrix of dimension m × 2n given by
S = {s1 , s2 , . . . , sm }. The stabilizer generators h i
form the m = n − k rows of the corresponding Hb = HX | HZ , (1)
stabilizer matrix Hp whose entries are the single-
qubit Pauli matrices I2 = [ 1001 ], X = [ 01 10 ], Z = where HX and HZ represent binary parity check
1 0 0 −ı matrices used for error correction. Each row in
0 −1 , and Y = ıXZ = ı 0 . Kronecker prod-
ucts of n single-qubit Paulis and scalars ıκ , where Hb denotes a stabilizer generator, and a pair of
κ ∈ Z4 = {0, 1, 2, 3} forms the n-qubit Pauli corresponding columns in HX and HZ represent a
group Pn , of which the stabilizer group S is a qubit. Equivalent to the commutativity relation
commutative subgroup that contains only Hermi- defined for Pauli operators, the stabilizer genera-
tian Paulis and excludes −In . The weight w(P ) of tors commute with each other based on the sym-
a Pauli operator P ∈ Pn is the number of qubits plectic inner product (SIP) in their binary repre-
on which it applies a non-identity Pauli matrix. sentation [25]. Any
h two rows
i p = (pX , pZ ) and
A QLDPC code is a stabilizer code with all stabi- q = (qX , qZ ) of HX | HZ must satisfy p q :=
mod (pX qZT + p bT , 2) = 0. This leads to the
lizer generators having low weight [2]. Analogous Z X
to the classical minimum distance, Jn, k, dK code condition
have logical operators L ∈ Pn \ S that commutes
with all si having minimum weight d. Like the HX HZT + HZ HX
T
= 0, (2)
codeword generators, the logical operators of the
where the right hand side (0) is an m × m zero
code map an n-qubit codeword to another. Logi-
matrix, T denotes the transpose of a matrix, and
cal group L is generated by k logical X generators:
operations (addition and multiplication) are done
LX = {lx1 , lx2 , . . . , lxk } and k logical Z genera-
modulo-2. We will refer to Eq. (2) as the SIP
tors: LZ = {lz1 , lz2 , . . . , lzk } obtained by using
constraint.
either Gottesman’s [26] or Wilde’s algorithm [27].
The stabilizers commute with each other fol-
2.3 Decoding Problem
lowing the commutativity relation between two
n-qubit Pauli operators P and Q defined as fol- Following the approach in [2], to assess perfor-
lows: mance of binary syndrome decoding of CSS codes,
n
Y it is sufficient to consider the two independent bi-
P ◦ Q := Pj ◦ Qj , nary symmetric channels (BSCs) rather than the
j=1
depolarizing channel, thus ignoring the correla-
where Pj ◦ Qj = ±1 if Pj Qj = ±Pj Qj . The Pauli tion between bit flip (X) and phase flip (Z) errors.
operators P and Q commute if P ◦ Q = +1 and In this case, the BSCs for X and Z errors have a
anti-commute if P ◦ Q = −1. Every logical gener- cross-over probability of 2p/3, decoded using HZ
ators commute with the stabilizers, and Lxi com- and HX , respectively.
mutes with every other generators except with Let e = (eX , eZ ) be the binary representation
Lzi ∀i ∈ {1, k}. of a Pauli error acting on the n qubits. The cor-
responding syndrome is computed as
2.2 Stabilizers as binary parity checks σ = [σX , σZ ]
=[ mod (HZ .eT
X , 2), mod (HX .eT
Z , 2)].
An alternative binary representation maps Pauli
matrices to binary tuples as follows: I2 → All-zero syndrome σ = 0̄ indicates that all the
(0, 0), X → (1, 0), Z → (0, 1), Y → (1, 1). stabilizers commute with the error pattern (unde-
More generally, binary representation of an n- tectable error), whereas non-zero entries/ones in
qubit Pauli operator P will be a binary vector σ indicate that some stabilizer generators anti-
of length 2n of the form p = (pX , pZ ), where pX commute with the error pattern (detectable er-
and pZ are of length n each with ones at positions ror). A syndrome based decoder’s task is to es-
of X- and Z-Pauli components, respectively. Such timate the error pattern ê whose syndrome σ̂
a mapping aids in the construction of quantum matches with the initial input syndrome σ. If
stabilizer codes using extensive classical coding σ̂ = σ, the estimated error pattern ê is applied
literature. The binary representation Hb of the to reverse the error e introduced by the channel.
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 3Error correction process is successful if ê = e ⊕ h, set of CNs connected to a VN vj by N (vj ), and
where h ∈ rowspace(Hb ), i.e., if the code word is |N (vj )|, where | · | denotes cardinality, is referred
recovered up to a stabilizer (ê ⊕ e is a stabilizer, to as the degree of the VN vj . Similarly, we can
where ⊕ denotes pairwise XOR). Error correc- define the neighbor set and the degree of a CN
tion fails when the decoder is unable to find an ci as N (ci ) and |N (ci )|, respectively. A (γ, ρ)
error pattern that matches the syndrome σ or QLDPC code have a sparse stabilizer matrix with
when the decoding process results in a logical er- the variable and stabilizer degree upper-bounded
ror, also referred to as miss-correction in classical by γ and ρ respectively. For a subset of VNs, say
coding theory literature. A logical error occurs if K ⊆ V, N (K) denotes the set of CN neighbors.
