Ultrafast and Fault-Tolerant Quantum Communication across Long Distances

Quantum repeaters (QRs) provide a way of enabling long distance quantum communication by establishing entangled qubits between remote locations. We investigate a new approach to QRs in which quantum information can be faithfully transmitted via a noisy channel without the use of long distance teleportation, thus eliminating the need to establish remote entangled links. Our approach makes use of small encoding blocks to fault-tolerantly correct both operational and photon loss errors. We describe a way to optimize the resource requirement for these QRs with the aim of the generation of a secure key. Numerical calculations indicate that the number of quantum memory bits required for our scheme has favorable poly-logarithmic scaling with the distance across which the communication is desired.

Quantum communication across long distances (10 3 -10 4 km) can significantly extend the applications of quantum information protocols such as quantum cryptography [1] and quantum secret sharing [2,3] which can be used for the creation of a secure quantum internet [4].Quantum communication can be carried out by first establishing a remote entangled pair between the sender and the receiver and using teleportation to transmit information faithfully.However, there are two main challenges that have to be overcome.First, fiber attenuation during transmission leads to an exponential decrease in entangled pair generation rate.Second, several operational errors such as channel errors, gate errors, measurement errors and quantum memory errors severely degrade the quality of entanglement used for secure key generation.In addition, quantum states cannot be amplified or duplicated deterministically in contrast to classical information [5].Establishing quantum repeater (QR) stations based on entanglement distribution is the only currently known approach to long-distance quantum communication using conventional optical fibers without exponential penalty in time and resources.
A number of schemes have been proposed for long distance quantum communication using QRs [6][7][8][9][10][11][12], most of which could be broadly classified into three classes.The first class of QRs [6][7][8][9] reduces the exponential scaling of fiber loss to polynomial scaling by introducing intermediate QR nodes.However, this scheme for long distance quantum communication is relatively slow [13], even after optimization [14], limited by the time associated with two-way classical communication between remote stations required for the entanglement purification process needed to correct operational errors [15].In contrast, the second class of QRs introduce quantum encoding and classical error correction to replace the entanglement purification with classical error correction, han-dling all operational errors [10,16].As a consequence, the entanglement generation rate further improves from 1/O(poly(L tot )) to 1/O(poly(log(L tot ))) where L tot is the total distance of communication.Recently, the approach to the third type of QRs was proposed, which uses quantum encoding to deterministically correct photon losses [11,12].By entirely eliminating two-way classical communication between all repeater stations, the third class of QRs promise extremely high key generation rates that can be close to classical communication rates, limited only by the speed of local operations.
Besides high key generation rate, it is very important to consider the resource requirement and fault-tolerant implementation for this type of QRs.In the fault-tolerant surface-code proposal by Fowler et.al. [11], the resource for each station was estimated to scale logarithmically with the distance, while the exact resource overhead was found to be sensitive to the parameters for various imperfections.The proposal by Munro et.al. [12] focused on the correction of photon loss errors using quantum parity code (QPC) [17], but did not consider fault tolerance, as perfect gate operations were assumed in their analysis.In this Letter, we propose a fault-tolerant architecture for third class of QRs, where a teleportation-based error correction (TEC) protocol [18,19] is employed within each repeater station to correct both operational and photon loss errors using Calderbank-Shor-Steane (CSS) encoding.We quantitatively investigate the optimum resource requirements using a cost function and optimize it for different repeater parameters.A schematic view of the proposed architecture of the third class of QRs is presented in Fig. 1.
Fault-tolerant architecture.Analogous to faulttolerant quantum computers [20], fault-tolerant QRs should reliably relay quantum information from one repeater station to another in the presence of various The quantum state is encoded into an error correcting code with photonic qubits, which are multiplexed and transmitted through the optical fiber to the neighboring repeater station.The quantum state of photonic qubits is transferred to the matter qubits and error correction is performed.After the error correction procedure, the quantum state of the matter qubits is transferred to photonic qubits and transmitted to the next repeater station.This procedure is carried out until the information reaches the receiver.(b) The TEC procedure consists of Bell state preparation and Bell measurement at the encoded level.Each line in the circuit represents an encoding block and the CNOT gate has a transversal implementation for CSS codes.This TEC scheme can be potentially implemented in a cavity QED system [4,22].
