Serialized Quantum Error Correction Protocol for High-Bandwidth Quantum Repeaters
SSerialized Quantum Error Correction Protocol for High-Bandwidth QuantumRepeaters
A. N. Glaudell,
1, 2
E. Waks,
1, 3 and J. M. Taylor
1, 2 Joint Quantum Institute, University of Maryland,and the National Institute of Standards and Technology, College Park, MD 20742, USA Joint Center for Quantum Information and Computer Science,University of Maryland, College Park, MD 20742, USA Department of Electrical and Computer Engineering,Institute for Research in Electronics and Applied Physics,University of Maryland, College Park, Maryland 20742, USA (Dated: October 3, 2018)Advances in single photon creation, transmission, and detection suggest that sending quantuminformation over optical fibers may have losses low enough to be correctable using a quantum errorcorrecting code. Such error-corrected communication is equivalent to a novel quantum repeaterscheme, but crucial questions regarding implementation and system requirements remain open. Herewe show that long range entangled bit generation with rates approaching 10 ebits/s may be possibleusing a completely serialized protocol, in which photons are generated, entangled, and error correctedvia sequential, one-way interactions with a minimal number of matter qubits. Provided loss anderror rates of the required elements are below the threshold for quantum error correction, this schemedemonstrates improved performance over transmission of single photons. We find improvement inebit rates at large distances using this serial protocol and various quantum error correcting codes. I. INTRODUCTION
Quantum communication, from quantum key distribu-tion [1–5] to distributed quantum computing [6, 7], ex-amines how entanglement between distant particles en-ables new information applications. Photons provide anatural means for sharing quantum information acrosslong distances. However, optical fibers attenuate photontransmission, reducing entanglement generation rates ex-ponentially with distance. Quantum repeaters [8–10] usenearby entangled pairs of qubits (Bell pairs) to createlonger-range entangled pairs via entanglement swappingand nested purification [11]. Further improvements arepossible [12] by using Bell pairs to purify qubits encodedin a quantum error correcting code (QECC) [13, 14].However, these schemes suffer from low bit rates lim-ited by the speed of light [15]. When losses and othererrors are below the threshold for quantum error correc-tion [16, 17], a different approach emerges [18–21]: pho-tons can be sent as parts of encoded states of a QECC,and photon loss is recovered via quantum error correc-tion. Crucially, the probability that a photon is lost fromthe creation to detection stage must be lower than thethreshold of the associated QECC.Motivated by recent advances in high efficiency detec-tors [22–24] and high efficiency cavity-fiber coupling [25–30], here we consider a minimalistic approach to producelogical states and correct errors in photonic systems withthe aim of building a high bit rate quantum repeater. Weassume continuing progress in recent demonstrations ofsingle photon nonlinear gates [31–40] between light andmatter qubits, and consider the limits for quantum er-ror corrected quantum repeaters when we simultaneouslyminimize the number of matter qubits required, makefew assumptions about long-term matter qubit coher- ence, and require that each photon interacts sequentially,and singly, with each matter qubit. We call this designa serial quantum repeater, and show in this paper thatwith current or near-future performance, one-way quan-tum communication at 1000 km ranges may approach thegiga-entangled bit/second range using a stream of entan-gled photons in a narrow bandwidth over single fibers.The signal for our serial repeater is comprised of severalphotons encoded in a codeword of a QECC; we use eitherpolarization qubits or time-bin based qudits for the pho-tons to distinguish loss errors from other errors. Our goalis to implement QECC-based teleportation of this signalto a new set of photons with sequential, single-shot inter-actions using stationary (matter) qudits. We will furtherrestrict ourselves to using a cavity QED-based controlledphase gate [35, 37, 39, 40], arbitrary local operations andmeasurements on single matter qudits, and Hadamard-type photon gates. These choices are designed to be con-sistent with recent progress in single-photon phase gatesand on-chip photonics.The procedure for implementing serial teleportationbased error correction involves replacing each portion ofthe protocol for state teleportation [1, 9, 41] with a newcomponent commensurate with a QECC that keeps seri-alization intact. Incoming and outgoing states of photonsare replaced by logical codewords of the QECC, whichamounts to selecting how to represent the k logical qu-dits with the D k codewords of the QECC.Just before the arrival of the signal photons at a re-peater node, the node will first generate a fresh set ofphotons encoded in a codeword of the QECC using asmall set of matter qudits as single photon sources andfor controlled phase gates. These outgoing photons arethen entangled with a matter qudit ‘memory’. The out-going photons are released toward the next node, while a r X i v : . [ qu a n t - ph ] A ug l l Classical Pauli Frame InformationInput OutputFast Optical SwitchesMatter QuditsLocalized in Cavities PhotonDetectorPhotons Encoded in Logical [[4,2,2]] State | ψ L i Z C S Z S X |↑ Stab i|↑