ê ⊕ e is a logical operator such that post error The induced sub-graph G(K) is the graph con-
correction state is a codeword different from the taining the nodes K ∪N (K) along with the edges
original codeword. We can detect a logical error if {(x, y) ∈ F : x ∈ K, y ∈ N (K)}. The girth, g, of
Pauli representation of ê ⊕ e anti-commutes with the Tanner graph G is the length of the shortest
any of the 2k logical generators. cycle in G. Denote the number of cycles of length
g, g +2, . . . by χg , χg+2 , . . ., respectively. If G has
2.4 Iterative Decoding of CSS codes χg , χg+2 , . . . cycles of length g, g + 2, . . ., then the
χr xr de-
P
cycle enumerator series CYC(x) =
Although the syndrome decoding paradigm is ap- r≥0
plicable to any class of quantum codes, trapping fines the cycle profile of G.
set analysis in this paper is focused on QLDPC
families: hypergraph product (HP) codes [8], bi- The goal of a syndrome-based iterative decoder
cycle codes [2] and generalized bicycle codes [15] Ds is to output an error pattern that matches the
representing the CSS class of codes [24]. An at- input syndrome. This is different from the tra-
tractive property of CSS codes constructed from ditional iterative decoder D that uses the chan-
two classical codes C1 and C2 , where C2⊥ ⊆ C1 is nel information as initial likelihoods to recover
that the parity check matrix can#be written in a the codeword matching to an all-zero syndrome.
Starting from an input syndrome σ and an all-
"
HX 0
separable form: Hb = . CSS-QLDPC zero error vector estimate, Ds performs a finite
0 HZ
codes have a sparse matrix Hb with the SIP con- number `max of iterations of decoding over the
straint: HZ .HXT = 0. Tanner graph. The messages are passed over the
We can perform error correction for the X edges of the Tanner graph from check nodes to
and Z errors separately using HZ and HX matri- their neighboring variable nodes and vice versa
ces, respectively. The corresponding input syn- at every iteration of message passing decoding.
dromes are obtained as σX = mod (HZ .eTX , 2) Decoder update rules and message alphabet size
and σZ = mod (HX .eTZ , 2), respectively. For can be of varying complexity ranging from the
simplicity going forward, we use H, L and σ, e simplest binary message passing algorithms such
for the parity check matrix, logical generator ma- as Gallager-B [28] to finite alphabet iterative de-
trix, input syndrome, and channel error vector, coders [23], and MSA or BP using floating point
respectively. messages [2]. Also, schedule of message passing in
Ds can be implemented with a flooding/parallel
The stabilizer generator matrix/parity check
schedule or a layered/serial schedule. Trapping
matrix H is the bi-adjacency matrix of a bipartite
set analysis presented here is applicable for all
Tanner graph G = (V ∪ C, E), where V repre-
such decoder implementations. We discuss a
sents the set of n qubit/variable nodes (VNs), C
generic syndrome-based iterative decoder in Ap-
is the set of m stabilizer generators/check nodes
pendix A for completeness. Based on the up-
(CNs) and E is the set of edges between them.
date rules, Ds outputs an error vector estimate
CN ci ∈ C and VN vj ∈ V are neighbors if (`) (`) (`)
there is an edge (vj , ci ) ∈ E between the nodes, ê(`) = (ê1 , ê2 , . . . , ên ) and corresponding out-
(`) (`) (`)
corresponding to the non-zero entry in the par- put syndrome σ̂ (`) = (σ̂1 , σ̂2 , . . . , σ̂m ). We re-
(`) (`)
ity check matrix Hci ,vj = 1. Diagrammatically, fer to êj /σ̂i as the value of the variable/check
Tanner graphs are drawn with circles represent- node vj /ci at iteration ` ≤ `max . We conclude
ing VNs, squares representing CNs, and solid- that Ds is successful if the output syndrome σ̂ (`)
lines representing the edges. Let us denote the is equal to the input syndrome σ (we also say that
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 4syndromes are matched ). Then, the n-length er-
ror pattern ê(`) is decided as the most likely er-
ror pattern. The iterative procedure is halted if
successfully matched or if `max number of itera-
tions is reached. At the end of iterative decoding,
the syndrome decoding process is successful if the
syndromes are matched. Otherwise, the decoding
is said to have failed.
2.5 Classical Trapping Sets
A classical trapping set example is shown in Fig. 1
illustrating the failure of a classical iterative de-
coder D on a small sub-graph inside the Tanner
graph. Let us consider a simple binary message
passing decoder - Gallager-B decoder which per-
forms XOR operation at the check nodes and a
Figure 1: An illustration of a failure configuration of
majority voting at the variable nodes. More pre- regular Gallager-B decoder, unable to converge to the
cisely, the outgoing check node message over an all-zero codeword when the input error pattern is a spe-
edge is computed as the XOR of extrinsic (all in- cific weight-three error pattern (v2 , v4 , v5 ) (shaded cir-
coming messages except the edge for the message •
cles ) among the five VNs in the sub-graph. Figures
is updated) variable node messages. The outgo- are marked with the binary messages (next to the ar-
ing variable node message is the majority value rows indicating the direction of the messages passed)
corresponding to the check/variable updates. Upper left
among incoming extrinsic check node messages
figure corresponds to the variable node update at the
and the channel value. The messages passed over zero-th iteration. The subsequent CN and VN updates
corresponding edges are marked next to the di- of the decoding process are indicated by the connecting
rected arrows in the figure. All-zero transmitted arrows. We assume that the rest of the Tanner graph is
codeword has errors only on three variable nodes correct.
•
(v2 , v4 , v5 - shaded circles ) as shown in Fig. 1.
The decoder is unable to converge to the all-zero
graphs observed in classical LDPC codes. Even at
codeword. In fact, the decoder oscillates from
low physical error rate levels, the presence of such
the error pattern (v2 , v4 , v5 ) to (v1 , v3 ) and back
small sub-graphs can result in decoding failures
as its output during the decoding iterations, thus
resulting in the characteristic error floors in their
failing to converge.
decoding performance (frame error rate (FER)
A classical iterative decoder is said to con-
vs. physical error rate) curves.
verge correctly if the decoder output word for any
Harmfulness of a TS is also closely linked to
` ≤ `max matches to the transmitted codeword
the decoder through their critical number µ and
and fails to converge correctly otherwise. A vari-
strength s defined as follows:
able node vj is eventually correct if there exists
a positive integer Ij such that for all iterations Definition 2 Critical number µ of a trapping set
` ≥ Ij , the decoder’s estimate of vj is equal to T is the minimal number of variable nodes that
the transmitted bit value. Then, trapping set is have to be initially in error for the decoder to fail
defined as to converge.
