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Superdense Coding Interleaved with Forward Error Correction Ronald J. Sadlier and Travis S. Humble Quantum Computing Institute, Oak Ridge National Laboratory, Oak Ridge, TN, USA Superdense coding promises increased classical capacity and communication security but this advantage may be undermined by noise in the quantum channel. We present a numerical study of how forward error correction (FEC) applied to the encoded classical message can be used to mitigateagainstquantumchannelnoise. BystudyingthebiterrorrateunderdifferentFECcodes, we identify the unique role that burst errors play in superdense coding, and we show how these can be mitigated against by interleaving the FEC codewords prior to transmission. We conclude that classical FEC with interleaving is a useful method to improve the performance in near-term demonstrations of superdense coding. 6 1 I. INTRODUCTION for protecting against sources of channel noise, including 0 demonstration using optically encoded qubits [14, 15]. 2 However,therequirementofusingadditionalancillaisat Entanglementisaversatileresourcethatenablesmany r odds with experimental limitations on the number of si- p new methods of communication including teleportation, multaneouslygeneratedphotonstates,whichmakeitdif- A quantum secret sharing, and superdense coding [1, 2]. ficulttopreparecodewordstatesforexistingQECcodes. Superdensecoding(SDC),inparticular,enablesasender 3 Alice to transmit a two-bit message to a receiver Bob by Classical forward error correction (FEC) offers an al- 1 sendingonlyonememberofasharedbipartiteentangled ternative to QEC that is more easily realized by exist- quantum state [3]. This is twice the classical capacity ing quantum communication technology. In this setting, ] h expected from direct transmission of a single unentan- FECcodesrelyonclassicalancillatoredundantlyencode p gledqubit. Inadditiontoincreasedcapacity,correlations the state of an input bit string. When used in conjunc- - withintheentangledstateprovidetheaddedbenefitthat tion with SDC, the resulting classical codewords are en- t n aneavesdroppercannotrecoverthemessagebysimplyin- codedintoquantumstatesandtransmittedacrossanoisy a tercepting the single transmitted particle. These unique quantumchannel. Thesereceivedstatesaredetectedand u features of SDC have been demonstrated by several dif- then decoded into classical strings representing the orig- q [ ferent experiments and shown to provide both greater inal message. An immediate advantage of this approach channel capacity and security relative to direct commu- is that it does not require additional entanglement in 2 nication techniques [4–8]. quantum resources. The use of FEC codes also does not v require changes to existing quantum hardware for im- 1 A limitation on SDC performance in realistic settings plementation, but rather can integrate with the software 2 isthepresenceofchannelnoise,whichcorruptsthetrans- systems that manage existing hardware [16]. 3 mitted state. Channel noise reduces the rate that infor- 6 mation is reliably transmitted and has the potential to However, the noisy quantum channels that lead to er- 0 completely undermine the expected benefits from SDC rors in the received codewords are not always equivalent . 1 [9]. Previously, Shadman et al. have investigated the to well-studied classical channels, and the performance 0 classicalcapacityforSDCwhentransmittinginthepres- of an FEC code for SDC depends on the details of the 6 ence of depolarizing noise [10–12]. Their results quanti- quantum channel. Indeed, this point has been empha- 1 fied the decrease in capacity for single-sided and double- sized in recent work by Boyd et al. who found that : v sided transmission, in which noise acts on one and both binary codes are limited in the capacity that is achiev- i particles, respectively. Notably, Shadman et al. found a able [17]. This limitation was traced back to the dis- X crossoverpointinthedepolarizingnoisemodelforwhich tinctionbetweenquantumstatesassoftsymbolsandthe r a double-sided SDC does no better than direct quantum measured bits as hard symbols. Boyd et al have shown transmission. This cross-over point represents