I Introduction
Enabling quantum communications among quantum devices within the Quantum Internet [1, 2, 3] will ultimately lead to various groundbreaking applications. These radically new applications do not necessarily have classical counterparts [4] and they are not limited to the already wellknown secure classical communications, blind computation, distributed quantum computing, and quantum secret sharing [5, 2, 6, 7, 8]. Naturally, the reliable transfer of quantum information is sought across the quantum network relying on quantum channels [9, 10]. However, the quantum channels inevitably impose deleterious quantum decoherence, which inflict quantum errors [11, 12]. In the classical domain, the errors imposed by the communication channels can be mitigated using errorcontrol codes [13]. The key idea of errorcontrol codes is to attach appropriately designed redundancy to the information bits by an encoding process, which is utilized by the decoder to correct a certain number of errors. However, observing and/or copying quantum information is not allowed in the quantum domain due to the nocloning theorem and the quantum measurement postulate. This motivates the carefully constructed design of quantum errorcorrection codes (QECCs) [14, 15, 16, 17].
QECCs constitute potent error mitigation techniques required for tackling the deleterious effect of quantum decoherence. Similar to the classical errorcorrection codes, QECCs rely on attaching redundant qubits to the logical qubits to provide additional information that can be exploited for quantum errorcorrection during the decoding step [18]. Interestingly, the whole encoding and decoding process can be completed without actually observing the physical qubits and thus, preserving the integrity of the quantum information conveyed by the physical qubits. In the quantum domain, the redundant qubits can be in form of auxiliary qubits initialized to the or states, or in the form of preshared maximallyentangled quantum states, which are assumed to be noisefree. For a twoqubit system, the maximallyentangled quantum states are represented by the EinsteinPodolskyRosen (EPR) pairs. The stateoftheart studies typically assume that the EPR pairs are preshared among quantum devices within the quantum networks before any quantum communication protocol is initiated. Hence, the EPR pairs can be considered as the primary resource within the Quantum Internet [10].
Having preshared entanglement offers several beneficial features for QECCs. Firstly, it can be used for conveniently transforming some powerful classical errorcorrection codes that do not satisfy the symplectic criterion^{1}^{1}1A pair of classical errorcorrection codes having paritycheck matrices and can be transformed to a quantum errorcorrection code if they satisfy . into their quantum counterparts [19, 20, 21]. Secondly, they can also be used for increasing the errorcorrection capability of quantum stabilizer codes (QSCs) [22]. Indeed, there are several types of QECCs in the literature that exploit preshared entanglement, such as entanglementassisted QSCs [23], entanglementaided canonical codes [24], as well as teleportationbased QECCs [25]. However, in all the abovementioned schemes, the preshared entanglement is considered to be noisefree.
In a scenario having realistic noisy preshared entanglement, QECCs are invoked for quantum entanglement distillation (QED) [26, 27, 28, 29, 17], which is followed by quantum teleportation [30] for transferring the quantum information. QED can be viewed as a specific application of QECCs, where several copies of noisy preshared EPR pairs are discarded to obtain fewer but less noisy EPR pairs. In this approach, QED and quantum teleportation have to be performed subsequently, which typically imposes excessive practical delay. Additionally, stateoftheart QED schemes will always have some residual quantum noise, unless infinitely many noisy preshared EPR pairs are discarded during QED. Unfortunately, this residual quantum noise is carried over to the logical qubits during the quantum teleportation process and hence it affects the integrity of the quantum information.