Ent i H •|↑ Enc i H •| Vac i S X X | ψ L i Classical Pauli Frame Information (a)(b)(c)
Figure 1. (a) Example of a serialized QECC-based quantum repeater node for [[4 , , (cid:96) . Additional Pauli frame information and error correction information are sent along the classical channel andaggregated for final correction at the end node of the repeater system. (c) Quantum circuit for QECC-based teleportation ofincoming photons carrying logical state | ψ L (cid:105) via serial interaction with three sets of matter qudits and newly created photonsfrom | Vac (cid:105) . The outgoing state is the error corrected incoming logical state when errors are sufficiently low. Individual quditsare not shown; lines represent sets of qudits as described in the main text. a switch inside the repeater enables incoming signal pho-tons to now interact with the same matter qudit. Afterthe interaction, measurement of the matter qudit tele-ports the signal state onto the outgoing photon state.However, the signal photons must also have their QECCstabilizers measured and losses detected. For CSS codes it suffices to first measure the Z -based stabilizers usingan additional matter qudit per stabilizer, then apply aHadamard to the photons and measure them with high-efficiency photon detectors. For an [[ n, k, d ]] D CSS code, < ( n − k ) qudits are necessary for the encoding stage,from 1 to k are necessary for the entangling stage, and < ( n − k ) are necessary for the Z -stabilizer readout. TheCSS codes considered in this work require only ( n − k ) / , , and [[4 , , F = C / (1+ C ) where C = 2 g /γκ is the atomic cooper-ativity, g is the cavity-quantum dot coupling strength, γ is the atomic dipole decay rate, and κ is the cavity energydecay rate. Fast rates are achievable in solid-state quan-tum dot systems, where one can attain a g/ π = 20 GHzand κ/ π = 6 GHz [44], and in which the dipole decayrate can be as low as γ/ π = 0.16 GHz. These numberscorrespond to a cooperativity of C = 416, which wouldprovide a maximum fidelity of 99.5%. The pulse dura-tion of the input photon in the strong coupling regime islimited by the coupling strength to τ = 1 / (2 πg ) = 8 ps,thus providing the possibility for both high fidelity andgigahertz bandwidths. II. SERIALIZED QUDIT-BASED ERRORCORRECTING TELEPORTATION
We now review QECCs using qudits [45–47]. Errorsconsist of two types: erasures, which either move quditsout of their D -level Hilbert space or reset the systemto a particular specified state, and gate errors, whichconsist of unitary operations acting within the qudits’Hilbert space. These unitary operations may in generalact on multiple qudits; however, it is assumed that theaction on one qudit is uncorrelated from the action onany other qudit. This assumption ensures that multiplequdit errors may be characterized using tensor productsof single qudit unitary operations. These single quditunitary operations may be constructed from members ofthe generalized Pauli group P [47], defined as P ≡ { σ a,b = ω c X a Z b ; ( a, b ) ∈ N D , c ∈ N D } , (1)where ω = e πi/D , (2a) Z = D − (cid:88) j =0 ω j | j (cid:105) (cid:104) j | , (2b) X = D − (cid:88) j =0 | ( j + 1) mod D (cid:105) (cid:104) j | . (2c)Multi-qudit Pauli operations on n qudits are composedof P ⊗ n , with the number of nontrivial single qudit op-erations defined as the weight of the operator. Another single qudit operator outside of the Pauli group but re-quired for this protocol is the higher dimensional analogof the Hadamard gate, the R gate, defined as R = D − (cid:88) j,l =0 ω jl | j (cid:105) (cid:104) l | . (3)An operator that is sufficient to entangle two qudits isthe CPHASE gate, which produces a differential phaseshift contingent on the state of both qudits:CPHASE = D − (cid:88) j,l =0 ω jl | j (cid:105) (cid:104) j | ⊗ | l (cid:105) (cid:104) l | . (4)The CPHASE gate may be raised to any power q ∈ N D to produce a related two qudit gate as well.An [[ n, k, d ]] D stabilizer code is a QECC for D -levelqudits [45] (when D is omitted, it is to be assumed that D = 2 and the code is referring to qubits). Here, n isthe number of physical qudits utilized for the code, and k is the number of logical qudits they represent. Thecollection of D k states making up the QECC are calledcodewords. The last parameter, d , is the lowest weightof any Pauli operator that projects one codeword ontoa different codeword. There are k logical phase and bitflip operators, { Z j } and { X j } respectively, associatedwith the k logical qudits. These operators, which are notunique due to representative freedom for the k logicalqudits using the codewords, obey the same commutationrelations as single qudit Pauli operations. Finally, thereare ( n − k ) measurements, belonging to the Abelian stabi-lizer group S , used to diagnose errors and which commutewith each other, { Z j } , and { X j } [45].For q located (erasure) and l unlocated (gate) errors tobe correctable using a stabilizer code, any two such errorsmust be distinguishable from one another when they acton any two codewords: (cid:104) w j | E † β, ( q,l ) E α, ( q,l ) | w m (cid:105) = C αβ δ jm , (5)where | w j (cid:105) are codewords of the QECC and the E operators possess the prescribed number of errors.This is guaranteed whenever the composite operator E † β, ( q,l ) E α, ( q,l ) has total weight less than d . Because anytwo such error operations must have support on the samesubset of q physical qudits, the combined weight of theoperator is 2 l + q . For a correctable error, the permissibleranges for located and unlocated errors is then0 ≤ l ≤ ( d − / , (6a)0 ≤ q ≤ ( d − − l. (6b)Setting q or l to zero reproduces the familiar results forhaving all located or unlocated errors [45].In order to perform measurements in S , we focus on us-ing quantum non-demolition (QND) measurements. Z qj QND measurements are performed using an R gate ap-plied to an ancilla qudit, followed by a CPHASE q gate | (cid:30) M R •| k (cid:30) M R • X X k Z C Z Ck (cid:31) D − j =0 | j (cid:30) (cid:30) M (cid:31) D − j =0 | j (cid:30) k (cid:30) M S S r U U n | k +1 (cid:30) M | k + r (cid:30) M (cid:30)(cid:30) · · · (cid:29) P (cid:30)(cid:30) ψ (cid:29) P (cid:30)(cid:30) ψ (cid:30) (cid:29) P | (cid:30) M R •| k (cid:30) M R • X X k Z C Z Ck (cid:31) D − j =0 | j (cid:30) (cid:30) M (cid:31) D − j =0 | j (cid:30) k (cid:30) M S S r U U n | k +1 (cid:30) M | k + r (cid:30) M (cid:30)(cid:30) · · · (cid:29) P (cid:30)(cid:30) ψ (cid:29) P (cid:30)(cid:30) ψ (cid:30) (cid:29) P · · · .... . . · · · .... . . · · · .... . . · · · .... . . · · · .... . . { (a)(b)(c)(d) | (cid:30) M R •| k (cid:30) M R • X X k Z C Z Ck (cid:31) D − j =0 | j (cid:30) (cid:30) M (cid:31) D − j =0 | j (cid:30) k (cid:30) M S S r U U n | k +1 (cid:30) M | k + r (cid:30) M (cid:30)(cid:30) · · · (cid:29) P (cid:30)(cid:30) ψ (cid:29) P (cid:30)(cid:30) ψ (cid:30) (cid:29) P · · · .... . . · · · .... . . { · · · .... . . · · · .... . . · · · .... . . · · · .... . . · · · .... . . | (cid:30) M R •| k (cid:30) M R • X X k Z C Z Ck (cid:31) D − j =0 | j (cid:30) (cid:30) M (cid:31) D − j =0 | j (cid:30) k (cid:30) M S S r U U n | k +1 (cid:30) M | k + r (cid:30) M (cid:30)(cid:30) · · · (cid:29) P (cid:30)(cid:30) ψ (cid:29) P (cid:30)(cid:30) ψ (cid:30) (cid:29) P (a)(b)(c)(d) { | (cid:30) M R •| k (cid:30) M R • X X k Z C Z Ck (cid:31) D − j =0 | j (cid:30) (cid:30) M (cid:31) D − j =0 | j (cid:30) k (cid:30) M S S r U U n | k +1 (cid:30) M | k + r (cid:30) M (cid:30)(cid:30) · · · (cid:29) P (cid:30)(cid:30) ψ (cid:29) P (cid:30)(cid:30) ψ (cid:30) (cid:29) P · · · .... . . | (cid:30) M R •| k (cid:30) M R • X X k Z C Z Ck (cid:31) D − j =0 | j (cid:30) (cid:30) M (cid:31) D − j =0 | j (cid:30) k (cid:30) M S S r U U n | k +1 (cid:30) M | k + r (cid:30) M (cid:30)(cid:30) · · · (cid:29) P (cid:30)(cid:30) ψ (cid:29) P (cid:30)(cid:30) ψ (cid:30) (cid:29) P · · · .... . . · · · .... . . · · · .... . . · · · .... . . (a)(b)(c)(d) Figure 2. General quantum circuit to be used at one node of the serial teleportation scheme, broken down into sections.Primes ( (cid:48) ) denote a state that is entangled with other qudits that are not pictured. (a) entangles the outgoing (cid:12)(cid:12) · · · (cid:11) logicalstate with k matter qudits and produces an equal weight superposition of every logical codeword entangled with the matchingset of logical matter qudit states. (b) entangles the incoming logical state (cid:12)(cid:12) ψ (cid:11) with the already entangled k matter qudits.Measurements of the matter qudits are performed, with the results being fed forward, and codewords contained in (cid:12)(cid:12) ψ (cid:11) areleft entangled with the codewords from the outgoing states. (c) demonstrates the measurement of the stabilizer generators onthe incoming photons. The first r “expensive” stabilizers are measured using QND measurements on ancilla matter qudits.The final ( n − k ) − r “cheap” stabilizers are measured by individually measuring the photons, which is permissible becausethe “cheap” stabilizers commute at the individual qudit level. This step simultaneously measures the stabilizers and checks forerasure errors, with results fed forward for Pauli