the con- that non-binary classical error correction codes can offer ditions for which the classical SDC capacity was smaller better performance and may be viewed though the lens than when using direct quantum transmission. of a d-ary memory-less classical model for entanglement- assisted communication [17]. The presence of noise in experimental realizations of SDC raise the question as to how this type of com- Inthispaper,weinvestigatetheperformanceofseveral munication can benefit from the use of error correc- binary FEC codes for mitigating errors from superdense tion techniques. Both quantum and classical error cor- coding over noisy quantum channels. In particular, we rection codes could in principle offer protection to the calculate the bit-error rate (BER) for finite-length mes- transmission of information. Quantum error-correction sages over a Pauli channel using numerical simulations (QEC) codes use ancillary qubits to construct a higher- of the transmission. We calculate the decoded BER for dimensional code space that can detect errors in the log- the repetition, Hamming, and Golay FEC codes in the ical code words [13]. There have been several experi- presence of depolarizing noise. We then investigate the mental demonstrations that show the benefits of QEC useofdatainterleavingtomitigate againstthestructure 2 of these Pauli operator errors. We present results from channel use is two bits. The idealized SDC protocol ne- numerical simulations of these different system designs glects several sources of noise that may be encountered andweidentifytherelativeperformanceunderthePauli in a realistic implementation. First, the preparation of noise model. the initial state shared by Alice and Bob may deviate The remainder of the paper is organized as follows. from the protocol specification. In particular, it may In Sec. II, we review the superdense coding protocol and be represented by a nonmaximally-entangled mixture of presentamodelfornoisytransmission. InSec.III,wein- bipartite states. This deviation from the protocol may troduce notation for FEC codes and provide an example be due to environmental induced decoherence of the ini- of how FEC coding affects the bit-error rate. In Sec. IV, tially pure state or to noisy preparation of the entangled wepresentnumericalsimulationresultsforFECencoded pair. Second, Alice’s encoding operations may be faulty SDC, and in Sec. V we analyze the impact of interleav- from imprecise application of the Pauli operators X,Z, ing for protecting entangled pair states from some Pauli and XZ. This also leads to less than maximally entan- errors. Our conclusions are then presented in Sec. VI. gled Bell states with the effect that the resulting measurement statistics will not be perfectly correlated with the input message. Third, the transmission of subsys- II. NOISY SUPERDENSE CODING tem A from Alice to Bob may encounter channel noise that introduces errors in the amplitude and phase of the The superdense coding (SDC) protocol begins by as- state. Additional possible errors include loss, whereby suming that users Alice and Bob share a pair of qubits thetransmittedparticleisnotreceivedbyBob,andleak- prepared in a known entangled state. We will consider age, in which the particle is perturbed into a state not this state to belong to one of the four Bell states, detected by Bob’s measurement apparatus. We will consider a model of noisy SDC given by the |Φ(±)(cid:105)= √1 (|0 ,0 (cid:105)±|1 ,1 (cid:105)) one-sided Pauli channel AB 2 A B A B (1) 3 |Ψ(±)(cid:105)= √1 (|0 ,1 (cid:105)±|1 ,0 (cid:105)) E(ρb)=(cid:88)pjOA(j)ρbOA(j)† (4) AB 2 A B A B j=0 where0and1denotetheeigenstatesofthePauliZ oper- with p ∈ [0,1] and p ∈ [0,1/3] constrained by 0 1,2,3 (cid:80) atorandAandB labelthedifferentsubsystems[18]. For p = 1 and the operator O (j) defined in Eq. (2). j j A concreteness, let the shared initial state be Φ(+) and the The quantum channel E serves as an effective model to corresponding initial density matrix be defined as ρ(+). accountforavarietyofnoisesourcesincludingtransmis- Alice uses this fiducial state to transmit a two-bit mes- sion noise, state preparation noise, and detection errors. sage b to Bob by encoding the binary