Branching MERA codes: a natural extension ofclassical and quantum polar codesAndrew James Ferris

Institut de Ciencies Fotoniques,Parc Mediterrani de la Tecnologia, Barcelona, Spain

Max-Planck-Institut fur Quantenoptik,Hans-Kopfermann-Str. 1, 85748 Garching, Germany

Email: andy.ferris@icfo.es

David PoulinDepartement de Physique, Universite de Sherbrooke,

Sherbrooke, Quebec J1K 2R1, CanadaEmail: David.Poulin@Usherbrooke.ca

Abstract—We introduce a new class of circuits for construct-ing efficiently decodable quantum and classical error-correctioncodes, based on a recently discovered contractible tensor networkknown as branching multi-scale entanglement renormalizationansatz [1]. We perform an in-depth study of a particular examplethat can be thought of as an extension to Arikan’s polar code[2]–[4]. Notably, our numerical simulation show that these codespolarize the logical channels more strongly while retaining thelog-linear decoding complexity using the successive cancellationdecoder. These codes also display improved error-correctingcapability with only a minor impact on decoding complexity.Efficient decoding is realized using powerful graphical calculustools developed in the field of quantum many-body physics.

I. INTRODUCTION

Tensor networks (TNs) are powerful a graphical tool usedover the last decade to develop efficient representations ofentangled quantum many-body states [5]–[7]. In this formal-ism, the evaluation of physical quantities of interest (localobservables, correlation functions, etc.) reduces to the problemof contracting (summing over) tensor indices. Quantum errorcorrection codes encode information non-locally into the long-range entanglement of quantum many-body systems, resultingin a protection against local errors, and can be understood asa non-commutative generalization of classical correction. Theresults presented in this submission relate the two seeminglydistinct disciplines of error correction and quantum many-bodyphysics. Our results are summarized below:

1) We establish a formal equivalence between the problemof decoding a classical or quantum error-correcting codeand the contraction of TN.

2) We provide many examples of this equivalence withexisting codes (cf. Fig. 2).

3) We propose a new contraction scheme for the TN asso-ciated to quantum polar codes [3], [4] which enhancestheir error-correction performance.

4) We propose a new class of classical and quantumerror correcting codes—that we name branching MERAcodes—that rely on recently introduced TNs [1] andgeneralize classical [2] and quantum [3], [4] polar codesin a natural way.

5) We conduct numerical simulations with branchingMERA codes that, in both classical and quantum set-

tings, display an improved error-correcting capabilitycompared to polar codes with only a minor impact ondecoding complexity.

II. METHODS

Abstractly, we can view a gate, such as a CNOT, as a tensorAαβγ... with a certain number of indices denoted α, β, γ, . . .,each taking values in a finite set, e.g. the set Z2 in case ofthe CNOT. The number of indices is the rank of the tensor.For instance, the CNOT gate is a rank-four tensor Nαβγδ

with indices α and β representing the two input bits and γand δ representing the two output bits, and the value of thetensor given by Nαβγδ = 1 if γ = α and δ = α ⊕ β, andzero otherwise. We can graphically represent a tensor as avertex and its indices as edges, with the degree of the vertexequal to the rank of the tensor. In that setting, an edge linkingtwo vertices represents a tensor contraction defined by thefollowing equation

= Cµ1µ2...⌫1⌫2... =

A B

C

⌫1 ⌫2µ2

µ1

↵

...

⌫1⌫2

µ2µ1

...

... ...

=X

↵

A↵µ1µ2...B↵⌫1⌫2...

. (1)

It follows from this definition that a graph represents a TNwith all edges contracted, and hence a scalar.