Definition 1 ([16]) A trapping set T for an iter- Let failure inducing set be the set of variable
ative decoder D is a non-empty set of variable nodes that have to be initially in error for the
nodes in a Tanner graph G that are not eventually decoder to fail to converge.
correct. If the sub-graph G(T ) induced by such a
set of variable nodes has a variable nodes and b Definition 3 Strength s of a trapping set T is the
odd degree check nodes, then the trapping set T is number of failure inducing sets of cardinality µ.
conventionally labeled as an (a, b) trapping set.
Two important assumptions are used in the def-
Fig. 2 shows examples of TS induced sub- inition of the critical number and strength of a
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 5where supp denotes the support set (indices of
non-zero elements). During iterative decoding, a
check node ci is eventually satisfied if there ex-
ists a positive integer Ii such that for all ` ≥ Ii ,
(`)
σ̂i = σi . We say that the variable node vi has
(a) (4,2) TS (b) (5,3) TS eventually converged if there exists a positive in-
(`) (`−1)
teger Ii such that for all ` ≥ Ii , êi = êi .
Figure 2: Graphical (Tanner graph) representations of a (`)
Note that the êi is not necessarily the correct
(4, 2) TS and a (5, 3) TS. Odd degree/unsatisfied checks estimate of error on the ith -variable node. With
are shown using black squares.
these definitions, we define quantum TSs as fol-
lows:
TS. The first one is that the minimum distance
Definition 4 A trapping set Ts for a syndrome-
of the code is large compared to the size of the
based iterative decoder Ds is a non-empty set of
TS. The second is an isolation assumption [23]
variable nodes in a Tanner graph G that are not
which ensures that messages from outside the TS
eventually converged or are neighbors of the check
are correct for the TS failure analysis. For ex-
nodes that are not eventually satisfied
ample, from Fig. 1, the critical number µ for
the (5, 3) TS with Gallager-B decoder is 3 and
Remark 1 If the sub-graph G(T s ) induced by
the number of weight-µ error patterns that fail is
such a set of variable nodes has a variable nodes
s = 1. Note that the decoder also fails to cor-
and b unsatisfied check nodes, then the trapping
rect weight-4 error patterns and the weight-5 er-
set Ts is conventionally labeled as an (a, b) trap-
ror pattern in the TS. Error floor of the decoder is
ping set.
dominated by the minimum critical number µmin
The QTSs similar to the TSs in classical LDPC
and the number of weight-µmin failure inducing er-
codes have exactly the same definition as Def.
ror patterns. Analytical/semi-analytical estima-
1, and we refer to them as classical-type trap-
tion of error floors of QLDPC codes using critical
ping sets. The second class of trapping sets
number and strength of TSs is beyond the scope
are specifically the harmful degenerate errors ob-
of this paper and we refer the reader to classical
served within the stabilizers classified as sym-
LDPC literature [16, 22].
metric stabilizer trapping sets. We will see that
in such trapping sets, even though the variable
3 Quantum Trapping Sets nodes eventually converge to some error pattern,
there exist check nodes that are not eventually
As our focus is on the error floor regime, we are in- satisfied. The definitions and assumptions for
terested in error patterns with small weight, well critical number and strength of the QTS remain
below the maximum likelihood (ML) error correc- the same as for the classical trapping set.
tion capability of the QLDPC code for which the In the next two paragraphs, we give examples
syndrome-based iterative decoder Ds fails to con- of these two classes of trapping sets. We assume
verge. Such low-weight error patterns are either that Ds is the well-known Gallager-B decoding
part of a classical-type TS or a symmetric stabi- algorithm. This assumption is made mostly for
lizer, defined as a quantum trapping set (QTS). pedagogical reasons, but also because some trap-
ping sets of Gallager-B are also trapping sets of
3.1 Definition of a Quantum Trapping Set other decoders such as BP or MSA.
After pre-defined number of iterations, `max , of
3.1.1 Classical-type trapping set
iterative syndrome decoding, we declare that the
decoder Ds failed for a particular input syn- We will first show in an illustration why classical-
drome/error pattern if the decoder is not able to type TSs as shown in Fig. 2 are also failure con-
find an error pattern with a syndrome equal to figurations of syndrome decoders. The same er-
the input syndrome. More precisely, a decoder ror pattern of the (5,3) TS given in Fig. 1 is re-
failure is said to have occurred if there does not drawn for a syndrome based Gallager-B decoder
exist ` ≤ `max such that supp(σ̂ (`) + σ) = ∅, in Fig. 3. The syndrome input is all-one vector,
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 6(a) (b)
Figure 4: The Tanner graph representations of the (4, 0)
and (10, 0) symmetric stabilizers with • and • repre-
senting the disjoint sets of variable nodes of the stabi-
lizer.
3.1.2 Symmetric stabilizer trapping set
Figure 3: An illustration of a failure configuration
of syndrome-based Gallager-B decoder [28]. The Recall the quantum decoding problem in Section
shaded squares represent the anti-commuting stabi- 2.3, wherein the decoder needs to identify any re-
lizers/checks. Syndrome iterative decoder starts mes- covery operator such that ê ⊕ e = rowspace(Hb ).
sage passing with an all zero error pattern trying to find This is in contrast to the classical decoding prob-
the true error pattern that matches with the all-one syn-
lem where an exact match of error ê = e is re-
drome but is unable to converge successfully. The mes-
sages passed within the sub-graph in consecutive itera-
quired. In quantum decoding, we say error vec-
tions oscillates showing that the decoder is trapped. tors e and f are degenerate errors if e⊕f is a sta-
bilizer, which makes it equivalent to output any
one of the degenerate errors as the candidate error
pattern for matching the syndrome. However, in
QLDPC codes whose minimum distance is higher
indicated by the black squares and the decoder than their stabilizer weight, some degenerate er-
starts from the all-zero error pattern (no error). rors can be detrimental to iterative decoding. A
The outgoing check node message over an edge is symmetric topology of the stabilizer sub-graph
computed as the XOR of extrinsic variable node that contains degenerate error patterns e and f
messages and the syndrome input value at the of equal weight will result in a decoding failure.