Scheme  Noisy Entanglement  Direct Transfer 

Entanglementassisted quantum errorcorrection codes  No  Yes 
Quantum entanglement distillation + Quantum teleportation  Yes  No 
Proposed  Yes  Yes 
Having said that, in this treatise, we propose a novel solution for achieving noiseless quantum communication, despite using noisy preshared entanglement. Firstly, we eliminate the idealized simplifying assumption of having noisefree preshared EPR pairs. Secondly, we devise a scheme for avoiding the undesired delay imposed by the consecutive steps of QED and quantum teleportation. More explicitly, our proposed scheme can be viewed as a direct quantum communication scheme, which exploits the quantum noise experienced by the preshared EPR pairs for improving the reliability of quantum communications. As it will become more evident later in this treatise, our proposal may be deemed philosophically reminiscent of trainingbased equalization techniques in classical communications, which rely on pilot sequences for estimating the channel and then eliminating its impairments. Table
I boldly and explicitly contrasts our proposed scheme to the existing techniques of amalgamating preshared entanglement and QECCs. In Table I, we explicitly indicate that our proposal relies on realistic noisy preshared entanglement. Naturally, our proposal is also suitable for the scenario of noisefree preshared entanglement, similarly to the EAQECC schemes. In Section VI, we formally show that our proposed scheme outperforms the stateoftheart. Our novel contributions can be summarized as follows:
We propose a new scheme for achieving noiseless quantum communications despite relying on noisy preshared entanglement.

We carry out the performance analysis of the proposed scheme for both errordetection and errorcorrection based schemes in the face of quantum depolarizing, bitflip and phaseflip quantum channels. The results show that the proposed scheme offers competitive performance in terms of its qubit error ratio, yield, and goodput despite requiring fewer quantum gates than the existing stateoftheart quantum entanglement distillation schemes.

In case of noisefree preshared entanglement, the proposed scheme outperforms even the existing entanglementassisted quantum stabilizer codes.
The rest of the treatise is organized as follows. In Section II, we commence by presenting the quantum communication model. In Section III, we detail the explicit formulation of our proposed scheme for direct noiseless quantum communication over noisy preshared entanglement. In Section IV, we exemplify our scheme proposed for errordetection, while in Section V, we conceive its counterpart for errorcorrection. In Section VI, we show the suitability of our proposal for quantum computing applications. Finally, we conclude in Section VII by also discussing some future research directions.
Ii System Model
As discussed in [10], both entanglement generation and distribution are key for the Quantum Internet. The specific “location” of the device implementing these functionalities – a.k.a. the entanglement generator and distributor – varies among the different schemes and solutions [10]. However, there is a general agreement in the literature that the employment of the socalled “at both endpoints” scheme is vital for the Quantum Internet by enabling ondemand communication capabilities at the quantum nodes. According to the “at both endpoints” scheme, the entanglement generator and distributor is embedded within both the transmitter and the receiver [10]. In this light, we consider the quantum communication model depicted in Fig. 1. The model includes a transmitter , a receiver , the entanglement generator and distributor, a noisy quantum channel and a classical channel. Without loss in generality, in the figure we only highlighted the entanglement generator and distributor used at the receiver, since it is exploited by the proposed scheme. The quantum communication session commences with the generation of the EPR pairs, whose quantum state is
(1) 
In the rest of this treatise, we assume that the preshared EPR pairs are initialized to the quantum state of , where the subscript indicates that the first qubit of each EPR pair is held by and the second qubit is held by . In Fig. 1, the entanglement generator is located at . Hence, the first qubit of the EPR pairs has to be sent by through the quantum channel, while the second qubit of the EPR pairs is available immediately at . After obtains the first qubit of the EPR pairs , it can be exploited for transmitting the quantum information embedded within the logical quantum qubit . In addition to the preshared EPR pairs, and are also connected via a classical communication channel, which is considered to be noisefree^{2}^{2}2This assumption is not restrictive since we focus our attention on the quantum noise only. In case of a realistic noisy classical channel, the wellknown classical errormitigation techniques can be implemented..
The main goal of the quantum communication model of Fig. 1 is to faithfully transfer the quantum state from to assisted by the preshared EPR pairs and also by classical communications. To achieve this goal, may exploit the noisy preshared EPR pairs for appropriately encoding the logical qubits into , which is sent to . In addition to the received encoded quantum state , also obtains the classical bits gleaned from the measurement of the EPRpair members at . Finally, performs a decoding procedure to reconstruct the original quantum state of the logical qubits by utilizing the qubits of the EPRpair members at .