frame correction at the last node. with the j th physical qudit and the ancilla, proceeded byan R − gate on the ancilla. Likewise, X qj measurementsmay be performed by first performing an R gate on boththe physical and ancilla qudit, performing a CPHASE q gate between them, and rotating both the physical andancilla qudits back to the original basis using two R − gates. Once all operations from a stabilizer have beenperformed using the same ancilla qudit, this ancilla ismeasured in the Z basis to complete the stabilizer mea-surement. If, however, the state is not needed anymore, astabilizer measurement may be performed destructivelyusing measurements directly on the n physical qudits.Furthermore, if multiple stabilizers are strictly composedof single qudit operators that mutually commute, theymay be measured simultaneously in this manner.Creation of an outgoing photon in state | (cid:105) is replacedwith generation of the QECC codeword representing allzeros for the k logical qudits, (cid:12)(cid:12) · · · (cid:11) . Efficiently gen-erating codewords of QECCs has been a topic of interestsince the inception of QECCs [48–51]. The schemes thatare permissible for a serial approach become even morerestrictive, owing to the requirement of at most one in-teraction between any photon and matter qudit and thecorresponding prohibition of multi-photon gates.One can simply measure all stabilizers with ( n − k )matter qudits to produce a codeword. However, we wouldprefer to use fewer, For CSS codes, Steane’s Latin rect-angle method [49] and related techniques [51] are used toefficiently produce the outgoing state with little overheadin non-serialized settings. These approaches would relyupon using photon-photon gates, and thus are not gener-ally useful to this serial teleportation scheme. However,Steane’s procedure of employing the generator matrix ofone of the QECC’s dual classical codes as a gate map isstill viable, except that each line of the matrix must beused as a map for gates with an ancilla dot. In this man- ner, a (cid:12)(cid:12) · · · (cid:11) codeword may be created using < ( n − k )stabilizers in general and ( n − k ) / k logical qudits of the QECC by employing k matterqudits. Using each matter qudit as a control qudit andfollowing the same guidelines used for imprinting stabi-lizer measurements on ancillas, each operator X j is per-formed as a controlled operation. This is demonstratedin panel (a) of Fig. 2. The incoming logical state maythen be entangled with the k matter qudits, this timeusing the { Z j } operators to determine which gates toapply. Because the photons are used as controls ratherthan targets in this case, these phase controlled flip op-erations undergo the substitutions Z qj → R − CPHASE D − qj R M , (7a) X qj → CPHASE D − qj . (7b)The j subscript now refers to the photonic control qu-dit, and R gates are now performed on the matter quditsrather than the photons. These new operators control-ling the matter qudits are called { Z Cj } . The entanglingoperations are followed by measurements in the Z basisof every matter qudit with results fed forward for Pauliframe correction [17, 52]. The circuit required to carryout this procedure is pictured in panel (b) of Fig. 2.Rather than strictly completing a bell measurement,this is the point at which stabilizer generators for theQECC must be measured. Constructing subsets of thestabilizers which consist of strictly commuting single qu-dit Pauli operators, selecting the subset that maximizesthe amount of nontrivial gates therein, and selectingthese ( n − k ) − r stabilizers in the subset to be measureddestructively from the individual photons minimizes thenumber of QND stabilizer measurements required. Fromthis standpoint, it is beneficial to select QECCs that havelarge subsets of stabilizers that commute at the individ-ual qudit levels, such as CSS codes. The first r stabilizersare then measured using QND measurements, the final( n − k ) − r stabilizer measurements are inferred fromthe destructive photon measurements along with the lo-cations of any missing qudits, and the results are fedforward for Pauli frame correction. This stabilizer mea-surement step is pictured in panel (c) of Fig. 2.Finally, amassing the results of measurements madeto create (cid:12)(cid:12) · · · (cid:11) , measurements of the k matter qu-dits, the QND measurements used to measure the first r stabilizers, and the destructive measurements of all n photons at each repeater node prescribes one final errorcorrection step to be made after the last node. Due tothe use of state teleportation, gate and erasure errors onindividual photons manifest themselves as errors in thelogical codeword basis of the outgoing photons and canbe understood as a change of the Pauli frame. As longas no individual node registers more than the permittedamount of located and unlocated errors, the final outputstate is guaranteed to be correctable. III. NUMERICAL ANALYSIS OFPERFORMANCE