check node, and the variable node message com- We will see more examples of such decoder failure
putation remains the same as in regular Gallager- when the iterative decoder attempts to converge
B decoder. The messages passed over correspond- to error patterns e and f simultaneously, thus not
ing edges are marked next to the directed arrows. matching the input syndrome. This failure can
Note that even though the messages passed in be attributed to the symmetry of the both the
Fig. 3 are different from that in Fig. 1 of the reg- stabilizer and the decoder message update rules.
ular Gallager-B decoder, the syndrome decoder is Hence, such errors are referred to as symmetric
also unable to converge, and its output oscillates degenerate errors and corresponding sets of vari-
from the all-zero error pattern to errors in v2 , v4 , able nodes as symmetric stabilizer trapping sets
and v5 and back. Hence, the (5, 3) TS in Fig. 3 is or just symmetric stabilizers, for short. Although
classified as a QTS for the syndrome decoder as degenerate errors are typically classified as harm-
well. less for quantum decoding, from the above dis-
cussion it follows that some (not all) degenerate
In addition to classical-type TSs, iterative de- error patterns in a symmetric stabilizer are harm-
coders on QLDPC codes fail for specific degener- ful for iterative decoders.
ate errors. Our quantum trapping set Definition 4
captures such failure configurations as well. This Definition 5 A symmetric stabilizer is a stabilizer
distinctive difference from classical codes deserves with the set of variable/qubit nodes, whose in-
further analysis in the next section. duced sub-graph has no odd-degree check nodes,
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 7(a) (b)
(a) (b)
Figure 6: Degenerate errors e and f located on • and •
Figure 5: The induced sub-graphs from • and • variable variable nodes, respectively in the symmetric stabilizer
nodes of a (10, 0) symmetric stabilizer trapping set. The in 6(a) result in an iterative decoder failure. Introducing
sub-graphs 5(a) and 5(b) are isomorphic and have the asymmetry during the QLDPC code design can lead to
same odd-degree checks represented using dark squares decoder success taking advantage of degeneracy of the
. QLDPC codes. As an example, for the sub-graph in 6(b),
a BP decoder is able to match to the syndrome (dark
squares represent unsatisfied checks) correctly with the
and that can be partitioned into an even number of red error pattern.
disjoint subsets, so that: (a) sub-graphs induced
by these subsets of variable nodes are isomorphic,
and (b) each subset has the same set of odd degree patterns. It is not difficult to see that such “am-
check node neighbors in its induced sub-graph. biguity” happens for all decoders for which: (a)
the check and message update rules are symmet-
Example 1 Consider the Fig. 4(b) with the sta- ric functions in incoming messages, and (b) in
bilizer sub-graph induced by ten variable nodes the same iteration all variable/check nodes in the
that are partitioned into two disjoint sets with graph apply in parallel the same variable/check
the coloring • and •. The induced sub-graphs update function, respectively. For example, dur-
from • and • variable nodes are shown in the ing the iterations of the Gallager-B decoder, every
Fig. 5. The sub-graphs in Fig. 5(a) and Fig. 5(b) unsatisfied CN sends the binary message, one
are isomorphic and have the same odd-degree back to the VNs. Because of the symmetry, the
checks represented using dark squares . Hence, VNs in both e and f receive exactly the same
the stabilizer shown in Fig. 4(b) satisfies the def- messages, thus converging to e ⊕ f , the symmet-
inition of a symmetric stabilizer. ric stabilizer.
Based on the Definition 4, the set of VNs in-
volved in the symmetric stabilizer form a QTS
Remark 2 The symmetric stabilizer shown in and the sub-graph in Fig. 6(a) is a (10,0) TS
Fig. 4(b) is present in generalized bicycle codes by convention. We can prove as in the following
given in [15]. Surface codes are highly degen- lemma pertaining to the general case.
erate, and symmetric stabilizers, for example as
shown in Fig. 4(a), are ubiquitous in them. Lemma 1 A symmetric stabilizer is an (a, b = 0)
Now, we discuss how degenerate errors within trapping set, and a is even.
the symmetric stabilizer are harmful for iterative
decoders. As a non-trivial example of a symmet- Proof Let the cardinality of the set of VNs of
ric degenerate error, let the error pattern e be lo- the stabilizer be a. By definition, the induced
cated on the • variable nodes in Fig. 6(a). They sub-graph having no odd-degree check nodes im-
result in unsatisfied check shown as . Note, plies that b = 0. Also, according to the symmet-
however, that the sub-graph is symmetric with ric stabilizer definition, the disjoint VN sets that
respect to the vertical axis, and therefore each partitions the stabilizer must have the same odd
erroneous node has a • twin. The set of all • degree check node neighbor set. This implies that
twins form an alternative error pattern f . The there can only be even number of such disjoint
existing iterative decoders fail as they simulta- sets which further implies that the parameter a
neously attempt to converge to both these error is even.
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 8When there are more than a pair (an even number techniques are utilized in the search for classical-
greater than two) of disjoint sets of VNs, the sym- type TSs. Note that there can be more than one
metric stabilizer can be split into smaller symmet- non-isomorphic sub-graphs with the same (a, b)
ric stabilizers. parameters. For example, a (5, 3) TS can have
From the discussion earlier, it follows that sym- non-isomorphic sub-graphs as in Fig. 2(b) and
metric stabilizers are trapping sets not only for Fig. 3. Observe that they all have different com-
the syndrome BP decoder, but for many other binations of short cycles of length six, eight and
iterative decoders, such as bit-flipping, Gallager- ten. Enumeration of cycles and their combina-
B and MSA with different critical number and tions also allows to find harmful classical-type
strength. Harmfulness of symmetric stabilizers TSs in the QLDPC code. Unlike these classical-
associated with decoders is distinct from classical- type TSs, the search for symmetric stabilizer TSs
type trapping sets, as summarized in the follow- requires a different approach of finding low-weight
ing lemma. codeword sub-graphs [31] with additional symme-
try constraints. In the case of CSS codes, the HZ
Lemma 2 For an (a, 0) symmetric stabilizer TS even-weight stabilizer generators are examples of
with any iterative decoder with a critical number symmetric stabilizer TSs for iterative decoding
a/2, no error pattern on more than a/2 nodes of over the Tanner graph of HX matrix and vice-
the symmetric stabilizer is a trapping set. versa. After obtaining the list of relevant QTSs,
we can perform decoder simulation with an iter-
Proof Consider an (a, 0) symmetric stabilizer ative decoder Ds to verify their relative harmful-
with critical number a/2 for a syndrome decoder ness. In the next section, we find and enumerate
Ds . By the definition of the critical number, any QTSs in some prominent QLDPC code families
error pattern of weight smaller than the critical presented in the literature. We also provide the
number a/2 with support on the symmetric sta- harmfulness analysis of “dominant” QTSs present
bilizer is corrected by the decoder Ds . Error pat- in these code families.