Before outlining the mathematical details of our proposal, let us briefly discuss the conventional quantum communication model that relies on the consecutive steps of QED and quantum teleportation, as portrayed in Fig. 2. Similar to the model of Fig. 1, the “at both endpoints” scheme is adopted. In contrast to our proposal, QEDbased schemes rely on numerous copies of EPR pairs, first generated and then distributed. At the transmitter and receiver , the entanglement distillers discard the several copies of the noisy EPR pairs to obtain fewer but lessnoisy EPR pairs. Once the QED step is completed, quantum teleportation is used for faithfully transferring the quantum state of the logical qubits from to . Since the QED and the quantum teleportation are performed subsequently, the conventional quantum communication model seen in Fig. 2 imposes a longer communication delay than the mdoel of Fig. 1. Additionally, the stateoftheart QED schemes always have some residual quantum noise unless infinitely many noisy preshared EPR pairs are discarded during QED. This residual quantum noise is carried over to the logical qubits during the quantum teleportation step and it affects the integrity of the quantum information.
Iia Error Model
In this treatise, we consider one of the most general quantum channel model, namely the quantum depolarizing channel , a type of quantum Pauli channel. For a singlequbit system, the quantum depolarizing channel is described by
(2) 
where are the Pauli matrices and denotes the density matrix of the input quantum state. The Kraus operators of are given by , , , [11], where
is the depolarizing probability of
.We also consider two other types of quantum Pauli channel, i.e. the bitflip and the phaseflip quantum channels. The other types of quantum Pauli channels are unitarily equivalent to bitflip and phaseflip quantum channels. Hence, the ensuing analysis can be readily extended by considering suitable preprocessing and postprocessing operations [31]. The quantum bitflip channel is defined as
(3) 
with the Kraus operators given by and , where is the bitflip probability. By contrast, the quantum phaseflip channel, which is characterized by the phaseflip probability and by the Kraus operators and , is defined as
(4) 
Iii Quantum Communication with Noisy PreShared Entanglement
In this section, we present the general concept of our proposed scheme for performing both errordetection and errorcorrection. The schematic of the proposed scheme is depicted in Fig 3. Its operation commences by preparing the initialized quantum state as follows:
(5) 
where represents the quantum state of logical qubits, while represents pairs of preshared EPR pairs between and . The subscripts and indicate that half of the EPR pairs are held by and the other half by .
As we elucidated in Section II, the generation and the distribution of the EPR pairs to is contaminated by the quantum noise imposed by the quantum channels. Let us denote the tuple Pauli operator inflicted by the quantum channel as . Then, we have
(6) 
The quantum state of the logical qubits is encoded by a quantum encoder , where we exploit the noisy EPRpair members at . The encoded state of the logical qubits is then sent through the quantum channel . Let us denote the tuple Pauli operator inflicted by the quantum channel as . Then, we have
(7) 
At the receiver side, the quantum decoder of Fig. 3 decodes the corrupted quantum state with the aid of the EPRpair members at . To design the quantum encoder and the quantum decoder , we impose the reversible property^{3}^{3}3We note that in Fig. 3 there is a little notationabuse, since we use the symbols and to denote the encoding and decoding performed on the qubits available at and , respectively. Instead, in Eq. (8), and denote the encoder and decoder acting on the global quantum state . However, this notation abuse can be tolerated since and in Eq. (8) leave the qubits unavailable at and , respectively, unchanged. on the initialized quantum state in Eq. (5), which is formulated as
(8) 
Remark.
We note that in conventional QECCs, the reversible property of a noisefree scenario can always be guaranteed, since the quantum encoder and decoder act on the same physical qubits. By contrast, in our scheme, the quantum encoder only processes the logical qubits and the EPRpair members at , whilst the quantum decoder only processes the logical qubits received via the noisy quantum channel and the EPRpair members at .