We now consider optimization of this protocol for arealistic set of errors. We will consider approaches tomaximize the entangled bit rate per photon generated byoptimizing the number of repeater nodes per attenuationlength denoted by the repeater density η . Successful en-tangled bit generation using this serialized teleportationprotocol hinges on each node registering a correctablesubset of errors. The optimal repeater spacing (cid:96) maxi-mizing this rate is determined by balancing photon lossesduring the transmission between nodes with the erasureand gate errors that may occur in components used forthe serial repeater. Using this optimal repeater spac-ing, the performance of the serial repeater protocol forvarious QECC assuming different error rates may be cal-culated and compared to sending single photons downa lossy transmission line without a repeat, henceforthcalled “bare transmission.” First, an error model must bedefined to quantify this performance. The following as-sumptions are made about the operation of componentsused within the protocol:1. All single qudit operations (other than photonpropagation and photon measurement) are as-sumed to be perfect.2. Whenever missing qudits were supposed to be actedupon by a logical gate, that gate instead behaves asan identity operation, leaving the photon missing asbefore and the state of any control or target ancillaqudit unchanged.The errors that may occur within the different compo-nents utilized for the protocol may be split into locatable erasures and unlocated gate errors. Locatable erasureerrors include:3. During photon generation, there is a probability p C that no photon is created. However, if a photon isproduced, it is precisely the intended state withoutlogical error.4. At two qudit logical gates, the photon may interactwith the matter qudit as intended, but go missingwith probability p G after completing a successfullogical operation.5. During transmission between nodes, we assume thephoton is free of logical errors. However, the photonencounters a constant probability to be lost fromthe fiber. This gives a probability u = exp( − α(cid:96) ) oftraveling a distance (cid:96) , where α is the characteristicrate of photon loss with distance.6. The photon may hit a detector without registeringa measurement, producing an erroneous null resultwith probability p M .Components that may introduce unlocated errors in-clude:7. Two qudit gates produce a Pauli group error on thephoton with probabilities p X , p Y , and p Z for bit fliperrors, simultaneous bit and phase flip errors, andphase flip errors, respectively.8. Photons may hit a detector while registering an er-roneous readout for the measurement with proba-bility p F .Finally, detector dark counts may manifest as unlocatederrors, reproduce the errorless syndrome, or prevent era-sure errors from registering:9. Dark counts are assumed to occur independent ofwhether a photon was present at the detectionstage. D − D times, this dark count willregister as a logical error. One out of every D times,it produces a syndrome that will yield the properrecovery operation.From these assumptions, a performance function forthe operation of an [[ n, k, d ]] D based node may be con-structed, giving the probability that the output state willproduce a correctable outcome after traveling a distance (cid:96) and being repeated at a single node. In this case, itproves beneficial to use the probability u rather thanthe distance (cid:96) . This single node performance functionrequires the enumeration of all possible ways in whichthe number of located and unlocated errors is below thethreshold for correctability. Each individual qudit mayencounter a different number of logical gates dependenton the circuit used for the given QECC, so in generalthis requires analyzing each photonic qudit’s probabilityof success separately and combining the results such that Figure 3. Plots of the performance of the serialized repeater scheme compared to bare transmission for various CSS codes,namely the (a) [[4 , , , , , , , , codes. Coloring of the surfaces is done to highlight thenumber of repeater nodes required to reach that indicated R value, with the color scale given by (e). P tot values have been fixedat 0 . P DC has been set to zero, P F has been set to 2 P L /