In general, not all TNs can be efficiently contracted. Re-ferring to Eq. (1) where tensor A has rank 6 and tensorB has rank 5, we see that the tensor resulting from theircontraction has rank 6 + 5 − 2 = 9. Thus, while tensor Ais specified by 26 entries and tensor B is specified by 25

entries, tensor C contains 29 � 26 + 25 entries. While aTN composed of constant-bounded-rank tensors (e.g. a circuitwith only two-bit gates) can be specified efficiently, the tensorsobtained at intermediate steps of the TN contraction schedulecan be of very high rank r, and so its contraction will producean intractable amount of data 2r. The contraction schedulethat minimizes the intermediate tensor rank defines the tree-width of the graph, so generally the cost of contracting a TNis exponential with its tree-width [8], [9]. However, it cansometimes be possible to contract TNs with large tree-width

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E E E E E E E

U

u1 u0u1

uy7uy6

uy5uy4

uy3uy2

uy1

E E E E E E E

U

uy7uy6 uy5

uy4uy3

uy2 uy1

ux1ux2

ux3ux4ux5

ux6ux7 uxi

a) b)

U

c)

E E E E E E E

bz bz bz bz

x

Qn(E)

e e e e e e

Fig. 1. (a) A simple TN diagram of the generic decoding problem. The n input bits xi are a combination of k data bits and n− k frozen bits, which arepassed through the encoding circuit U . Given the measurements yi and the symmetric noise model E , we wish to determine the most likely configurationof data bits. The unnormalized probability P (x|y) is given by contracting the above TN, but it is not feasible to repeat for all 2k possible inputs. (b) Thesuccessive cancellation decoder iteratively determines input bits in a right-to-left order. To determine the relative probabilities of bit i, we freeze the bits tothe right using prior knowledge, while remaining completely ignorant about the states to the left, where “e” represents the uniform mixture (1, 1).

by making use of special circuit identities. An example isprovided by the fact that a CNOT gate with a 0 entry on thecontrolled bit is equivalent to the identity (see Fig. 5 (b) forthe corresponding graphical identity). The combination of suchcircuit identities provide a powerful graphical calculus thatcan be used to contract highly complex TNs. In particular, wewill show that Arikan’s sequential cancellation decoder can berecast in this graphical calculus as an efficient TN contraction.

The relation between decoding and the problem of TNcontraction is simplest to describe in the classical setting.A (n, k) code can be defined by an encoding circuit U—areversible transformation on Zn2—as follows C = {y ∈ Z2

n :y = Ux, x = (l, 0n−k), l ∈ Zk2} where n− k of the input bitsare frozen to the value 0. An bit-flip error E ∈ Zn2 takes acodeword x to x′ = x+E. The value of the the frozen bits onthe string U−1x′ yields the error syndrome, from which theerror E can (hopefully) be inferred.

The encoding circuit U can be viewed as a rank-2n ten-sor, with n indices representing n input bits and n indicesrepresenting n output bits, where some of the input bits arefixed to 0. A single bit channel E is a stochastic matrix,and hence a rank-two tensor. Finally, we can represent theprobability distribution of a bit as a rank-one tensor, with thetensor u0 = (1, 0) representing the bit value 0 and u1 = (0, 1)representing the bit value 1. Given these, the probability of theinput bit string x = (x1, . . . , xn) given the observed outputy = (y1, . . . , yn) can be represented by Fig. 1 (a).

The TN of Fig. 1 (b) represents another probability, whichis particularly useful to implement Arikan’s successive cancel-lation scheme for polar codes. In this decoder, the goal is todetermine a single bit at the time, moving from right-to-left,by assuming complete ignorance of the input bits to the left[we define the tensor “e” = (1, 1)], and total confidence inthe value of the input bits to the right (either because theyare frozen in the code, or because we have decoded those bitsalready). Successive cancellation decoding for polar codes canbe recast as an efficient contraction of the resulting TN.

The stabilizer formalism enables us to transfer the relationbetween error correction and TNs to the quantum setting,although a subtle distinction is caused by error degeneracy.Much like in the classical setting, a code encoding k qubitsinto n qubits can be defined as the image of a unitary encoding

circuit U acting on an n-qubit state, where k “data qubits” canbe in an arbitrary state and the other n−k “syndrome qubits”are frozen to the state |0〉,

C = {|ψ〉 = U |φ〉k ⊗ |0〉⊗n−k : |φ〉k ∈ (C2)⊗k}. (2)

Subjected to an error E, the encoded state |ψ〉 is transformedto |ψ′〉 = E|ψ〉. Measuring the syndrome qubits on the stateU†|ψ′〉 yields the error syndrome, and the decoding problemconsists in identifying the optimal recovery given the syn-drome. In this article, we focus on the case where the encodingcircuit is a Clifford transformation and the noise model is aPauli channel; the general case can be treated similarly so longas we restrict our recovery to Pauli operations. Recall that thePauli group is generated by the four Pauli matrices I , σx, σy ,and σz acting on each qubit and that Clifford circuits mapPauli operators to Pauli operators.