coefficients b and Equation (4) also has the property that it maps every 0 b ofthemessageintothestate. Themessageisencoded Bell-state to a mixture of Bell states and, consequently, 1 by applying the unitary operator this is a covariant channel with respect to the Pauli basis defined by Eq. (2). We will limit subsequent analysis OA(b)=OA(b0,b1)=XAb0ZAb1 (2) to the case p0 = 1 − p and p1 = p2 = p3 = p/3 for p ∈ [0,1], which is known as depolarizing noise limit for tosubsystemA. Eachofthefourpossibleoperatorsmaps noise parameter p [18]. theinitialstateuniquelyintooneofthemaximallyentan- While the probability to receive the incorrect quan- gled Bell states and we will use the shorthand notation tum state is 1−p , the probability to receive the incor- 0 ρb = OA(b)ρ(A+B)OA(b)† to denote the state encoding the rect classical message is slightly less. This is because the message b. message encoded in a received Bell state will have either After encoding the message, Alice transmits subsys- one, two, ornobitscorrupted. Giventheoutputstateof tem A to Bob, who performs a projective measurement thechannelinEq.(4),theprobabilitythatBobobserves in the Bell basis to identify the state prepared by Alice the message b(cid:48) when Alice sent the message b is formally [19]. The complete set of Bell-state measurements may be modeled by the set of projection operators Π(b)=ρb, ProbSDC(b(cid:48)|b)=Tr[Πb(cid:48)E(ρb)] (5) where b = {0,1,2,3} uniquely labels the measurement For the depolarizing channel, the corresponding transi- outcome. TheprobabilityofmeasuringBellstateb(cid:48)given tion probabilities are given as the encoded value b is given as  1−p b(cid:48)b(cid:48) =b b Prob(b(cid:48)|b)=Tr[Πb(cid:48)ρb]. (3) p b(cid:48)0b(cid:48)1 =b¯0b1 Prob (b(cid:48)|b)= 3 0 1 0 1 (6) mFoarytthreannsolaisteeletshsepdroetteocctoeld, Pstraotbe(bt(cid:48)o|b)rec=ovδebr(cid:48),bt,heantdwoB-boibt SDC pp3 bb(cid:48)0(cid:48)bb(cid:48)1(cid:48) ==bb¯0bb¯¯1 3 0 1 0 1 message according to map generated by Eq. (2). Thus, intheabsenceofnoise,eachtwo-bitmessageisfaithfully with b¯ the binary conjugate of b . The bit error rte j j transmitted using a pure state and the capacity of each (BER)measurestheratioofincorrectbitstothenumber 3 of bits transmitted. For the one-sided SDC depolarizing the sender, making forward error correction codes useful channel, the frequency with which the binary message is on simplex (i.e., one-way) communication channels. For incorrectly decoded is given by a SDC channel, these codes are applied to the classical message before being encoded to quantum symbols. In 2p BER = (7) addition,themeasuredclassicalsymbolsbyBobarethen SDC 3 decoded according to the FEC code. An FEC code with distance d can detect up to (d−1) errors in the received As a point of comparison, consider the case that Al- messageandcorrectanymessagewith(cid:98)(d−1)/2(cid:99)errors ice and Bob use direct quantum transmission instead of or less. entanglement for communication. Alice then uses the Formally, we will represent an [n,k,d] code by the n- noisy quantum channel twice to transmit a single qubit by-k generator matrix G, which transforms an input bi- correctly with probability p. However, the action of the nary vector b into a codeword c, i.e., PauliZ operatorchangesonlythephaseofthequbitand, therefore, does not lead to an incorrect bit. The corre- c=Gb (12) sponding transition probabilities for Bob to successfully recover both of Alice’s bits are encodes k input bits into an n-bit codeword c. These (1− 2p)2 b(cid:48)b(cid:48) =b b n bits will then be transmitted pairwise using the SDC 2p(1−3 2p) b0(cid:48)b1(cid:48) =b¯0b1 protocol. If n is odd, we concatenate two consecutive Prob (b(cid:48)b(cid:48)|b b )= 3 3 0 1 0 1 (8) codewords for transmission of a 2n-bit message. Alter- Direct 0 1 0 1 (1− 2p)2p b(cid:48)b(cid:48) =b b¯ natively,thecodewordmayalsobepaddedwithanextra (cid:0)2p(cid:1)23 3 b0(cid:48)b1(cid:48) =b¯0b¯1 bit. Ineithercase, consecutivebitpairsaremappedinto 3 0 1 0 1 the Bell states according to Eq. (2). Applying FEC to In this case, the BER also evaluates to the original classical message increases the size of the transmitted information by a factor n/k, the inverse of 2p the code