3, and P C , P G , and P M have been set equal. The max achievabledistance for bare transmission for the given P tot is plotted in gray on every plot. the condition for correctability is met. Defining the prob-ability of receiving the j th qudit without error, the proba-bility of producing a correctable unlocated error, and the probability of the qudit having gone missing as P A,j ( u ), P B,j ( u ), and P C,j ( u ) respectively (see Appendix A), asingle node performance function may be defined as S ( u ) = (cid:98) ( d − / (cid:99) (cid:88) l =0 ( d − − l (cid:88) q =0 (cid:88) Permutations of q located and l unlocated errors n (cid:89) j =1 P ξ,j ( u ) , (8)where ξ takes on the indices A , B , and C depending onwhich permutation of errors is examined. This functiondefines the probability the quantum repeater node willproduce a final state from which the correct state is re-coverable with a known set of operations.If there is to be any benefit for using multiple re-peaters, there must be some length scale (cid:96) > defined by u = exp( − α(cid:96) > ) for which S ( u ) is greater than the baretransmission case. This will always be possible as longas S ( u ) > u (9) for some u ∈ (0 ,
1) provided the error rates lie below thethreshold for quantum error correction. In this manner,eliminating for u using the set of equations S ( u ) = u, (10a) dSdu = 1 (10b)provides an equation defining a “threshold relationship”between all of the error probabilities. Given a set ofprobabilities that solve this threshold relationship, onemay guarantee that the serial protocol is better than baretransmission for some value of u by decreasing any of theerror probabilities by any amount.For transmission over long distances, the use of multi-ple repeaters will provide a benefit to achievable successrates as long as error probabilities lie below the thresh-old for error correction. It is then pertinent to optimizethe distance (cid:96) between adjacent nodes to maximize thechance of successfully teleporting a state over some longerdistance L . Working with the dimensionless parameter R = αL (representing the ratio of the length L to theattenuation length of a photon in the optical fiber), thisamounts to maximizing the function P tot ( u ) = [ S ( u )] − R log u , (11) where the exponent represents the (approximate) numberof nodes covering the distance L . The value of u thatmaximizes P tot is independent of R (and vice versa) andmay be found numerically by solving the transcendentalequation S ( u ) log [ S ( u )] = u dSdu log u (12)for u . An equivalent form of Eq. (12), useful for shortdistances when u is expected to be very close to 1, maybe written as an equation for (cid:15) = (1 − u ):ln [ S (1)] = ∞ (cid:88) j =2 (cid:15) j j ! (cid:34) ( − j ( j − d j du j [ln S ( u )] + j − (cid:88) m =0 (cid:18) jm (cid:19) ( j − m − − m d m +1 du m +1 [ln S ( u )] (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u → . (13)In general, higher order terms may be neglected whenthe error rates are very low and provided higher orderderivatives of S ( u ) are of O (1) near u = 1.Solving Eq. (11) for R and evaluating it at the value u solving Eq. (12) provides an output representing thefurthest possible (dimensionless) distance at which onecould still expect to receive the correct quantum stateat the last node with probability P tot . Assuming thatapproximations to Eq. (13) may be made to lowest order, R may be estimated as R ≈ √ S (1) | ln P tot | [ | ln S (1) | [ S (cid:48) (1) − S (1) ( S (cid:48)(cid:48) (1) + S (cid:48) (1))]] (14)for a repeater density η = 1 /α(cid:96) of η ≈ S (1) (cid:115) S (cid:48) (1) − S (1) ( S (cid:48)(cid:48) (1) + S (cid:48) (1))2 | ln S (1) | . (15)Eqs. (14) and (15) are useful for finding limiting behavioronce the full form of S ( x ) is known for a given QECC. Tolowest order, R depends on error probabilities through apower law relationship, but this behavior breaks downwhen the error rate being varied is of the same order asother fixed error probabilities.The analysis here is restricted to CSS codes due to theaforementioned property of having cheap stabilizer mea-surement costs and simple codeword generation schemes,as documented in Table I. Four codes are analyzed inboth Fig. 3 and Fig. 4. The [[4 , , , , , , , , (panel (d)) qutrit code has been chosen both because it is the simplest QECC that can correct one error us-ing three qudits, and because it has an extremely simplegenerating circuit and set of stabilizers.Fig. 3 examines the maximum distances achievablegiven a fixed entangled bit rate over a wide range of errorprobabilities for the selected codes. P tot was selected tohave a value of 0 . P DC was set to zero, and P C , P G ,and P M were set equal to one another. The probabilitiesfor logical errors in gates were set equal and to a sumtotal of P L (i.e. p X = p Y = p Z = P L / p F wasset to a value of 2 P L /