We model the noise by assigning a probability Pn(E =E1 ⊗ E2 ⊗ . . . ⊗ En) = E(E1)E(E2) . . . E(En) to eachelement E of the Pauli group, where E(Ei) is a probabilitydistribution. Hence, E is represented by a rank-one tensorof bond dimension 4, i.e., E = (pI , px, py, pz). Considerthe distribution Qn(E) = Pn(U

−1EU) corresponding to thedistribution of errors after the de-encoding circuit U−1. Thisdistribution is obtained by contracting the encoding circuit(viewed as a rank 2n tensor) with the E’s, c.f. Fig. 1 (c).To decode, we need to condition this probability distributionon the observed error syndrome, and we typically decode onequbit at a time for efficiency reasons.

In the classical setting, a syndrome bit measured in the statesj is represented by the tensor equal to the indicator functionon s — that is, either u0 = (0, 1) or u1 = (1, 0). When a qubitis prepared in the state |0〉, subject to a Pauli channel, and latermeasured along σz , a +1 measurement outcome doesn’t implythat the qubit has not suffered an error. Instead, it indicates thatit either had no error or a σz error. Thus, the TN representingsuch a measurement is the indicator function on I and σz , i.e.a bimodal indicator function bz = (1, 0, 0, 1). Similarly, a −1outcome is consistent with a σx and a σy error, and equivalentrelations hold for a qubit prepared and later measured in a σx,yeigenstate. The action of the CNOT on these bimodal indicatorfunctions obey the circuit identities shown in Fig. 5.

Given this, we recognize the TN of Fig. 1 c) as the

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(a) (b) (c) (d) (e)

Fig. 2. Graphical definitions of various unitary TNs and encoding circuits of quantum (and classical) codes: wires represent one or a few qubits (bits),rectangles represent unitary (reversible) transformation, with time running from top to bottom. The size of each circuit can be varied in an obvious way.For coding applications, location of the data qubits (bits) are indicated by triangles, other qubits (bits) are initialized to 0. a) MPS TN and encoding circuitof convolutional codes. b) Tree TN and encoding circuit of concatenated quantum block codes. c) MERA TN and encoding circuit of topological code. d)Branching tree TN and encoding circuit of polar codes. e) Branching MERA TN and encoding circuit of codes introduced in this article.

probability that the 4th qubit (from the right) has not beenaffected by a x-type error, given that among the first threequbits, only the third one has been flipped, and ignoring anyinformation about the last three qubits. In this TN, we modelthe quantum channel E as a probability over the single-qubitPauli group, so it corresponds to a rank-one tensor. This type ofTN is the ingredient needed to estimate the bit-flip probabilityon each qubit in a successive cancellation scheme. To completethe decoding, we repeat the same procedure, sweeping fromleft to right and evaluating the probability of phase-flip errorsσz (the error σy is viewed as a combination of σx and σz).

We have unveiled the relation between TNs and errorcorrecting codes (classical and quantum) through the similarityof the methods used in the two distinct disciplines. Figure2 (a-d) presents existing classical and quantum codes in thelanguage of TNs. The last example Fig. 2 (e) has been studiedin the context of quantum many-body physics [1] but, to thebest of our knowledge, has not appeared in coding theory. Thenext sections are devoted to its study.