rate. An input message b of size 2k is encoded BER = (9) Direct 3 by FEC as the codeword vector c of size 2n and then transmitted by the ensemble of states Although the BER for SDC and direct transmission n−1 are the same in the case of a depolarizing channel, the (cid:89) ρ = O (c ,c )ρ O (c ,c )† (13) classical channel capacities are not. In particular, the c A 2i 2i+1 0 A 2i 2i+1 i=0 classicalcapacityforSDCinthepresenceofdepolarizing noise has been given previously as [9] where c is the i-th bit as defined by Eq. (12). i DetectionofnquantumstatessentbyAliceleavesBob p C (p)=2+(1−p)log(1−p)+plog (10) with the received message d=c+e, where e represents SDC 3 the error imparted by the noisy channel. Forward er- whilethecorrespondingcapacityforasingleuseofdirect ror correction codes will permit decoding of the error transmission is provided its weight (Hamming distance) is within the boundpreviouslymentioned. Ifso,thentheerrorcanbe C (p)=1+(1−p)log(1−p)+plogp, (11) decoded using the parity matrix H for the [n,k,d] code, Direct which satisfies which is less than half of the former. Finally, it is useful to compare the capacity for super- b(cid:48) =Hd. (14) dense coding with that of a 4-ary classical communication system. Both communication methods are capable Whenever Hd(cid:54)=0, an error has occurred. However, the of transmitting 4 distinct symbols, although SDC lever- error can be identified only if the weight is less than the ages the non-local correlations between Alice and Bob distance of the code. Thus, the probability for a given for this purpose. Bennett et al. have previously shown bit pair to be correctly decoded is given by the number that the capacity for both systems is the same under the of ways in which a correctable error may be received by symmetric channel [9]. Bob. For example, an FEC code with distance d = 3 cor- rectsuptoasingleerrorinthereceivedcodeword. From III. SDC WITH FORWARD ERROR Eq. (6), the corresponding probability that the original CORRECTION message is correctly received following error correction underthedepolarizingchannelisthebiterrorrate(BER) Forward error correction (FEC) codes increase the re- 2p liability of transmission over a noisy channel by encod- BER =(1−p)n+ (1−p)(n−1), (15) SDC 3 ing the message such that a transmission contains re- dundant information. The receiver may detect a trans- which measures the number of incorrectly decoded code- mission error and correct it without the involvement of words. TheBERisusefulforassessingtheimpactofthe 4 TABLE I. Properties of FEC codes tested Coding Distance Rate No FEC 0 1 Repetition [n={3,5,7},k=1] n k 2 n Hamming [n=7,k=4] 1 4 7 Golay [n=24,k=12] 3 1 2 FIG. 1. (top) Alice’s qitkat application implements FEC and SDC encoding steps before pushing data to the middleware layer to transmit the specified Bell states. (bottom) FEC as it measures errors remaining after error correc- Similarly, Bob’s application gets measurement results from tion. Italsopermitserrorstobeisolatedtotheindividual thereceivermiddlewarelayerandperformstheSDCandFEC codewords that contain them. We have tested the per- decoding operations. formanceofSDCwhenusingseveraldifferentFECcodes to protect against depolarizing noise. We will present numerical results of these studies in the next section but munication systems that has been presented previously we first provide an example using the Hamming [7, 4, [16]. Briefly, the QITKAT library is an extension of the 3] code. Consider the explicit example that Alice sends GNU Radio real-time signal processing framework that Bobthe4-bitmessageb={1,0,0,1}usingtheHamming adds primitives for quantum information applications. [7,4] code. The FEC encoder using the Hamming [7, 4] The GNU Radio framework provides a run-time man- encoding prepares the codeword c={0,0,1,1,0,0,1,0}, ager and a large variety of conventional communication where we have appended an extra 0 to make the code- methods that can be easily integrated into a graphical word even. These 8 bits are then sent to the quantum workflowanddeployedtointeractwithhardwarecompo- encoder, which translates the each pair into a Bell-state symbol, i.e., Φ(+),Ψ(−),Φ(+), and Ψ(+). nents[20]. WeleveragetheGNURadioframeworkusing