3. Three dimensional plots wereproduced, plotting the obtained R value versus both thetotal logical error probability P L and the combined era-sure errors P C,G,M . Also plotted is a surface representingthe bare transmission distance at an identical P tot value.The surfaces for the serial protocol are colored according Codes Matterqudits Elements usedfor encoding Est. ebitrate (GHz) Eff. att.length ( /α ) [[4 , , , , , , , , (cid:12)(cid:12) · · · (cid:11) preparation, and the maximumestimated entangled bit rates rates achievable if P tot = 1 byassuming 100 ps per photon gate for quantum dot-based ap-proaches as discussed in the main text. These base rates arecalculated assuming QND measurements may be performedon the matter qudits using, e.g., CPHASE gates with twophotons and multiplexed high-efficiency photon detection. Ef-fective attenuation lengths are calculated under the same as-sumptions used in Fig. 4 and with a joint error rate of 10 − . Figure 4. Plot of effective code rate versus unitless distance R = αL for the (a) [[4 , , , , , , , , codes. Effective code rate is defined as P tot × k/n . The colored lines correspond to different error rates of individualgates as pictured in (e), with the black line corresponding to bare transmission. The error rates listed correspond to the valuesof every error probability 3 P F / P L = P C = P G = P M . Dark counts are neglected here. Multiplying the probability valueshere by the rates in Table I and n yields true entangled bit/second rates at the specified distance and error rate. to how many repeaters are required to reach the plotted R value. The number of repeaters utilized may be de-creased, but this will naturally also decrease the R valueas the function will no longer be maximized.The plots in Fig. 3 demonstrate improvement in theachievable distance at a fixed rate compared to baretransmission for low error probabilities. The larger thecodes, the more it becomes beneficial to space out thenodes, both because the larger distance codes can handlemore photon loss in transmission and because the morenodes are used, the more chance for uncorrectable errorsto occur during stabilizer measurements. The [[7 , , , , code possessing the least stringent error threshold.The scaling of P tot × k/n with distance R is plotted inFig. 4 at a variety of joint error rates. These error ratescorrespond to the values of both the logical and erasureerror probabilities, with 3 P F / P L = P C = P G = P M representing the listed joint rates. Dark count rates areagain neglected. True entangled bit/second rates for aQECC at a given distance for the different error proba- bilities may then be found by taking values from Table I,multiplying by P tot × k/n values taken from Fig. 4, andfinally multiplying by n . The rate lines are evaluatedfor their optimum repeater spacings η found using the u values solving Eq. (12), with this optimal value listed ad-jacent to its respective curve. In both Fig. 3 and Fig. 4, itis evident that the larger the code, the greater the depen-dence of both R and P tot (assuming the other parameteris fixed) on the error probabilities. Limits of performancefor a QECC are determined by the number of entanglinggates between the incoming photonic qudits and the mat-ter qudits used to establish entanglement and measurethe “expensive” stabilizers, as any errors in these gatesthat do not commute with the operations will produceerroneous syndromes (see Appendix A for further discus-sion). IV. SUMMARY
We have shown that quantum repeaters can mediatecommunication within a quantum network using a com-pletely serialized approach, where photons encoded in aquantum error correcting code interact once per photonwith each matter qudit in a repeater node before beingdetected with a near unit-efficiency photodetector. Thisscheme is particularly well suited to achieving high bitrates and does not require long-term quantum memory.Looking at available physical systems, high bandwidthsingle photon generation and photon-matter qudit gatesremain challenging but rudimentary demonstrations in-dicate that near giga-entangled bit/second rates may beachieved for near-future device performance. We notethat CSS codes seem optimal for serialized approachesbecause of their simple codeword preparation and thepermissibility of joint stabilizer generator measurements.This does not preclude non-CSS codes from also perform-ing well. However, in a comparison between a CSS qubitcode and a general non-CSS stabilizer code with similar n and k parameters, the CSS code will have lower overheadafforded by the two aforementioned properties. Finally,we emphasize the crucial challenge for this approach tobuilding a quantum repeater: one needs extremely lowloss, end-to-end single photon generation, transmission,CPHASE gates, and detection. While individual demon-strations of such performance now exist, integrating thesepieces into a comprehensive package remains a formidabletask. ACKNOWLEDGMENTS