By denoting the density matrix of as , it is possible to reformulate the proposed general scheme of Fig. 3 as the following supermap :
(9) 
In Eq. (9), we take into account the effects of the quantum noise inflicted by the quantum channels utilized for both the distribution of the EPRpair members at and for the transmission of the encoded state of the logical qubits. Furthermore, in Eq. (9), , represent the Kraus operators of the quantum channels^{4}^{4}4To be more precise and with a little notationabuse, , denote the extended Kraus operators of the quantum channels, which account for the specific qubits affected by the quantum channels and for the increased dimension induced by the supermap of Eq. (9), acting on the global state ., while and are the matrix representations of the quantum encoder and decoder, respectively.
The scheme proposed in Fig. 3 is completed by local measurements on the EPR pairs whose outcomes control the operator depending the particular errorcontrol strategy implemented. Specifically, to perform the associated errorcontrol procedure, local measurements of the EPR pairs are performed for obtaining the classical bits^{5}^{5}5When EPR pairs are considered, the local measurements of the EPR pairs produce
outcomes. To denote the associated vectors, we utilize the notation
. and . Since no joint measurements are applied to the EPR pairs for the sake of reducing the number of quantum channels utilization, a syndromelike quantity may be constructed from the modulo2 addition of the classical measurement results as follows:(10) 
It is important to note that both and have chosen the appropriate predetermined measurement basis for each of the EPR pairs.
In the case of the proposed errordetection schemes, the operator of Fig. 3 acts as a discardretain unit based on the syndrome of Eq. (10). More specifically, if the syndrome values of Eq. (10) indicate the presence of errors, i.e. the syndrome values are not zeros , the operator will decide to discard the logical qubits , otherwise it will retain the logical qubits. By contrast, in the case of the proposed errorcorrection schemes, the operator represents an errorrecovery procedure based on maximumlikelihood decoding relying on the syndrome values of Eq. (10). Specifically, the errorrecovery procedure can be formally expressed as
(11) 
where denotes the probability of experiencing the logical error imposed on the logical qubits , given that we obtain the syndrome values .
Iv ErrorDetection Scheme
In this section, we consider the errordetection of either a single logical qubit or of two logical qubits and carry out its performance analysis. We rely on Definition 12 and 2 for characterizing the performance of the proposed errordetection schemes.
Definition 1.
The success probability of the proposed errordetection schemes is defined as the conditional probability of obtaining the legitimate quantum state of the logical qubits, given that we obtain the allzero syndrome values :
(12) 
The relationship between the qubit error ratio (QBER) and the success probability can simply be defined as .
Definition 2.
The yield of the proposed errordetection schemes is defined as the ratio of logical qubits retained after the detection using noisy preshared EPR pairs, i.e.:
(13) 
Readers from the classical communication field may notice the relationship between the yield and goodput metrics. While yield has been widely used in the QED literature, goodput is a common metric utilized for normalizing the performance of classical coded communication systems with respect to the associated coding rate. The notion of goodput in the quantum domain is clarified in [32], where it is used for comparing the performance of various QECCs exhibiting different quantum coding rates and for determining their performance discrepancies with respect to the quantum capacity also known as the quantum hashing bound. We underline that yield and goodput are not the same metric, although they are intimately linked. More specifically, the goodput is defined as the product of the success probability of a given QECC by its quantum coding rate [32], i.e. . Therefore, the goodput of our proposed scheme may be reformulated as in Definition 3.
Definition 3.
The goodput of the proposed errordetection schemes is defined as the product of the success probability by the ratio of logical qubits to the noisy preshared EPR pairs:
(14) 
By comparing Def. 2 and Def. 3 we can observe the intrinsic relationship between the yield and the goodput.
Iva ErrorDetection for A Single Logical Qubit
Let us consider the proposed single qubit errordetection scheme depicted in Fig. 4, which utilizes only a single noisy EPR pair. Specifically, the encoding and decoding circuit of Fig. 3 is detailed in Fig. 4. We design the quantum encoder and decoder for ensuring that the reversible condition of Eq. (8) is satisfied. The quantum encoder and quantum decoder of Fig. 4 can be represented using unitary matrices as follows:
(15) 
By scrutinizing Eq. (15), it is readily seen that the reversible property is indeed satisfied, i.e. . The performance of the scheme proposed in Fig. 4 is characterized by Proposition 1.