III. BRANCHING MERA CODES

Fig. 2 e) proposes a new class of classical and quantum errorcorrecting codes based on the recently introduced branchingMERA TN [1]. Like polar codes, the fact that they can beefficiently decoded using a sequential cancellation schemefollows from the fact that the TN, once simplified using thecircuit identities of Fig. 5, has a constant tree width. Anexample of the two encoding circuits with 16 bits (4 layers)is shown in Fig. 3, where the layout is organized in a moreconventional way for coding theory. For n bits, the polar codecontains n log2 n/2 CNOT gates, while the branching MERAcode contains twice as many. The branching MERA codeincludes all of the gates of the polar code in addition to analternating layer of gates, highlighted in blue. The depth ofthe circuit increases from log2 n to 2 log2 n. Encoding timesfor these circuits grows log-linearly with n, and thus they canbe considered efficient encoders for all practical purposes.

Applying the circuit identities of Fig. 5 (a-c) to the polarand branching MERA codes results in a vast simplification. Infact, most of the CNOT gates are removed, and the numberof remaining gates drops from O(n log n) to O(n). This isillustrated at Fig. 4 (a) for the polar code and (b) for thebranching MERA code. In both cases, the remaining tensor has

(a) (b)

Fig. 3. The encoding circuits of (a) the polar code, and (b) the branchingMERA code for 24 = 16 sites. The polar code contains half of the gates of thebranching MERA. The extra gates are highlighted in blue in (b), while the dotsrepresent periodic boundaries. Branching MERA codes can be defined withour without periodic boundary conditions, the difference being the presenceor absence of these dotted gates.

a constant tree-width, and so it can be contracted efficiently.This demonstrates that sequential cancellation decoding canbe realized as an efficient tensor network contraction in boththe polar and branching MERA code.

Decoding in the quantum setting proceeds similarly bymaking use of the circuit identities of Fig. 5 (d-g). TheTN formalism enables us to go beyond the usual two-phasesequential decoder (corresponding to x-type and z-type errors)and to take advantage of existing correlations between xand z errors, e.g. mediated by y errors on a depolarizationor erasure channel (remember σy = iσzσx). Indeed, it ispossible with polar codes as well as branching MERA codes tosimultaneously sweep for x errors from the left and for z errorsfrom the right with only doubling the TN’s tree-width. Theworking principle behind this symmetric decoder is the ideathat the circuit identities in Fig. 5 can be used to remove gateseverywhere from the circuit, except at domain walls where, forexample, a sequence of bz tensors encounter a sequence of “e”tensors. The enhanced decoder has at most two such domainwalls at any given time during the decoding.

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(a)

(b)

Fig. 4. The simplified tensor networks for successive cancellation decodingof (a) the polar code and (b) the branching MERA code, using the identitiesillustrated in Fig. 5. These tensor networks contain open (non-contracted)indices as proxies for any tensor that could be placed at these location. For thebranching MERA, it is natural to determine the joint probability distributionof three neighboring bits, because of the iterative scheme in [1]. Both tensornetworks can be contracted from the bottom-up in a time linear in the totalnumber of bits, n.

= =

= =(e)

(d) (f)

ee ee

bz

xb

bz

xb

�

�

''

(g)

=(a)

=(c)

e e=

(b) 0u 0u

1u 1u

Fig. 5. Classical (a-c) and quantum (d-g) circuit identities. (a) We use thesymbol e for both the classical uniform mixture e = (1, 1) and quantumuniform mixture e = (1, 1, 1, 1). The truth table for (d) (σ, σ′)→ (κ, κ′) isgiven by (I, σx) → (I, σx), (σx, I) → (σx, σx), (I, σz) → (σz , σz),and (σz , I) → (σz , I). (e-g) Action of the CNOT gate on the bimodalindicator functions. Identities (e-g) follow from the application of (d) tobimodal indicator functions.

IV. NUMERICAL RESULTS

A. Classical

We have numerically compared the performance of the polarand branching MERA codes at protecting data from a varietyof channels on codes between 256 and 8192 bits. Here wereport our results for the bit-flip channel. These simulationsuse a simplified channel selection scheme that is independentof the details of the error model (see [10]).