the QITKAT library to construct quantum communica- During transmission, each of these states will expe- tion applications that can be either deployed directly to rience noise according to the probabilistic model given hardware or simulated using numerical methods [21]. In by Eq. 4. The effect of the noise depends on both the the present study, we use numerical simulation methods state and the error operator. For example, an X error will transform Ψ(+) to Φ(+) and Bob’s measurement re- to investigate the behavior FEC codes under the Pauli noise model. sults will be mapped to the bit pair (0,0) according to Eq. 2. This binary error can be corrected however us- Example QITKAT applications are shown for Alice ing the Hamming code. Assuming no other errors dur- and Bob in Fig. 1. These applications represent pre- ing the transmission of four consecutive Bell states, the processing steps that prepare messages to be sent to the complete message that Bob receives in this example is communication middleware layer. For Alice’s applica- {0,0,1,1,0,0,0,0}. Following decoding of the received tion,aninputfilesourcefeedsastreamofbitstotheFEC message, the original message is recovered as {1,0,0,1} code block (the repetition code is shown in the exam- and the resulting BER is 0. The FEC succeeds in this ple). The output from the FEC encoder passes through example because the single binary error resulting from the SDC encoder, which converts each pair of consecu- corruption of the transmitted quantum codeword is cor- tivebitstoatransmissionsymbolthatisthensenttothe rectable when using the Hamming code. Similarly, any quantumtransmitter. Thequantumtransmitterconsists other weight-one binary error in the message can be cor- of both hardware that prepares and measures the entan- rected following measurement of the complete encoded gledquantumstatesaswellasmiddlewarethatmanages message. However, weight-two and higher binary errors these processes. In our case studies, we have replaced cannot be corrected by the Hamming code. Moreover, hardware instances with a numerical simulator that in- these errors may be mistakenly characterized as weight- terprets the middleware control signals and prepares a one errors and the resulting attempt to correct the mes- numerical representation of the quantum state [16]. The sage will corrupt it further. It is notable that the binary simulator uses a model of the quantum channel to gen- errorsinthereceivedmessagetracebacktothequantum erate a representation of the measurement results that is noise operations that act on the transmitted Bell state. observed by Bob’s receiver. The receiver middleware in- terprets the measurement results and performs decoding of the transmission. As shown in Fig. 1, the decoding IV. NUMERICAL SIMULATIONS OF NOISY steps mirror the encoding performed by Alice. SDC TRANSMISSION We use a numerical simulator that takes advantage of the algebraic structure of both the Bell states and the Numerical simulations of the noisy SDC transmis- Pauli channel for noisy transmission. In particular, we sion model presented in Sec. II were used to investi- modelBob’sdetectorsasprojectivemeasurementsinthe gate the influence of FEC codes on BER. We use the Bell basis as given by Eq. (5). These measurements are QITAKT framework for software-defined quantum com- thereforeequivalenttosamplingofthePaulichannel. We 5 then use a Monte Carlo method to sample the simulated BER variance that arise from finite sampling. transmission and chose a specific outcome for each measurement. The sampling is biased according to the error rates {p ,p ,p } with random instances drawn using x y z a pseudo-random number generator. For the symmetric depolarizing noise, we consider the noise parameter p within the range [0,0.1] as suggested by recent experimental results [22]. We calculate the BER after simulated transmission by performingacomparisonbetweenthedecodedbitstream and the original input message. We use a nominal input ofapproximately1millionbits,whereslightadjustments are made to accommodate the number of bits required per classical FEC symbol. The baseline case of SDC transmissionthroughthedepolarizingnoisechannelwith no FEC coding is shown