imperfections.Unlike quantum computers, QRs do not require a universal gate set and it is sufficient to have CNOT gates and state initialization/measurement associated with the complimentary basis of {|0 , |1 } and {|+ , |− }.However, QRs are confronted by an important challenge from transmission loss, which is less severe in most models of quantum computation.To design fault-tolerant third class of QRs, we consider the CSS codes for their fault-tolerant properties [20], in particular the compatibility with the TEC protocol that can efficiently correct not only operational errors, but also photon loss errors [18,19].The (n, m)-QPC [17] is a class of CSS codes with encoded qubits where |± L are given by The (n, m)-QPC consists of n sub-blocks, and each subblock has m physical qubits.First, we define the Pauli operators, X i,j , Y i,j , Z i,j associated the (i, j)-th qubit, where i = 1, • • • , n is the row (sub-block) label and j = 1, • • • , m is the column label for the qubit.There is one logical qubit encoded in the (n, m)-QPC, with logical operators Z ≡ n i=1 Z i,j and X ≡ m j=1 X i,j , where we may choose any j = 1, • • • , m for Z and any i = 1, • • • , n for X [20].The encoded states {|0 L , |1 L , |+ L , |− L } can be prepared fault-tolerantly with with suppressed correlated errors [5,22,23].The encoded state is transmitted via an optical fiber to the neighboring repeater station followed by error correction and transmission to the next repeater station (Fig. 1).
Suppose that each transmitted physical qubit can reach the next QR station with probability η, meanwhile suffering from depolarization errors.We apply TEC [18,19] to correct both photon loss and depolarization errors.The TEC procedure consists of Bell state preparation and Bell measurement at the encoded level (Fig. 1 (b) ), and both operations can be achieved fault-tolerantly without propagating errors within each encoding block [20].The Bell measurement of two encoded blocks (received block R and local block S) can be achieved by an encoded CNOT gate followed by measurement of logical operators XR and ZS .More specifically, it consists of nm pair-wise CNOT gates between R i,j and S i,j , followed by 2nm individual qubit measurements.Besides erasure errors, TEC can also correct operational errors from qubit depolarization (ε d ), imperfect measurement (ε m ), and noisy quantum gates (ε g ), which can be captured by an effective error probability ) acting on single qubit for our fault-tolerant circuit designs [22].
In the presence of photon loss errors, each measurement may have three possible outcomes {+1, −1, 0}.Each qubit R i,j in the R block is measured in the X basis with outcome X R i,j taking value +1 for qubit state |+ , −1 for qubit state |− , and 0 if the qubit is erased due to transmission loss.Similarly, each qubit S i,j from the S block is measured in the Z basis with outcome Z S i,j taking value +1 for qubit state |0 , −1 for qubit |1 , and 0 if the corresponding qubit in the R-block (R i,j ) is erased.The logical measurement outcomes depend on individual qubit measurement outcomes with three possible values {+1, −1, 0}.Here sign [ • • • ] is associated with majority voting between {±1}, and • • • is associated with the product of trinary outcomes.Ideally, in the absence of errors, the outcomes should be M R X = XR and M S Z = ZS , which determine the Pauli frame [18,19] of the encoded block after teleportation.In the presence of errors, however, the outcomes become M R X = α XR and M S Z = β ZS , with α, β = +1 for correct measurement, −1 for incorrect measurement, and 0 for heralded failure of measurement.We obtain the probability distribution (see Fig. 2a) [22] p α,β ≡ Pr M R X = α XR , M S Z = β ZS , which can be used to evaluate the QR performance.
Quantum bit error rate and success probability We use the probability distribution to compute the success probability and quantum bit error rate (QBER) that characterizes the QR.Since the encoded qubit passes through N repeater stations, there are N pairs of measurement outcomes ( MX and MZ ).The success probability with no heralded failure of measurements is Given that all measurement outcomes have no heralded failure, there might be an odd number of incorrect measurements of MX (or MZ ), which induces an error if the receiver decodes the information by measuring X (or Z) of the received block.We define the QBER at the encoded level of the QR as the ratio of the probability of having an odd number of incorrect measurements of MX (or MZ ) to the probability of having no heralded failure.The QBER for X (or Z) measurement by the receiver is (3) Key generation rate.For our QR, the raw key generation rate is 1/t 0 , where t 0 is the time taken for TEC.For simplicity, we may use t 0 as a time unit in our analysis.The raw keys can be converted to secure keys through classical communication protocols involving error correction and privacy amplification [1].Due to finite success probability and non-vanishing QBER, the asymptotic secure key generation rate is given by [25,26] where the binary entropy function.In Fig. 2, we show that R can approach 1/t 0 for reasonable encoding size (n × m) with an appropriate repeater spacing (L 0 ), because it is possible to achieve high P succ and low Q.The range of (n, m) that yields a high key generation rate varies with L 0 and the total distance of communication L tot (= N × L 0 ).Hence, we need to optimize the repeater parameters, including the size of encoding block, repeater spacing, and secure key generation rate.
For each secret bit generated by the QR, we should consider the cost of both time and qubit resources [27]: (1) the average time to generate a secret bit is 1/R, and (2) the total number of memory qubits needed for the QR scheme is 2mn× Ltot L0 [49].We introduce the cost function, C to be the product of these two factors 2nm R × Ltot L0 , in units of [qubits •t 0 /sbit].Here the rate R implicitly depends on the control parameters of {n, m, L 0 }.For given L tot , we can achieve the minimum cost:

Measurement Outcomes
among all possible choices of (n, m)-QPC and repeater spacing L 0 .We assume the following imperfections as we search for the optimal scheme: (1) operation error with probability ε, and (2) finite photon transmission with probability η = (1−p c )e −L0/Latt due to fiber attenuation (L att = 20km) and coupling loss (p c ). Numerical search for optimized strategy.We search for optimized choices of {n, m, L 0 } for different values of L tot with fixed imperfection parameters of {ε, p c }.We run a numerical search for L 0 and for different number of qubits to obtain C (L tot ), which should increase at least linearly with L tot .In Fig. 3, we show the variation of cost coefficient (C = C/L tot ) with L tot , to illustrate the additional overhead associated with L tot .The cost coefficient can be interpreted as the resource overhead (qubits×t 0 ) for the creation of one secret bit over 1km (with target distance L tot ).
For imperfection parameters of = 10 −3 and p c = 0, the algorithm picks only four different codes up to L tot = 10, 000km.When the code chosen by the algorithm changes (for example at 4500km in Fig. 3), the product of L 0 and R also jumps to an appropriate higher value, so that the cost coefficient varies continuously with L tot .In the presence of coupling loss p c < 10%, the optimized values of L 0 is within the range 1.4 ∼ 2km (Fig. 3) with total loss errors up to 20%; R • t 0 is high (0.6 − 0.85) because of the favorable QBER associated with the chosen codes.
The optimized cost coefficient for different operational error probabilities is shown in Fig. 3.When ε decreases below 10 −3 , the cost coefficient is dominated by photon loss errors rather than operational errors, and does not decrease by a significant amount as ε decreases further.In a realistic scenario, photons are lost due to finite coupling losses besides fiber attenuation.In Fig. 3, we show that the QR scheme can tolerate coupling losses up to 10% for a reasonable overhead in the number of qubits.Numerical calculations indicate that the cost coefficient increases by O(poly(log(L tot ))) [22].Table I provide an estimate of the resource requirements of our code under different scenarios.
Experimental considerations.To implement our QR scheme, it is crucial to fulfill the following two experimental requirements: (1) The coupling loss should be sufficient low (p c 10%), because if the transmission probability η < 50%, then the probability that the receiver decodes the logical qubit will be exponentially small [50].
(2) Quantum repeater station should have hundreds of qubits with high fidelity operations.For ion trap systems, single qubit gate error probability of 2 × 10 −5 [29], two-qubit gate error probability of 0.007 [30], and measurement error probability of 10 −4 [31,32] have been demonstrated.There are also promising developments in micro-fabricated ion traps for coherent control of hundreds of ion qubits [33].
In addition to these two requirements, efficient downconversion to telecom wavelength (using similar techniques as described in [34], where conversion efficiency of up to 86% was reported) can be used to minimize fiber attenuation.The collection efficiency of the photon from the ion (enhanced by adequate cavity QED effect [35,36]), wavelength conversion efficiency, and coupling of the resulting photons into the propagating media (fiber) should all be maximized to 90% levels, which remains an experimental challenge.
The techniques of time and wavelength-divisionmultiplexing will enable us to transmit multiple photons through a single optical fiber, increasing the communication rate by as much as four orders of magnitude (100 wavelengths, with 100 ions transmitting in sequence).
The operation of TEC can be achieved with cavity QED systems [4,22].The performance of the QR scheme introduced here depends crucially on the range of input parameters (ε, p c , t 0 ).The key generation rate R depends on the TEC time of t 0 .Since it is possible to have subnanosecond quantum gates [37,38] with trapped ions, the TEC time will be mostly limited by the relatively slow measurement (10 ∼ 100µs) [39] due to finite photon scattering rate and collection efficiency, which can be significantly improved by enhancing the ion-cavity coupling strength.For instance, if the TEC time is improved to t 0 = 1 µs, a secure key generation rate over 0.5 MHz can be achieved over 10, 000 km with the (41, 8) code for ε = 10 −3 , p c = 10% and L 0 = 1.2km.Besides trapped ions, neutral atoms in cavities [40,41], NV centers [42,43], quantum dots [44,45], and Rydberg atoms [46,47] are also promising systems for quantum repeater implementations.Furthermore, with the advance of coherent conversion between optical and microwave photons [48], superconducting qubits may become an attractive candidate to realize our scheme as they can achieve both ultrafast quantum gates and high coupling efficiency.
Summary and Outlook.We have presented a new QR scheme belonging to the third class of QRs, which considers both fault tolerance and small encoding blocks for ultrafast quantum communication over long distances.In comparison with the first and second classes of QR schemes, our QR scheme uses TEC within each QR station to correct both photon loss and operation errors.In particular, our QR scheme can tolerate finite coupling loss (p c 10%) and achieve fault-tolerant operation with approximately hundreds of qubits per repeater station.This enables improved key generation rate that is limited only by local gate operations.Our scheme requires smaller QR spacing compared to the previous classes of QRs and consequently the number of QR stations is higher by roughly an order of magnitude.But it is important to note that the key generation rate increases by more than three orders of magnitude, by eliminating the communication time between the repeater stations.In addition, we have introduced a cost function to optimize the control parameters of our QR scheme, which can potentially be used as a criterion to compare all three classes of QRs as well as to search for more efficient quantum error correcting codes for quantum communication.[49] There is an overhead of ancillary qubits to enable the fault tolerant preparation of the encoded Bell pair.This overhead depends on the fault tolerant preparation scheme as discussed in [22].To fix ideas, we will use the number of memory qubits 2mn in our calculations.
[50] Our repeater scheme is effectively a sequence of quantum erasure channels with forward-only communication between neighboring repeater stations, which has capacity max [0, 2η − 1] [28].If η < 1/2, the channel capacity vanishes, which implies that the probability of faithful transmission between neighboring repeater stations cannot approach unity, and consequently the probability of faithful transmission over many repeater stations will decrease exponentially with the number of repeater stations.