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The performance function defined in Eq. 8 depends onthe probabilities of the physical qudits to have three dif-ferent outcomes: to be measured without incurred error,to be measured with a registered gate error, and to notregister a photon (null measurement) with the photondetector. The probabilities for the j th qudit to produceone of these outcomes are denoted by P A,j , P B,j , and P C,j , respectively.For a qudit to be measured without any incurred er-rors, it must proceed through every element of the proto-col without any gate or erasure errors. There are, how-ever, a few exceptions to this rule for the CSS codes con-sidered here. In particular, once the stabilizer measure-ments for detecting bit flips have been completed, bitflips may occur on the incoming qudits without intro-ducing any error to the outgoing logical state. Further-more, even if a phase error occurs (or a bit flip/missingqubit error after the bit flip-detecting stabilizers), a darkcount can produce an output syndrome indicating thatthe node has functioned without error. That can yieldthe proper syndrome for recovery 1 out of every D times.Thus: P A,j ( u ) = (1 − p C )(1 − p G − p X − p Y − p Z ) N γ,j + N δ,j − u (1 − p G − p X − p Y )(1 − p M − p F )(1 − p DC )+ (1 − p C )(1 − p G − p X − p Y ) N γ,j + N δ,j − u p DC D , (A1)where N γ,j and N δ,j are the number of gates used in theoutgoing codeword preparation and entanglement stepsand the incoming entanglement and stabilization steps,respectively.For a qudit to register a correctable gate error, at leastone gate error needs to occur somewhere during the op-eration of the repeater. In general, errors cannot occurand be correctable if they happen between two opera-tions with which the error does not commute. For ex-ample, if a particular qudit is needed for two differentstabilizer measurements that help detect bit flip errors,a bit flip error occurring on this qubit in between thetwo measurements will produce a syndrome that yieldsan incorrect recovery operation. This is due to a lim- ited ability to correct errors occurring in certain parts ofthe circuit. Specifically, in ideal quantum error correc-tion, errors are assumed to be incurred during operationof some quantum circuit, and perfect error detection isable to detect and correct them afterwards. Account-ing for errors happening in the stabilizer measurementsthemselves creates scenarios where a particular (normallycorrectable) syndrome may be produced either by errorsduring the operation of the quantum circuit or in an errorduring the stabilizer measurements, each requiring differ-ent recovery operations. In this case, the simple strategyadopted here is to assume that the most likely scenariooccurred, i.e. that the error happened during operationof the quantum circuit.1Using this assumption, we cannot correct bit flip errorsoccurring during the entangling gates between the incom-ing photons and the dots used to mediate state telepor-tation as well as the dots used for stabilization. Further-more, during these gates in which some gate errors aredetrimental to proper operation, it is still permissible toincur any error that commutes with all operations eitherbefore or after the error occurred.In addition, dark counts that occur when a qudit hasotherwise undergone errorless operation, or dark counts that occur when a qudit is in fact lost, may register asgate errors. In the case when a qudit has gone missing (aslong as it has not gone missing between stabilizer opera-tions that would detect the change), dark counts producea syndrome that appears to be an unlocated gate error.For the case of otherwise errorless operation with a darkcount, the additional error will appear to be an unlocatedgate error whenever the result is precisely any dark countresult different than those accounted for in Eq. (A1). In-corporating these additional errors, and subtracting offthe dark counts producing errorless operation, yields P B,j ( u ) = (1 − p C )(1 − p G ) N γ,j +1 u (1 − p G − p X − p Y ) N δ,j − (1 − p M ) − (1 − p C )(1 − p G − p X − p Y − p Z ) N γ,j + N δ,j − u (1 − p G − p Y − p Z )(1 − p M − p F )(1 − p DC ) − (1 − p C )(1 − p G − p X − p Y ) N γ,j + N δ,j − u p DC D + p DC − (1 − p C )(1 − p G ) N γ,j u p DC + (1 − p C )(1 − p G ) N γ,j u (1 − p G − p X − p Y ) N δ,j − p G p DC + (1 − p C )(1 − p G ) N γ,j u (1 − p G − p X − p Y ) N δ,j − (1 − p G ) p M p DC . (A2)Finally, this leaves the probability that the qudit hasgone missing in such a way that it is detectable and cor-rectable while registering strictly as a null measurementat the detector. Much like the case for gate errors, era-sure errors do not commute with the stabilizer measure-ment stage. This again prohibits correcting for errors in which the photon goes missing in between these stabi-lizer measurements and the final photon detection stage.However, as the loss is known, gate errors incurred beforethe photon has been lost are correctable as well. Pars-ing out the correctable possibilities, one arrives at theexpression P C,j ( u ) = (cid:104) − (1 − p C )(1 − p G ) N γ,j u + (1 − p C )(1 − p G ) N γ,j u (1 − p G − p X − p Y ) N δ,j − p G + (1 − p C )(1 − p G ) N γ,j u (1 − p G − p X − p Y ) N δ,j − (1 − p G ) p M (cid:105) (1 − p DC ) . (A3)Inserting these results into the performance functiongiven by Eq. (8) for the nn