Proposition 1.
The success probability of the errordetection scheme depicted in Fig. 4 in the face of quantum depolarizing channels relying on a single noisy EPR pair is given by:
(16) 
while the yield is given by:
(17) 
Proof:
Please refer to Appendix A. ∎
We compare our proposed scheme to the stateoftheart QED followed by quantum teleportation schemes of Fig. 2. Specifically, we compare the scheme proposed in Fig. 4 to the singleround recurrencebased QED of [27] and to the quantum stabilizer code (QSC)based QED of [28, 29, 17] having the stabilizer operator of . We assume that the quantum teleportation step is noisefree and therefore the QBER of the benchmark schemes is directly determined by the QBER of the associated QED scheme. Note that both the benchmark schemes require two noisy preshared EPR pairs, while our proposed scheme only needs a single noisy preshared EPR pair.
The QBER is portrayed in Fig. 5, where we label the performance of the scheme presented in Fig. 4 as ‘Proposed 1’. We observe that the QBER of the scheme presented in Fig. 4 matches that of the recurrencebased and QSCbased schemes without requiring the additional quantum teleportation step, which also relies on the idealized assumption of being noisefree for both benchmarks. Furthermore, we observe that all the schemes considered are only capable of detecting a single error. This will be confirmed later in Fig. 7(a), where we report the QBER in the presence of quantum bitflip channels.
In Fig. 6(a), we report the performance of our proposed scheme in terms of its yield. We observe that our proposed scheme provides a significantly better yield than the benchmark schemes. Two noisy preshared EPR pairs are used for obtaining a single less noisy preshared EPR pair for both the recurrencebased and the QSCbased schemes. This means that during the process one of the noisy preshared EPR pairs is discarded. By contrast, our protocol only needs a single noisy preshared EPR pair for achieving the same QBER performance. Finally, the goodput of our proposed errordetection scheme is presented in Fig. 6(b), which confirms again the intrinsic relationship between the yield and the goodput. Specifically, our proposal that provides a better yield, gives us also an improved goodput. Apart from its benefit of having a higher yield and goodput, our proposed scheme also offers a pair of additional advantages:

It does not suffer from long communication delay, since it does not require the consecutive steps of performing QED followed by quantum teleportation.

It requires fewer controlledNOT (CNOT) quantum gates. Quantitatively, the proposed scheme of Fig. 4 requires a total of only two CNOT gates. By contrast, the recurrencebased scheme requires a total of three CNOT gates: two for a singleround recurrence and one for quantum teleportation. As for the QSCbased scheme, we need a total of seven CNOT gates: four for the measurement of stabilizer operators, two for the quantum inverse encoder, and one for quantum teleportation. The number of CNOT gates provides a reasonable estimate of how severe of the quantum error proliferation is expected to be when the realistic quantum encoder and the decoder are errorinfested as demonstrated in [33, 34]. In this case, the overall proliferation of quantum errors is heavily dependent on the number of twoqubit quantum gates. In our future work, we will carry out the performance analysis of our scheme in the presence of noisy quantum gates.
Remark.
By invoking the simple scheme presented in Fig 4, we can attain both an improved yield and a reduced delay, despite relying on a reduced number of CNOT gates compared to the benchmarks, which is achieved without degrading the QBER.
In the following, we extend the above performance analysis of the proposed errordetection scheme to both bitflip and phaseflip quantum channels. Specifically, in Corollary 1, we evaluate the performance in the presence of quantum bitflip channels, while in Corollary 2, we assume the presence of quantum phaseflip channels.
Corollary 1.
The success probability of the errordetection scheme depicted in Fig. 4 in the face of quantum bitflip channels by relying on a single noisy EPR pair is given by:
(18) 
while the yield is formulated as
(19) 
Proof:
Please refer to Appendix B. ∎
Corollary 2.