SUPPLEMENTAL MATERIAL
In the Supplemental Material, we first provide an overview of all three classes of Quantum repeaters (QRs).Then we present key procedures of fault-tolerant preparation of the encoded quantum states, teleportation-based error correction (TEC) and its implementation in cavity-QED systems.After that we give an in-depth analysis of various errors and calculate the probability distribution of measurement outcomes at each repeater station.Finally, we provide the optimization algorithm and discuss the scaling of the cost function with respect to the long distance of communication.

INTRODUCTION
The first two classes of QRs require generation of heralded EPR pairs between neighboring repeater stations, and entanglement purification or quantum error correction steps to generate an EPR pair of high fidelity between distant repeater stations.If a photon is lost in the procedure, the heralded outcome will be a failure and the procedure will be repeated until it succeeds.Hence, apart from a constant time overhead, photon loss events do not have a major role to play in the success or the failure rates of the protocol.However, the heralded outcome requires two-way classical communication, which limits the key generation rate of the first two classes of QRs.In our new scheme for QRs, entanglement purification steps (in the first class of QRs) and the heralded entanglement generation steps (in the second class of QRs) are replaced by quantum error correction at individual repeater stations, eliminating the need to establish entangled links between any two repeater stations.Such a procedure makes use of one way classical communication which can be very fast and only limited by the speed of local operations.On the other hand, photon losses during the transmission of the encoded state may lead to failure in the secret key generation.Therefore, it is important that the error correction procedure at individual repeater stations can correct both loss and operational errors.
Keeping this in mind, we choose the Calderbank-Shor-Steane (CSS) encoding because they have properties of fault-tolerant state preparation and gate implementation.For instance, an encoded CNOT gate between the codewords can be achieved by simply applying transversal CNOT gates between the physical qubits.As we will see later, this is essential to perform a fault-tolerant quantum error correction at individual repeater stations.