terns of weight larger than the critical number
a/2 with support on the symmetric stabilizer are
4 Trapping Set Analysis of CSS codes
decoded correctly, converging to their respective
low-weight degenerate error pattern. A myriad QLDPC code families have been pro-
posed over the years. They include the CSS-
In Fig. 4(b), if a syndrome decoder Ds is able to
based constructions (bicycle codes [2], hyper-
correct all error patterns of weight smaller than
graph product (HP) codes [8] and their general-
five, it can also correct error patterns of weight
izations [15], expander codes [9]), non-CSS based
six and more by converging to their respective
QLDPC codes [32, 33] and quaternary QLDPC
low-weight degenerate error patterns.
codes [34]. In this section, we analyze trapping
The strength of an (a, 0) symmetric stabilizer sets of CSS based QLDPC codes, the generalized
TS with critical number a/2 is given by the twice bicycle codes, and HP codes, in particular. Simi-
the number of possible partitions into two disjoint lar analysis may be extended to the general class
subsets of VNs that satisfy the symmetric stabi- of stabilizer codes.
lizer definition. Each of such partition (distinct
by their unsatisfied syndromes) contributes two
4.1 Generalized bicycle codes
error patterns each to the decoder failure in the
TS. Bicycle codes [2] were generalized by Kovalev and
Pryadko in [35] as follows: Consider two binary
3.2 Searching for Quantum Trapping Sets n/2×n/2 matrices A and B that commute (AB =
BA). Let
Using the definition of a QTS, one can search
HX = [A, B] and HZ = [B T , AT ].
for small sub-graphs in the Tanner graph of
the QLDPC code to identify and enumerate the The SIP condition is clearly satisfied by defini-
QTSs. There are efficient algorithms for TS tion, and in [35], A and B are chosen as binary
search widely used in classical literature [29, 30] circulant matrices so that they commute. Bicy-
to identify sub-graphs that are (a, b) TSs. Such cle codes are dual containing CSS codes where
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 9B = AT . Compared to the HP codes, these
codes generally have a wider range of parameters;
in particular, they can have a higher rate while
preserving the estimated error threshold [35]. In
[15], Panteleev and Kalachev use binary polyno-
mials over rings to define the circulant matrices
for constructing [[n, k]] family of generalized bi-
cycle codes. The choice of circulant matrices de-
termines the properties of both the classical-type
TSs and the symmetric stabilizers in the code. Figure 7: A (5, 5) TS with 5 variable nodes and 5 odd
Hence, the observations on QTSs in the example degree check nodes (the shaded squares represent the
we discuss next can be generalized to the code odd-degree checks). The degree of every variable node
family. is 5. The blue and red shaded circles for the variable
nodes indicate their relative position in the HX matrix,
from A and B, respectively.
Example 2 For illustration, in our TS analysis,
we chose the A1[[254, 28]] code, where the circu-
lant size is 127, a(x) = 1 + x15 + x20 + x28 + x66 the Gallager-B algorithm is µ = 3. For stronger
and b(x) = 1 + x58 + x59 + x100 + x121 as given decoders such as BP and MSA decoder, the be-
in Appendix B in [15]. The girth of the Tanner havior is more complex and interesting. Any
graph is six, CN degree ρ = 10 and VN degree weight-5 error pattern in the TS in the circulant
γ = 5. matrix A indicated by blue qubits results in a
failure, whereas similar error patterns in the TSs
4.1.1 Classical-type trapping sets in the circulant matrix B (indicated by the red
qubits) are decoded correctly. Such a behavior is
Based on our QTS definition, we search for QTSs typically attributed to the neighborhood of the
of small size (upto a = 5) present in the A1 code TS in the Tanner graph. Whether a TS is harm-
in Ex. 2. As noted in [15], based on the circu- ful depends not just on the graphical configura-
lant matrices in the A1 code, it does not have tion, but also on the neighborhood and the de-
(a, b) trapping sets with b ≤ 5. The (5, 5) TS coder. This “true” behavior of all such quantum
is the most harmful small sub-graph present in TSs can be systematically analyzed and charac-
the Tanner graph, making BP based iterative de- terized by extending the sub-graph expansion-
coders to fail for low weight error patterns. The contraction algorithm [22] developed for classical
critical number and strength of this TS are deter- LDPC codes to quantum codes, and it is left for
mined by the specific decoder used and the neigh- future work.