The success probability of the errordetection scheme depicted in Fig. 4 in the face of quantum phaseflip channels by relying on a single noisy EPR pair is given by:
(20) 
while the yield is formulated as
(21) 
Proof:
Please refer to Appendix C. ∎
Observe from Fig. 7(a) and 7(b) that similar trends are also valid for bitflip and phaseflip quantum channels. Furthermore, our proposal significantly outperforms the stateoftheart in both scenarios in terms of its yield, as depicted in Fig. 8(a) and 8(b).
In order to further generalize our analysis, let us compare the aforementioned schemes by using the same number of noisy preshared EPRpairs. More specifically, we assume having two noisy preshared EPRpairs for all the schemes considered. Specifically, we modify the scheme proposed in Fig. 4 as seen in Fig. 9, where the first EPR pair is measured in the basis , while the second pair in the basis . Let us distinguish the components of the syndrome vector in Eq. (10) according to the observation basis used for the measurement. Specifically, let us denote the syndrome component obtained when the first EPR pair is measured in the basis by and that obtained when the second EPR pair is measured in the basis by . The operator acts as follows: if , the measurement of the second EPR pair is performed to obtain . Otherwise, the logical qubit is discarded immediately, since there is no need to measure the syndrome value , if the syndrome value already indicates that the logical qubit is corrupted. The aforementioned decision strategy is summarized as a lookup table (LUT) in Table II(a). The performance of the errordetection scheme depicted in Fig. 9 is quantified in terms of its QBER and yield presented in Proposition 2.


Proposition 2.
The success probability of the proposed errordetection scheme of Fig. 9 operating in the face of quantum depolarizing channels by utilizing two noisy EPR pairs is:
(22) 
while the yield is expressed as
(23) 
Proof:
Please refer to Appendix D. ∎
The QBER, yield, and goodput of the proposed scheme of Fig. 9 are portrayed in Fig. 5, 6(a), and 6(b), respectively, where it is labeled as ‘Proposed 2’. Observe in Fig. 5 that the errordetection scheme proposed in Fig. 9 outperforms all the benchmark schemes, while in Fig. 6(b), the proposed errordetection of Fig. 9 exhibits a better goodput than the benchmarks. We conclude that upon maintaining the same maximal yield, the scheme of Fig. 9 can provide better QBER. For the sake of conciseness, we refrain from reporting the performance results for the proposed scheme of Fig. 9 in the presence of bitflip and phaseflip quantum channels.
Remark.
By maintaining the same maximal yield and goodput as the stateoftheart schemes, our proposed scheme provides better errordetection performance.
IvB ErrorDetection for Two Logical Qubits
Let us now shift our focus to the scheme presented in Fig. 10, where we use two noisy EPR pairs for constructing an errordetection scheme for two logical qubits. Similar to the previous subsection, we commence by determining the performance of our scheme in the face of quantum depolarizing channels. Then, we evaluate the performance of the proposed scheme in mitigating the errors imposed by quantum bitflip channels and the errors inflicted by quantum phaseflip channels. As mentioned in Section II, the quantum encoder and decoder are designed for satisfying the reversible property. The resultants quantum encoder and decoder are seen in Fig. 10. Additionally, the decision block of Fig. 10 is represented by the LUT of Table II(b). We summarize our performance results in Proposition 3.
Proposition 3.
The success probability of the proposed errordetection scheme of Fig. 9 operating in the face of quantum depolarizing channels is given by:
(24) 
while the yield is expressed as
(25) 
Proof:
Please refer to Appendix E. ∎
To benchmark the performance of the proposed scheme, we have chosen the following QED schemes. Firstly, for the recurrencebased method, we carry out two singleround distillations to obtain two less noisy EPR pairs from four noisy EPR pairs. Secondly, for the QSCbased scheme, we choose the stabilizer operators of and to apply errordetection to a set of four noisy EPR pairs. The uncoded QBER is given by , which means that any error experienced by any logical qubit within the two qubits is considered as an error. The resultant QBER is portrayed in Fig. 11(a), while the yield is quantified in Fig. 11(b).