FAULT-TOLERANT PREPARATION OF THE ENCODED QUANTUM STATES
In this section, we will provide key procedures to prepare encoded states of quantum parity code (QPC) within each repeater station.We assume that within each repeater station there are long range interconnects for state preparation, making the physical location of the qubits irrelevant (e.g., this can be achieved in an anharmonic linear ion trap [1]).In principle, the standard procedure for fault-tolerant preparation of CSS codes [2] can be applied to our QPC encoding, because QPC is a special class of CSS code.For completeness, we will provide the procedure of fault-tolerant preparation of QPC, because the logical operators and stabilizer of QPC have special structures which enables efficient state preparation.
We define the (n, m)-QPC using the stabilizer formalism [3].We use the Pauli operators X i,j , Y i,j , Z i,j for the (i, j)-th qubit, where i = 1, • • • , n is the row (sub-block) label and j = 1, • • • , m is the column label for the qubit.The stabilizer operators for the (n, m)-QPC are Given the above nm − 1 independent stabilizer operators, there is one logical qubit encoded in the (n, m)-QPC, with logical operators The distance of the code is given by d = min(n, m).In the following, we will focus on fault-tolerant preparation of three encoded quantum states -|0 L , |+ L , and 1 √ 2 (|00 L + |11 L ) -which are needed for our new scheme of QRs.
First, we can fault-tolerantly prepare the encoded state |+ L = 1 2 n/2 (|00...0 12...m + |11...1 12...m ) ⊗n , which is simply a tensor product of n copies of m-qubit GHZ states (also called cat states).There are many approaches to prepare the m-qubit GHZ states faulttolerantly.To fix ideas, we outline the preparationverification procedure provided by Brooks and Preskill [5]: (1) prepare the product state |+ ⊗m ; (2) measure the m ZZ stabilizer operators using m ancilla qubits as illustrated in Fig. 4 [7]; (3) repeat step r times to suppress measurement errors; (4) determine the syndrome of the prepared GHZ state by picking the syndrome that occurs most frequently (or performing a perfect matching algorithm) based on the space-time history of the syndrome measurement.The syndrome associated with X errors need not be corrected, because we can track their propagation as the computation proceeds, by updating the "Pauli frames" [6] of the individual physical qubits.(A detailed error analysis of the GHZ state preparation is presented in [5].)Following the above procedure, we can prepare n independent copies of m-qubit GHZ states, and obtain the fault-tolerant preparation of the logical state FIG. 4: Fault-tolerant preparation of a GHZ state, following the scheme by Brooks and Preskill [5] .The syndrome measurements in the circuit are repeated r times.
We can also fault-tolerantly prepare the encoded state |0 L .Different from |+ L discussed earlier, |0 L cannot be decomposed as a tensor product of some simple GHZ states.Hence, we follow the standard procedure of fault-tolerant preparation for CSS codes: (1) prepare the produce state |0 ⊗nm to ensure S i,j = 1 and Z = 1; (2) repeatedly measure the stabilizer operators S i,0 using 2m-qubit GHZ states 5, which can be fault-tolerantly prepared as discussed earlier; (3) repeat step r times to suppress measurement errors; (4) determine the syn-  drome associated with stabilizer operators S i,0 based on the space-time history of the syndrome measurement.Note that gate errors during the syndrome extraction will not cause correlated errors in the encoding block, as each quantum gate can affect at most one physical qubit from the encoding block.The syndrome need not be corrected, as we can track their progagation by updating the Pauli frames.Therefore, we can prepare the logical state |0 L fault-tolerantly.Following the analysis of Brooks and Preskill [5], upper bounds on the errors in the preparation of a GHZ state P err (GHZ) and the probability that atleast one of the stabilizers is decoded wrongly P err (Stabilizer) can be determined for different values of r and r as shown in Table II.
The measurement of the stabilizer Si,0 using a GHZ state.One needs a 2m qubit GHZ state to measure the stabilizers which are associated with two-consecutive rows of the QPC.This measurement is repeated r times.
Finally, we can fault-tolerantly prepare the encoded Bell state ) by applying encoded CNOT gates (i.e., transversal CNOT gates between the k th qubit of the first block and the k th qubit of the second block for all k) between two encoding blocks |+ L and |0 L .