borhood of the (5, 5) TS. Fig. 7 shows the dense
(5, 5) TS present in both the circulant matrices
4.1.2 Symmetric stabilizer trapping sets
A and B. From their cyclic property, we can lo-
cate 127 (equal to the circulant size) isomorphic Failures of syndrome-based iterative decoding on
(5, 5) TSs in each of them. In the Fig. 7, blue generalized bicycle code also consist of the degen-
and red shaded circles for the VNs indicate their erate error patterns in symmetric stabilizers dis-
relative position in the HX matrix, from A and cussed in the Section 3.1.2. An example of a pair
B respectively. of symmetric degenerate error patterns of weight
The (5, 5) trapping set in Fig. 7 has five vari- five in the Ex. 2 code is shown in Fig. 5(a) and
able nodes. Every variable nodes have exactly the Fig. 5(b). Together, they form a (10, 0) TS shown
same VN degree ρ = 5 and one odd-degree check in Fig. 4(b) referred as a symmetric stabilizer. In-
node neighbor (black squares). The number of terestingly, the blue and red shaded circles indi-
small cycles within the trapping set and their cates the variable nodes relative position as before
symmetry makes this a hard configuration to de- in case of the (5, 5) TS coloring. Also, these er-
code. For simple binary decoders like syndrome ror patterns induce isomorphic sub-graphs - trees
based Gallager-B, any weight three or more er- without any cycles, quite distinct from the error
ror patterns will result in a failure inducing set. patterns in classical-type TSs which are usually
Hence, the critical number for the (5, 5) TS with composed of one or more cycles. Using the cyclic
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 10property of the circulant matrices in the code, we Lemma 3 Tanner graphs of hypergraph product
can easily locate 127 isomorphic symmetric sta- codes have girth at most 8.
bilizers present in the Tanner graph of HX , and
similarly for HZ . Proof The Kronecker product of the constituent
LDPC code graphs results in unavoidable 8-cycles
Remark 3 The input syndrome (dark squares in the Tanner graphs of HX and HZ matrices,
representing odd-degree checks) in both Fig. 5(a) upper bounding the girth of HP codes by 8 [8].
and Fig. 5(b) is not matched correctly using iter- Let the two constituent code Tanner graphs be
ative decoder using a parallel or flooding sched- (V ∪ C) and (V 0 ∪ C 0 ), then one can find a cycle
ule. Breaking such TSs requires use asymmetric of length 8 in the Tanner graph of HX by looking
update of variable node decisions such as used at HZ rows. Consider a variable node v 0 ∈ V 0
in a layered/serial decoder. Since such symmet- and a check node c ∈ C. Let v 0 be connected
ric trapping sets have clear distinction of red and to check nodes c01 and c02 in the graph (V 0 ∪ C 0 ).
blue nodes with respect to the cyclic matrices A Similarly, let the check node c has variable node
and B, we can identify the layered decoder sched- neighbors v1 and v2 in the graph (V ∪ C). Based
ule that can break such trapping sets. on the Kronecker product, the following length-8
cycle is formed involving variable node (v1 , v 0 ) in
4.2 Hypergraph product codes the hypergraph product: (v1 , v 0 )-(v1 , c01 )-(c, c01 )-
(v2 , c01 )-(v2 , v 0 )-(v2 , c02 )-(c, c02 )-(v1 , c02 )-(v1 , v 0 ).
HP codes by Tillich and Zemor [8] and their
improvements by Kovalev and Pryadko [36] are Our TS search procedure identified thirty
constructed by taking Kronecker product (de- (4, 2) trapping sets and ten (5, 1) trapping
noted as ⊗) of two classical LDPC codes. Us- sets in the classical code. Also, there are two
ing two classical parity check matrices H1 and non-isomorphic topologies of (5, 3) TSs: one
H2 of dimensions m1 × h n1 and m2 × n2 respec-
i hundred and seventy (5, 3) TSs whose induced
tively, we have HX = H1 ⊗ In2 | Im1 ⊗ H2T and graph has a six, eight, and ten-cycle, and fifteen
h i (5, 3) TSs having three eight cycles. All these
HZ = In1 ⊗ H2 | H1T ⊗ Im2 .
trapping sets manifest themselves in the HP
Example 3 For our trapping set analysis, we use code with their count scaling as in the case of
the example of a [[900, 36, 10]] HP code given in small cycles. In Table 2, we enumerate all the
[17] using a symmetric Kronecker product of a smallest QTSs (with a ≤ 5, b ≤ a) present in the
single (n = 24, k = 6, d = 10) classical code. [[900, 36, 10]] HP code having no CN with degree
> 2 in their induced sub-graphs. These values
The classical LDPC code determines the HP code for a and b are chosen as such classical-type
properties and its trapping sets. As we analyze TSs are typically the most harmful for iterative
the cycle profile of the Tanner graph of the classi- decoders. The QTS enumeration of HX and HZ
cal code used in Ex. 3, we observe that there are for symmetric HP codes is the same. Also, the
χr xr for
P
54 cycles of length six (6-cycles) and 160 8-cycles. cycle enumerator series CYC(x) =
r≥0
Table 1: HPG code parameters and number of cycles each QTS sub-graph in the parameters column
in the Table indicates the number of small cycles
Code n m g χg χg+2 present, which we refer to as its cycle profile.
[24,6,10] 24 18 6 54 160 Observe that the (5, 3) QTS with γ = 3 has two
[[900,36,10]] 900 432 6 2268 14496 non-isomorphic topologies with different cycle
profiles. Another interesting observation specific
These cycles appear in the HP codes, multi- to these HP codes having VNs with degree 3
plying according to the size of the classical parity and 4 is the presence of (5, 3) and (5, 5) QTSs
check matrix (m = 18, n = 24) as given in Table with the same cycle profile - CYC(x) = 3x8 .
1. For example, 54 six cycles in the classical code The (5, 5) QTSs are indeed the result of the
gives rise to (54 × 24) + (54 × 18) = 2268 6-cycles Kronecker product in the HP codes. The QTSs
in both HX and HZ matrix of the [[900, 36, 10]] in Table 2 are the main reason for poor iterative
code. This behavior is consistent across the sym- decoding performance of such family of codes.
metric HP code family.