in Fig. 2 alongside the case of direct quantum transmission. As expected from our ear- lierresults,therateatwhicherrorsoccuristhethesame for both protocols. FIG. 3. BER averages versus noise parameter p for FEC- encoded,non-interleavedtransmissionwitheitherSDCordi- rect transmission. Curves correspond to Hamming [7,4], Go- lay [24,12], and repetition [3,1] codes. InFig.3,thebehavioroftheSDCanddirectcommuni- cationschemesarenotablydifferenteventhoughtheyare both subject to the same noisy channel parameters. The reason for this difference traces back to the effect of the PaulinoiseoperatorsontheBellstates. Inparticular,the transmissionofFECcodewordsunderdepolarizingnoise issusceptibletobinaryerrorswithweightgreaterthan1. For the single-sided Pauli noise mode, these errors arise when a transmitted Bell-state symbol is mapped into its exact opposite in terms of the 4-ary encoding. More- over, these Pauli error operators yield correlated bit errors with respect to the subsequent decoding. However, manyFECcodesworkbestwithanindependentrandom FIG. 2. BER average versus noise parameter p for non-FEC errormodelandthepresenceofcorrelatederrorscanun- encodedtransmissionsthroughadepolarizingchannel. Points dermineperformance. Thisunderperformanceresultsin correspond to single-sided superdense coding (SDC) and di- a higher BER than arises for direct transmission, which rect transmissions of single qubits. cannot incur such correlated errors. In comparison, simulation results for the average of FEC encoded transmissions are presented in Fig. 3. V. INTERLEAVING Thesecurvesshowdistinctbehaviorforthethreetypesof FEC codes considered in Table I, which all perform bet- terthantheunencodedtransmissionschemes. TheGolay In classical communication, data interleaving is fre- code, which can correct up to weight-3 errors, performs quently used to mitigate against correlated errors, also the best when transmitting by SDC. However, there is a known as burst errors. Prior to transmission, binary el- crossover with the direction transmission scheme using a ements of codewords output from the FEC encoder are repetition code. The Hamming code produces a similar interleaved with respect to transmission order while the curve for both SDC and direct transmission. Scatter in received symbols are deinterleaved in the reverse order. the plots for small values of p are due to the increases in Althoughbursterrorsstilloccurduringtransmission,any 6 correlatederrorsareeffectivelydispersedacrossthemul- tiple, interleaved codewords. As an example of interleaving, consider a two-bit input message {b ,b } protected by a repetition [3, 0 1 1] code. Encoding generates the binary string c = {b ,b ,b ,b ,b ,b } which maps to the joint quantum 0 0 0 1 1 1 state ρ =ρ ⊗ρ ⊗ρ (16) c (b0,b0) (b0,b1) (b1,b1) The simulation of transmission of this sequence using SDCthroughthedepolarizingchannelsamplesall64pos- sible combinations of Pauli noise operators. After detection and decoding, the resulting BER is 1 7 2 BER = p+ p2+ p3 (17) [3,1] 3 18 27 which has a leading order term that is linear in the noise parameter p. We contrast this non-interleaved result with a transmission scheme in which the output of the encoder is interleaved. Inthecurrentexample,wewillconsideranin- terleavingpatternthatalternativesbetweenthefirstand FIG. 4. BER average versus noise parameter p for repeti- secondcodewords,i.e.,wepreparetheinterleavedbinary tionencodedtransmissionwithandwithoutinterleaving. The sequence c(cid:48) ={b ,b ,b ,b ,b ,b }, which is mapped into non-FEC encoded transmission is plotted for comparison. 0 1 0 1 0 1 the joint quantum state ρc(cid:48) =ρ(b0,b1)⊗ρ(b0,b1)⊗ρ(b0,b1) (18) of the correlated errors by using interleaving, in which reordering of the transmitted data is found to improve Although the noisy channel operator is the same, the the ability of the code to protect against noise. This is probability for a burst error is greatly decreased by the shown to be due to the combined effects of Pauli errors interleaving and the overall BER is on FEC codewords and the decoding method. 