TELEPORTATION-BASED ERROR CORRECTION
We now consider teleportation-based error correction (TEC) at each repeater station.As illustrated in Fig. 1(b) of the main text, the TEC protocol consists of preparation of encoded Bell state and Bell measurement at the encoded level.As discussed in the previous section, we can fault-tolerantly prepare the encoded Bell state |Φ + L .The Bell measurement at the encoded level can be achieved by fault-tolerant transversal CNOT gates followed by measurement of logical X and Z operators.
Consider the simple case with only photon loss errors.When an encoded block of photons reaches a repeater station, missing photons are detected through a quantum non-destructive measurement and the remaining photons are error corrected by the TEC protocol.As illustrated in Fig. 6, a (3, 3)-QPC is used to correct loss of one photon in the absence of operational errors.In order to have successful recovery of quantum information encoded in the (n, m)-QPC, both of the following two conditions should be satisfied: 1.At least one qubit must arrive for each sub-block; 2. At least one sub-block must arrive with no loss.
In a realistic scenario, there are also operational errors from imperfect memory, measurement, and quantum gates.The TEC protocol can protect the qubits from operational errors as well as photon loss errors, by TABLE III: For (5, 4)-QPC, the measurement strategy of X based on majority voting among the following procedure of measuring the logical X and Z operators.
For logical X measurement, we uses the definition of logical operator X ≡ m j=1 X i,j for i = 1, • • • , n. Ideally, one complete sub-block is sufficent for X measurement.However, in the presence of photon loss and operational errors, we need to perform majority voting among all outcomes from complete sub-blocks.For example, in Table.III with (n, m) = (5, 4) encoding, we rearrange the encoding blocks such that the first n = 2 sub-blocks (rows) contain missing qubits, while the remaining n − n sub-blocks are complete sub-blocks.All the qubits are measured in the X basis.The i-th complete sub-block can infer the X operator by computing X * i ≡ m j=1 X i,j .Finally, we use majority voting among complete subblocks {X * i } i=n +1,n +2,••• ,n to obtain the true value of X.
For logical Z measurement, we may infer the encoded logical Z operator by calculating n i=1 Z * i , where Z * i is obtained by majority voting from the i-th sub-block.For example, Table .IV illustrates the computation of the value for logical Z operator in the presence of loss errors.
With the above procedure of measuring the logical X and Z operators, we can perform the TEC fault-tolerantly.The TEC circuit at the encoded level (Fig. 1(b) in the Letter) is very similar to the TEC circuit at the physical level (Fig. 7), consisting of Bell state preparation and Bell measurement.However, the determination of the Pauli frame is not based on the Bell measurement at physical level, but depending on the Bell measurement outcomes at encoded level.As shown in Fig. 7, we need to perform quantum gates that couple the incoming photon R i,j , local qubits S i,j , and outgoing photon T i,j .After that, we measure the incoming photon R i,j in X basis and the local atom S i,j in Z basis.Cavity QED systems can implement the TEC protocol.The key is to perform the CNOT gate, which can be decomposed into two Hadamard gates and a CPHASE gate, CN OT a,b = H b • CP HASE a,b • H b , with an efficiently implementation using cavity QED systems proposed by Duan and Kimble [4].For example, with polarization encoding {H, V } for the photon, a CHPASE gate can be achieved through an optical setup shown in Fig. 8. Using this implementation for a CPHASE gate, the TEC circuit can be effectively implemented for an atom inside a cavity as shown in Fig. 9.

ERROR MODEL & PROBABILITY DISTRIBUTIONS
We consider error model with both photon loss and operational errors (due to imperfect gates and measurement).Since the encoding blocks (R and S) are prepared fault-tolerantly and independetly, the qubits from these blocks have independent errors before we perform the Bell measurement.Before the application of the CNOT gate, the combined state of R and S can be written as ρ RS = ρ R ⊗ ρ S .In the absence of CNOT gate errors, the application of a CNOT operation (denoted by U ) on the state ρ RS is given by U ρ RS U † .In the presence of gate errors g , the action of a noisy CNOT gate can be denoted with the super-operator where {σ k } k=0,••• ,3 = {I, X, Y, Z} are Pauli matrices including identity.

FIG. 9: (color online). A schematic view of the implementation of the TEC protocol between a single atom and a single
There two high Finesse cavities with S atom and T atom and the outgoing photon from the cavity containing the T atom enters the cavity containing S atom.In addition, the the incoming photon from the previous repeater station and the input field to control and measure the atom in the cavity enters the cavity.At the output, an additional quarter wave plate (QWP) is added for the X measurement of the incoming photon.The Z measurement is carried out by taking the photon through a beam splitter (BS) and detection.
For qubit R i,j , the error channel can be characterized by the following super-operator: where η = (1 − p c ) e −L0/Latt is the transmission probability, 1 − η is the photon loss error probability, d is the probability of depolarization error for a transmitted qubit, is the effective qubit error which takes into account-measurement error m and the gate error g , respectively.Similarly, for the qubit S i,j , the error channel can be characterized as which only has depolarization error but no photon loss error because the S block consists of local qubits with no transmission loss.After the encoded Bell measurement (with transversal CNOT gates), the errors of the two encoding blocks become correlated.Hence, the measurement outcomes will become correlated between the two blocks, in particular the measurement outcome of the qubit pair (R i,j , S i,j ) will become correlated.For example, a Y R error on R i,j error will also induce a correlated error Y R X S on the qubit pair (R i,j , S i,j ).In order to compute the full distribution for the measurement outcome at the encoded level, we take the following three steps: 1.At the physical qubit level, we consider the errors associated with the qubit-pair measurement At the intermediate encoded level, we consider the errors associated with the row-pair measurement 3. At the logical encoded level, we consider the errors associated with the encoded-pair measurement XR , ZS .
In the following, we will compute three error probability distributions associated with these three different levels.