The results for the binary bit-flip channel with code-rate1/2 are given in Fig. 6. Finite-size effects are significant in

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

Probability of bitflip

Codeerrorrate

FER polarFER bMERA

BER polarBER bMERA

(a)

0 0.05 0.1 0.15 0.2 0.25

10−6

10−5

10−4

10−3

10−2

10−1

100

Probability of bitflip

Codeerrorrate

FER polarFER bMERA

BER polarBER bMERA

(b)

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

Probability of bitflip

Codeerrorrate

FER polarFER bMERA

BER polarBER bMERA

(c)

0 0.05 0.1 0.15 0.2 0.25

10−6

10−5

10−4

10−3

10−2

10−1

100

Probability of bitflip

Codeerrorrate

FER polarFER bMERA

BER polarBER bMERA

(d)

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

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0.8

1

Probability of bitflip

Codeerrorrate

FER polarFER bMERA

BER polarBER bMERA

(e)

0 0.05 0.1 0.15 0.2 0.25

10−6

10−5

10−4

10−3

10−2

10−1

100

Probability of bitflip

Codeerrorrate

FER polarFER bMERA

BER polarBER bMERA

(f)

Fig. 6. Comparison of the performance of rate 1/2 polar and branchingMERA codes of various sizes for the bit-flip channel. The encoded messagecontains (a,b) 256 bits. (c,d) 1024 bits and (e,f) 8192 bits. The capacity withbit-flip probability approximately 0.11 corresponds to the code rate 1/2.

both codes, with the waterfall region separating “perfect” and“useless” behavior being somewhat below the threshold of0.11 at encoding rate 1/2. Nonetheless, the threshold of thebranching MERA code is significantly closer to this valuethan the polar code. On a logarithmic scale, is it evident thatthe performance in the low-error region is significantly better.Neither code displays any evidence of an error floor (nor isit expected). Finally, both codes display a tendency for anyerror to be catastrophic — involving errors on many bits. Theratio between the bit error rate (BER) and frame error rate(FER) is very large for the polar code and even higher (closeto 0.5) for the branching MERA code. This corresponds toeither a perfectly decoded message or a completely scrambledone. Interestingly, this is the behavior expected of a “perfect”random code as Shannon envisaged, where the most likelymessages are completely uncorrelated.

B. Quantum

In Fig. 7 (a), we plot the performance of the polar code (withstandard decoder), the polar code with symmetric decoderand the branching-MERA code (with standard decoder) asa function of depolarization probability. Channel selection ismade similarly to [11] (see [12] for details). In all cases we seea relatively sharp crossover between a low-error rate regimeand a high-error rate regime, occurring somewhat below the

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0 0.02 0.04 0.06 0.08 0.1

0

0.2

0.4

0.6

0.8

1

(a)

Depolarization rate

Fra

me e

rror

rate

Polar

Polar

symmetric

Branching

MERA

4 6 8 10 1210−6

10−5

10−4

10−3

10−2

10−1

100(b)

Layers

Upp

er−b

ound

on

FER

PolarPolar symmetricBranching MERA

Fig. 7. Performance of the polar and branching-MERA codes under thedepolarizing channel. (a) Comparison of the various codes and decodingalgorithms for codes of size 212 qubits with encoding rate 1/2. (b) Upperbound on the frame-error rate for codes with depolarizing rate 9.92% andencoding-rate 1/8 as a function of system size. The probability of errordecreases strongly with code size. Both the symmetric decoder and thebranching-MERA improve beyond the standard quantum polar code.

coherent capacity at depolarizing rate 9.92%. We observe thatthe threshold approaches the capacity with increasing codesize (like the classical case, this approach is relatively slow),and that both the improved symmetric decoder and branching-MERA code perform better than the standard polar code.

In Fig. 7 (b) we study the performance as a function of code-size. These results are a simple upper-bound on the error rateachieved by summing the individual error rates of the datachannels (both quadratures) and the frozen channels (the non-frozen quadrature) that were generated in the channel-selectionphase. In all cases that the frame-error rate decreases rapidlywith the number of layers used in the codes.

V. CONCLUSION

We have demonstrated a general connection between theproblem of decoding an error correcting code and the graphicalcalculus of tensor networks used in the field of quantummany-body physics. Using a family of tensor network recentlyintroduced in that setting, we presented a new family oferror-correcting codes that generalize classical and quantumpolar codes in a natural way. Recasting the decoding problemas a tensor network contraction, we have demonstrated thatsequential cancellation decoding can be realized with log-linear complexity, requiring roughly twice the computationaleffort of polar codes sequential decoding.