4 16 OurresultsfortheBERconfirmthebehaviorexpected BERi[3n,t1e]rleaved = 3p2− 27p3 (19) for the entangled channel in the presence of isotropic Pauli noise, i.e., depolarizing noise. The effective bit- In particular, the leading order term is now quadratic in error rate is found to be 2p/3 with p the channel error thenoiseparameterp, whichisanimprovementoverthe rate. However, the effectiveness of the error correction non-interleaved results. Comparative plots of the BER depends subtly on the specific encoding and interleaving for interleaved and non-interleaved cases are shown in method used during transmission. For example, when X Fig. 5. errors on a transmitted entangled state lead to single-bit We also present results for the BER obtained using errors then Z errors will yield weight-two errors on the interleavingfortheHammingandGolaycodesinFig.(5). classical message. The relative significance of these errors depends not only on the distance of the FEC code but also the position of the errors in the codeword. We have shown that interleaving is effective for mitigating VI. CONCLUSION against those higher-weight errors that would otherwise lead to uncorrectable states of the code. We have presented a numerical study of how classical Our use of classical FEC was motivated by a need forward error correction techniques can be used with su- to improve the reliability of quantum communication. perdense coding. We have calculated the bit error rates While future QEC codes are likely to offer more error for one-sided SDC using a Pauli channel model. Our correction benefits, they will also require higher dimen- simulations show the relative change in the BER when sional entangled quantum resources. Current limitations usingHamming, Golay, andrepetitioncodesforbothdi- on quantum state preparation represent a bottleneck for rectandsuperdensecodingtransmissionswithrespectto using QEC to improve the quality of service in quantum channelnoise. Theseresultsareexplainedintermsofthe communicationsystems. WehaveshownthatFECcodes varying distance of the FEC codes as well as the corre- with proper interleaving offers an effective method that lated noise that arises within the context of superdense may be immediately employed with existing quantum coding. We have studied how to mitigate the influence communication hardware. We expect follow-on studies 7 as quantum key distribution, especially in the context of multi-user communication networks that may require codes specialized to different channel modes [23]. The FEC methods may also be useful for improving the de- tectionofunmodulatedstatesintamper-indicatingquan- tum seals [24]. However, we must expect that QEC codeswillultimatelybenecessaryforanyfull-scalequan- tum networking environment. Protocols that do not communicate classical information should perform better with QEC than with FEC, since only QEC can be used to correct errors in the quantum state itself. This includesapplicationsinmulti-userquantumcommunica- tionprotocolsthatteleportquantuminformation[25–27] or swap entanglement between communication channels [28]. A promising approach to reconfiguring networks between FEC and QEC encoding methods is to exploit programmable nodes [29], in which the error correction scheme is determined by an external application based on channel diagnostics. This work was supported by the Defense Threat Re- ductionAgencyBasicResearch,theArmyResearchLab- oratory, and the Department of Energy HERE program. ThismanuscripthasbeenauthoredbyUT-Battelle,LLC, FIG. 5. BER average versus noise parameter p for FEC- under Contract No. DE-AC0500OR22725 with the U.S. encoded, interleaved SDC transmission. Points correspond to FEC using Hamming [7,4], Golay [24,12], and repetition Department of Energy. The United States Government [3,1] codes. retains and the publisher, by accepting the article for publication, acknowledges that the United States Gov- ernment retains a non-exclusive, paid-up, irrevocable, of other FEC codes will prove especially interesting for world-wide license to publish or reproduce the published quantum communication systems, including polar codes form of this manuscript, or allow others to do so, for the forefficiencyanderasurecodesforadditionalnoisemod- UnitedStatesGovernmentpurposes. TheDepartmentof els. 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