Probability distribution for qubit-pair measurement
First, we consider probability distribution associated with the qubit-pair measurement X R i,j , Z S i,j .For the ideal case with no loss or operational errors, we will have outcomes X R i,j , Z S i,j = (r i,j , s i,j ) with r i,j , s i,j = ±1, but in the presence of errors the outcome will be X R i,j , Z S i,j = (αr i,j , βs i,j ) with (α, β) = (0, 0) , (±1, ±1) , (±1, ∓1), which corresponds to the following cases: 1. (α, β) = (0, 0): Erasure error on R i,j with probability e = Pr X R i,j = 0, , where the effective error probability is defined as , where the last step upper bounds the probability of the case of (α, β) = (−1, −1).

Probability distribution for row-pair measurement
We now consider the distribution associated with the row-pair measurement X * R i , Z * S i .Suppose the ideal case, we will have outcomes X * R i , Z * S i = (r i , s i ) with r i , s i = 0, ±1, but in the presence of errors the outcome will be X * R i , Z * S i = (αr i , βs i ) with (α, β) = (0, ±1) ⊗ (0, ±1), with the following probability distribution: Note that q α,,β does not depend on the row index i, because all rows have the same probability distribution.

OVERHEAD FROM FAULT-TOLERANT STATE PREPARATION
The qubit overhead and the time overhead are closely related for fault-tolerant state preparation.To understand this better, consider the syndrome measurement of the stabilizers S i,0 .The stabilizers can be measured in parallel with two time steps.For instance, suppose we have a (4, 4) QPC, we need one time step to measure the rows {1, 2} and rows {3, 4} simultaneously and another time step to measure the rows {2, 3}.To achieve this, we need to prepare two GHZ states of 8 qubits each simultaneosuly.So, it takes 16 qubits in total to prepare one GHZ state and 32 qubits to prepare two GHZ states.It is fairly straigtforward that it will take an additional overhead of 2mn qubits to measure the stabilizers within two steps (i.e, after the creation of the GHZ states).Similarly, one can also consider the overhead associated with the fault-tolerant preparation of the GHZ state.Using this procedure it will take an additional 4mn qubits for the fault-tolerant preparation of the encoded Bell pair.
Suppose, if we can achieve very fast quantum gates with a high efficiency, then we can further improve the overhead in the number of qubits by using the same 2m qubits to recreate a GHZ state and to measure all the stabilizers of the QPC.This can be achieved with a overhead of just 4m qubits, but the time-overhead is scaled by a factor of (n − 1) compared to the previous preparation scheme for the creation of an encoded EPR pair.
It is for this reason, the cost function introduced in the manuscript considers only the qubits required for the creation of the encoded Bell pair and does not consider the additional qubit overhead required for the fault-tolerant preparation as there is more than one way to do so.But the analysis of the cost function will be very similar to the one considered in the manuscript.While we discussed a specific fault-tolerant preparation scheme of Brooks and Preskill [5], it will take future work to determine the best fault-tolerant preparation scheme for the QPC given the overhead in terms of qubits and time.

FAULT TOLERANT PROPERTIES OF QPC
An important difference between fault-tolerant quantum computers and fault-tolerant quantum repeaters is that loss errors play an important role in the latter.For a single transmission of QPC between neighboring QR stations, we can define the effective encoded error rate to be en = (1 − p 1,1 ), which takes into account both heralded failure and quantum bit error rates.Analogous to the recent study of Brooks and Preskill [5] on Bacon-Shor codes [5], we show in Fig. 10 that it is possible to suppress the encoded error to en ≈ 2 × 10 −14 by choosing an appropriate encoding with a large number of qubits in a specific range for , (a) 1.5 × 10 −2 ≤ ≤ 2.5 × 10 −2 in the absence of loss errors and in the presence of low loss errors (1%).(b) 1 × 10 −3 ≤ ≤ 9 × 10 −3 in the presence of higher loss errors (5%, 10%).Alternatively, we confirm with numerical calculations that in the absence of loss errors and in the presence of loss errors (up to (10%)), it is possible to arbitrarily suppress the encoded error rate -which accounts for both the bit-error rate and the failure probabilities to en ≈ 10 −14 .Below 10 −14 , numerical errors begin to play an important role.The results are summarized in the Table V

DETAILS OF THE OPTIMIZATION ALGORITHM
A self explanatory flow chart of the optimization algorithm used in the Letter for the minimal cost coefficient of third class of QRs is shown in Fig. 11.We start the search with L tot = 500 mboxkm, L 0 = 1 km and m = n = 2.