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 11Table 2: QTS enumeration in HX /HZ of [[900, 36, 10]] HP code [17]
Parameters a Parameters
(a,b) (a,b)
Quantum TS Quantum TS
CYC(x) CYC(x)
Count Count
(4,2) (4,4)
2x6 + x8 4x6 + 3x8
720 72
(5,1) (5,4)
2x6 + 3x8 + 2x10 4x6 + 5x8 + 4x10
240 36
(5,3) (5,4)
x6 + x8 + x10 5x6 + 5x8 + 2x10
4080 90
(5,3) (5,5)
3x8 3x8
360 5184
ric property of the stabilizers of the HP code. In
Ex. 3, since the classical parity check code cho-
sen has γ = 3 and ρ = 4 in its Tanner graph,
the variable nodes of the HP code have VN de-
grees 3 and 4. For the stabilizer in Fig. 8, the
VNs with degree γ = 4 are shown as • and those
with γ = 3 as •. The stabilizer is not symmetric
according to the Definition 5. Suppose the input
syndrome corresponds to all the check nodes in
Figure 8: A stabilizer sub-graph in the [[900, 36, 10]] HP the sub-graph in error, then an iterative BP or
code [17] is not symmetric. Note that the red and blue
MSA decoder (based on the update rule) will be
variable nodes have four and three check node neighbors,
respectively. Thus, a BP decoder converges to the red able to successfully converge to the red error pat-
variable nodes as its output exploiting the asymmetry in tern. This happens as every • VN uses messages
the stabilizer. from four CNs compared to three CNs for the •
VNs in the decoding process to successfully con-
Observe that the node degree of the classical verge. Note that the above statement depends on
parity check Tanner graph influences the symmet- the decoder update rule chosen. For example, the
red and blue error patterns in the stabilizer are
1
In Table 2, the parameters-(a,b), CYC(x), and Count indeed failure configurations for a simple binary
are listed row-wise under the column header-Parameters decoding algorithm like Gallager-B algorithm.
for each QTS.
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 12This observation emphasizes the importance of qubit inter-connectivity to perform check opera-
the decoder in characterizing the harmfulness of tions. Using QC LDPC code brings structure to
QTSs as in the case of classical TSs [22]. In the the constituent codes and flexibility in improv-
subsequent section showing the applications of ing finite length QLDPC codes along with effi-
TS analysis, we will use the example of different cient implementation of decoders. Instead of the
decoding schedule that “breaks” such symmetry random codes which are only optimized for girth
of the stabilizer. g = 6, we construct QC [40, 10, 12] code with
girth 8, and make sure that small trapping sets
are not present. The QC code with circulant size
5 Using the QTS to design better Q = 10 is constructed by carefully choosing cir-
QLDPC codes and better decoders culants to build the Tanner graph that is free of
(4, 2), (4, 4), and (5, 1) harmful trapping sets. For
In this section, we explain practical importance of a fair comparison, we use a random code of the
QTS analysis by providing two approaches for fi- same size and minimum distance. The random
nite length QLDPC code and decoder design with [40,10,12] code is constructed to be cycle-4 free.
QTS knowledge. The HP codes constructed from these constituent
classical codes both have the same number (=
5.1 Improved Code Design 21600) of unavoidable length-8 cycles (see Lemma
3) in their respective Tanner graphs, as it only
While previous research has observed the issue of depends on the size of the component code and
symmetric degenerate errors [13, 20], there has their variable and check node degrees. For the
been no effort to fully characterize them. Iden- symmetric HP code fromregular LDPC code, we
can count these as m × ρ2 × n × γ2 , where m and
tifying symmetric stabilizers, particularly of low
weight in the QLDPC code is an important step n correspond to the number of check nodes and
in quantifying the effect of degenerate errors on variable nodes, respectively, and dv and dc are
iterative decoding. Removal of symmetry in low- variable and check node degrees, respectively of
weight stabilizers present in the Tanner graph, the constituent classical code. These 8-cycles can
especially during the QLDPC code design, sig- be classified as (4, 6) TSs and are not harmful for
nificantly reduces the number of instances of it- iterative decoders like min-sum or BP algorithm.
erative decoder failure. For example, during the
row removal step in the bicycle code [2] we care-
fully modify the original bicycle code to obtain
codes wherein the stabilizer is asymmetric as in Fig. 9 shows improved decoding performance
Fig. 6(b). The BP decoder will able to match to (flooding BP decoder with `max = 100 iterations)
the syndrome (dark squares represent unsatisfied in the error floor regime for the newly constructed
checks) correctly with the red error pattern, in QC HP code. Even though the curves start with
contrast to the symmetric stabilizer in Fig. 6(a). a similar waterfall performance (due to same min-
Similarly, in the case of HP codes, careful removal imum distance), the randomly constructed code
of TSs in the constituent classical LDPC codes performs worse in the error floor regime compared
helps to further optimize rate and minimum dis- to the QC code. For the same code parameters,
tance properties. Such code design improvements using the QC component code improves the FER
lead to performance gains especially in the error by nearly an order of magnitude at p ≈ 0.005.
floor region for QLDPC codes. Here, we show This improvement can be clearly attributed to
an example of such an improved HP code design the absence of harmful TSs which were removed
with quasi-cyclic (QC) LDPC [37] code for the in the constituent QC code construction. The ex-
constituent classical LDPC code. ample of the HP-LDPC code construction with
QC constituent code is chosen considering their
hardware friendly nature and simple bookkeep-
5.1.1 Improved HP codes without harmful TSs
ing of trapping sets. In general, no specific code
One of the disadvantages of QLDPC codes is that structure is required in our method, and the con-
the random code construction makes the stabi- cept, definition and analysis of trapping sets are
lizers highly non-local, requiring arbitrary qubit- applicable to random codes as well.