Our numerics show that this new code outperforms polarcodes in several ways, including stronger channel polarizationand enhanced error-correcting performance. Yet there clearly ismore room for improvement, so that finite-size performanceis closer to capacity. We have also analyzed the maximumlikelihood decoder for smaller polar and branching MERAcodes under the erasure channel and our results indicateperformance significantly closer to capacity than presentedhere. This difference arises because at all stages of decoding,every syndrome measurement is available to be used, unlikethe successive cancelation decoder which only has accessto previous bits. We speculate that the main advantage ofbranching MERA code compared to the polar code is thatthe syndrome bits are more tightly clustered to the edge —

thus increasing the information available to the successivecancellation decoder for the data bits.

The connection between tensor networks and coding opensthe door to many other encoding schemes. Only within thefamily of branching MERA networks, many different codescan be obtained by varying the elementary gates in the networkand increasing the number of bits in elementary gates (i.e.increasing the “bond dimension” in the TN language). Othertensor networks could also be considered along with theirheuristic contraction schemes, e.g. [6], [13]. In a similarvein, other decoders including belief propagation [14] andlist decoding [15] could also enhance the error-correctionperformance.

ACKNOWLEDGMENTS

The authors would like to Jean-Pierre Tillich for usefuldiscussions. AJF would like to thank TOQATA (Spanishgrant PHY008-00784), the EU IP SIQS, and the MPI-ICFOcollaboration for supporting this research. DP would like toacknowledge support from NSERC and FQRNT through thenetwork INTRIQ. Computational resources were provided byCompute Canada and Calcul Quebec.

REFERENCES

[1] G. Evenbly and G. Vidal, “A class of highly entangled many-body statesthat can be efficiently simulated,” 2012.

[2] E. Arikan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” IEEETransactions on Information Theory, vol. 55, pp. 3051–3073, 2009.

[3] J. Renes, F. Dupuis, and R. Renner, “Efficient quantum polar coding,”arXiv:1109.3195, 2011.

[4] M. Wilde and S. Guha, “Polar codes for classical-quantum channels,”IEEE Transactions on Information Theory, vol. 59, no. 2, pp. 1175–1187, 2013.

[5] G. Vidal, “Efficient classical simulation of slightly entangled quantumcomputations,” Phys. Rev. Lett., vol. 91, p. 147902, 2003.

[6] F. Verstraete and J. I. Cirac, “Renormalization algorithms for quantum-many body systems in two and higher dimensions,” 2004.

[7] G. Vidal, “Entanglement renormalization,” Phys. Rev. Lett, vol. 99, p.220405, 2007.

[8] I. L. Markov and Y. Shi, “Simulating quantum computationby contracting tensor networks,” SIAM Journal on Computing,vol. 38, no. 3, pp. 963–981, Jan. 2008. [Online]. Available:http://epubs.siam.org/doi/abs/10.1137/050644756

[9] I. Arad and Z. Landau, “Quantum computation and the evaluation oftensor networks,” 2008.

[10] A. J. Ferris and D. Poulin, “Branching mera codes: a natural extensionof polar codes,” arXiv:1312.4575, 2013.

[11] Z. Dutton, S. Guha, and M. M. Wilde, “Performance of polar codesfor quantum and private classical communication,” in Proceedings ofthe 50th Annual Allerton Conference on Communication, Control, andComputing, 2012, pp. 572–579.

[12] A. Ferris and D. Poulin, “Tensor networks and quantum error correc-tion,” arXiv:1312.4578, 2013.

[13] Z.-C. Gu, M. Levin, and X.-G. Wen, “Tensor-entanglement renormal-ization group approach to 2d quantum systems,” 2008.

[14] A. Eslami and H. Pishro-Nik, “On bit error rate performance ofpolar codes in finite regime,” Communication, Control, and Computing(Allerton), 2010 48th Annual Allerton Conference on, p. 188, 2010.

[15] I. Tal and A. Vardy, “List decoding of polar codes,” Information TheoryProceedings (ISIT), 2011 IEEE International Symposium on, p. 1, 2011.

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