SCALING OF THE COST COEFFICIENT
In the absence of a QR, the cost coefficient scales exponentially with the distance across which the commu- nication is desired.In the presence of our QRs, a numerical investigation (Fig. 12) of the cost coefficient indicates that it has a poly-logarithmic scaling with the total distance of communication up to L tot = 10 4 km in the absence and in the presence of coupling losses (up to pc = 10%), respectively.In the regime where ε is smaller than 10 −3 and there are no coupling losses, the QBER and the success probability are dominated by the photon loss errors and the cost coefficient scales as ≈ O(logD) 2 .As the contribution of ε to the final success probability and QBER increases, the quadratic scaling breaks, but the scaling of the cost coefficient still seems to be polylogarithmic with distance.

GENERALIZED COST COEFFICIENT
The cost coefficient introduced in the letter is defined for the case when the cost of the qubits are expensive, but it is possible to envision a scenario, where qubits may be cheap.Taking this into account, we can define the generalized cost coefficient to be where k is a constant satisfying 0 ≤ k ≤ 1.The choice of the above definition is guided by the constraint to obtain a unitless cost coefficient which scales polynomial in the number of qubits.For k = 0, qubits cost absolutely nothing and k = 1 corresponds to the case considered in the letter, which takes into account the cost of the qubits.A comparison of the cost coefficients for different k's is shown in Fig. 13.Interestingly, it is possible to have higher repeater spacings for the case where the qubits are cheap as shown in Fig. 13.For k = 0, one can have a higher repeater spacing by increasing the range of search.To provide an estimate, for = 10 −3 and p c = 0, to generate a secret key across 1000 km, with 800 qubits per repeater station, one can have a repeater spacing of 4.3 km and with 8500 qubits per repeater station, one can have a repeater spacing of 6.3 km.Similarly, to generate a secret key across 10, 000 km, with 1000 qubits per repeater station, one can have a repeater spacing of 4.1 km and with 9100 qubits per repeater station, one can have a repeater spacing of 5.5 km.

FIG. 1 :
FIG.1:(color online).(a) A schematic view of the third class of QRs showing individual matter qubits in the repeater stations connected by an optical fiber.The quantum state is encoded into an error correcting code with photonic qubits, which are multiplexed and transmitted through the optical fiber to the neighboring repeater station.The quantum state of photonic qubits is transferred to the matter qubits and error correction is performed.After the error correction procedure, the quantum state of the matter qubits is transferred to photonic qubits and transmitted to the next repeater station.This procedure is carried out until the information reaches the receiver.(b) The TEC procedure consists of Bell state preparation and Bell measurement at the encoded level.Each line in the circuit represents an encoding block and the CNOT gate has a transversal implementation for CSS codes.This TEC scheme can be potentially implemented in a cavity QED system[4,22].

FIG. 6 :
FIG. 6: (color online).An illustration of the TEC scheme using (3, 3)-QPC to correct the loss of two photons.The photons at positions (1, 3) and(3, 3)  are missing in the first block and consequently a CNOT gate is not applied between the first block and the second block at those positions.Subsequently, an encoded X and Z -measurements are carried out and the outcomes of the measurement are used to adjust the Pauli frame either at the same repeater station or transmitted to the receiver station and the Pauli frame is adjusted.The grey shading represents entanglement before the CNOT gate and the green shading represents measurement of encoded X and Z operators.

FIG. 7 :
FIG. 7:The TEC quantum circuit at the level of physical qubits.

FIG. 8 :
FIG.8:(color online).Implementation of a CPHASE gate between a photon and an atom.The polarizing beam splitter (PBS) splits the path of the input photon depending on its polarization and the atom interacts with only photons with a certain polarization.

FIG. 10 :
FIG. 10: (color online).Optimum logical error rate en vs physical error rate (a) for no losses and 1% loss errors.(b) for 5% loss errors and 10% loss errors.Numerical errors begin to dominate in the range en ≈ 10 −14 .
FIG. 11: (color online).The flow chart of the algorithm to find the optimized QR parameters.The units of Ltot and L0 are in km and are ignored in the figure for convenience.

FIG. 13 :
FIG.13:(color online).For a range of search 2 ≤ (n, m) ≤ 50, = 10 −3 and pc = 0, Optimal repeater spacing for various k's.Note that for k = 0, it is possible to have a larger repeater spacing by increasing the range of search.

TABLE I :
Optimized resource requirements for our faulttolerant QR scheme with (n, m)-QPC encoding for different coupling losses pc and operational error ε.

TABLE II :
An estimate of the upper bounds of Perr(GHZ) and Perr(Stabilizer) for different codes for different number of rounds of syndrome measurements with gate error g = 10 −3 and measurement error m = 10 −4 .

TABLE V :
QPC codes that are required to achieve an encoded error rate of en ≈ 2 × 10 −14 for different physical error rates in the presence of varying losses in %.