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 13100 100
10-1
10-2
-2
10
10-3 10-4
10-4
10-6
10-5
10-6 10-8
10-2 10-1 10-2 10-1
Figure 9: FER performance comparison between the Figure 10: FER performance comparison for the
symmetric HP codes constructed using random con- A1[[254, 28]] code using the min-sum algorithm (MSA)
stituent codes ([20,5,8] code and [24,6,10] code in [17], for two different schedules: flooding/parallel and lay-
and [40,10,12] code with girth g = 6) and the sym- ered schedule. The layered schedule is able to decode
metric HP code constructed using a trapping set aware all the symmetric stabilizer TSs and numerous classical-
QC [40,10,12] code. All curves are decoded using a type TSs correctly leading to two orders of magnitude
BP decoder for `max = 100 iterations with the flooding improvement in the error floor regime (low physical error
schedule. For the same code parameters, using the QC rates).
component code without harmful trapping sets shows
FER improvement (close to an order of magnitude) over
the random constituent code in the error floor region. metric stabilizers in the code as well as other
harmful trapping sets improves the error floor
decoding performance. As an intuitive example,
5.2 Novel Decoder Design suppose the symmetric stabilizer of weight 6 has
support on variable nodes v1 , . . . , v6 with sym-
An alternative or complementary approach is metric degenerate error patterns: e1 , e2 , e3 and
to devise iterative decoders that do not fail for e4 , e5 , e6 . A layered decoder with the update or-
the error patterns in the QTSs identified for der: starting with VN update of v1 , v2 , v3 , fol-
the QLDPC code. This approach, prevalent lowed by the check node updates, and then VN
in classical LDPC decoders (finite alphabet it- update of v4 , v5 , v6 converges to the correct error
erative decoding (FAID) algorithms such as in pattern without getting trapped. Since the sched-
[23]) do not ignore the topology of the TSs ule order is with respect to the variable nodes cor-
while devising decoder update rules. Breaking responding to the columns of the H matrix, we
the symmetry of messages by using non-linear refer to such schedule as column-layered. In ad-
message update rules leads to orders of mag- dition to fast decoder convergence in terms of the
nitude decoding error performance improvement number of iterations [38, 39], column-layered de-
[23]. For QLDPC iterative decoders, the typically coders break some harmful TSs in classical LDPC
used parallel/flooding message update schedule codes [40]. In the following section 5.2.1, we com-
(in the same iteration all variable/check nodes pare the two schedules: flooding MSA and layered
in the Tanner graph apply in parallel the same MSA decoders for the chosen QLDPC code.
variable/check update function, respectively) at-
tributes to decoders’ failure to symmetric degen-
5.2.1 Layered Decoding to break QTSs
erate errors. We devise decoder strategy that
corrects these errors by taking into account the We employ a specific layered decoding sched-
topology of the low-weight symmetric stabilizers ule to break the symmetric stabilizers in the
in the code. Specifically, we show that an MSA A1[[254, 28]] code in Fig. 10 using a column lay-
decoder with sequential message update sched- ered schedule. The layered schedule employed
ule (layered decoder as in classical literature [38]) here is based on the circulant-size of the cyclic
that uses the knowledge of location of the sym- matrices A and B. The symmetric trapping sets
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 14have a clear distinction of red and blue nodes 100
with respect to these cyclic matrices, giving a
straight forward update order: v1 , . . . , v127 fol- 10
-2
lowed by v128 , . . . , v254 . The column-layered de-
coder (MSA with `max = 20 iterations) is able
10-4
to decode all the symmetric stabilizer TSs and
numerous classical-type TSs correctly leading to
two orders of magnitude improvement in the error 10-6
floor regime (low physical error rates) compared
to the flooding MSA decoder. 2 10-8
Remark 4 (Random perturbation and post-
processing) With the QTS knowledge, we can 10-10
10-2 10-1
construct iterative decoders without compu-
tationally expensive post-processing steps,
Figure 11: FER performance comparison for the
and thus reduce the decoding complexity and
A1[[254, 28]] code using the belief propagation decoder
latency. Note that the two curves in Fig. 10 with the flooding schedule with/without perturbation
use same maximum number of decoding iter- rounds (dashed/solid lines). When the total number of
ations (`max = 20). In contrast, the heuristic iterations allowed are only 100 iterations, the dashed-
random perturbation approach in [13] requires blue-square marked and dashed-cyan-triangle marked
many rounds of decoding attempts (Dr ) with (BP with random perturbation) curves do not outper-
additional BP decoder iterations (`r ) each of form the solid-red-diamond BP-only curve for 100 itera-
tions. For a significant performance improvement, post
which are initialized with perturbed channel
processing step with random perturbation decoders re-
log-likelihood ratios on variable nodes that are quires much more decoding rounds as shown in dashed-
neighbors of the unsatisfied syndromes. If we al- red-diamond curve. Also, note that the heuristic choice
lot the same total number of decoding iterations of noise parameter value affects these decoding improve-
to compare the random perturbation method ments.
with BP, we observe that the advantage of using
random perturbation is only observed with very
enhanced feedback decoding [41] and augmented
large number of iterations. However, if the total
decoding [20] can be used as additional tools over
number of iterations is set to 100 iterations,
the iterative decoder based on QTS analysis.
the decoding performance improvement using
heuristic methods is not significant as illustrated
An interesting result to note is that using more
in Fig. 11. For a significant performance im-
iterations in the BP algorithm results in match-
provement, post processing step with random
ing with many degenerate error patterns in case
perturbation decoder requires much more de-
of quantum LDPC codes. FER versus iterations
coding rounds (using `max + (Dr × `r ) = 1700
of BP plotted in Fig. 12 shows that increasing it-
iterations which is >> 100) as shown in dashed-
erations of flooding BP do not lower the FER
red-diamond curve. The heuristic choice of the
in terms of purely classical errors (for the red
randomness parameter can be further improved
curve, the degenerate error patterns that match
by using feedback as in [41]. This is also shown
the corresponding syndrome are considered as er-
to improve the convergence speed of the post-
rors). However, such degenerate error patterns
processing step, but it does not eliminate this
that match the syndrome without introducing a
step completely. The performance improvement
logical error are harmless for QLDPC codes, re-
shown with the chosen layered schedule is to
sulting in lowering the FER with increasing itera-
demonstrate the usefulness of the QTS analysis.
tions of BP. In our future work, we will devise im-
If required, post-processing techniques as well as
proved message passing algorithms of lower com-
improvements over heuristic approaches such as
plexity than BP that exploits the degeneracy of
2
In the Monte-Carlo simulations presented in the pa- QLDPC codes systematically.
per, we collect at least 100 errors for simulation points
in the low FER region ensuring that they are statistically Remark 5 These alternative improved decoders
significant. are an attractive solution when the QLDPC code
Accepted in Quantum 2021-09-28, click title to verify. Published under CC-BY 4.0. 15You can also read