In Fig. 11(a), our proposed scheme can be seen to outperform the recurrencebased scheme for , while exhibiting an identical QBER to the QSCbased scheme. However, observe in Fig. 11(b) that our proposed scheme attains better yield than both the recurrencebased and the QSCbased schemes. Additionally, as shown in Fig. 10, the total number of CNOT gates required by the entire errordetection scheme is eight. Compared to the QSCbased scheme, which required a total of 25 CNOT gates, namely 16 CNOT gates for the stabilizer measurements, eight CNOT gates for the quantum inverse encoder, and one CNOT gate for quantum teleportation. Hence, our proposed scheme requires significantly fewer CNOT gates while offering an identical QBER and better yield.
Remark.
While providing an identical QBER performance to the QSCbased schemes, our errordetection scheme always provides better yield and requires fewer CNOT gates.
Next, we apply the above performance analysis of the proposed errordetection scheme to the bitflip and phaseflip quantum channels. Specifically, in Corollary 3, we evaluate the performance of our proposal for quantum bitflip channels, while in Corollary 4, for quantum phaseflip channels.
Corollary 3.
The success probability of the proposed errordetection scheme of Fig. 9 operating in the face of quantum bitflip channels is formulated as
(26) 
while the yield is given by
(27) 
Proof:
Please refer to Appendix F. ∎
Corollary 4.
The success probability of the proposed errordetection scheme of Fig. 10 operating in the face of quantum phaseflip channels is expressed as
(28) 
while the yield is given by:
(29) 
Proof:
Please refer to Appendix G. ∎
Observe from Fig. 12(a) that the QBER of the recurrencebased scheme is better than that of our proposed errordetection scheme as well as of the QSCbased scheme. This is due to the fact that the recurrencebased scheme is specifically tailored for detecting errors. Hence, it mitigates more effectively the errors imposed by the quantum bitflip channels. By contrast, we can observe from Fig. 12(b) that the recurrencebased scheme degrades the QBER in the face of the quantum phaseflip channels, instead of improving it, since the recurrencebased scheme is not designed for detecting the errors. In Fig. 13(a), we portray the yield of the proposed errordetection scheme against both the recurrencebased and the QSCbased scheme for quantum bitflip channels. Observe that our proposed scheme provides a better maximal yield. The same comment can be made for the yield obtained for quantum phaseflip channels as presented in Fig. 13(b).
V ErrorCorrection Scheme
Errordetection schemes provide dynamic yields, since they rely on a retainanddiscard action of the operator , while errorcorrection schemes provide a constant yield, since they attempt to recover the legitimate quantum state of the logical qubits from the received encoded state. Therefore, a modification of Def. 12 and 2 is required in order to accurately evaluate the performance of the proposed errorcorrection scheme.
Definition 4.
The success probability of the proposed errorcorrection scheme is defined as the sum of the conditional probabilities , i.e. the sum of the probabilities that the errorrecovery operator successfully applies based on the syndrome value :
(30) 
where the relationship between and the QBER can be expressed as .
Definition 5.
The yield of the proposed errorcorrection scheme is defined as^{6}^{6}6We note that while Def. 4 is the standard definition utilized in the QECC literature, Def. 5 is not standard. Indeed, in the available QSCbased QED literature, the yield is evaluated as the quantum coding rate. In our proposal, we cannot use such a definition since the yield does not equal to quantum coding rate. Hence, we evaluate the yield as the ratio between the number of logical qubits and the number of utilized preshared EPR pairs.:
(31) 
while its goodput is similarly defined in Def. 3.
Let us now consider the quantum encoder and decoder of Fig. 14. To investigate its errorcorrection performance, we have to check first that the scheme of Fig. 14 is capable of discriminating all the singlequbit error patterns based on the measured syndrome values. In Fig. 14, we can observe that the overall scheme requires six noisy preshared EPR pairs, which means that we have a sixbit syndrome string denoted by , where the indices represent the EPR pair starting from the top. Therefore, for each of the singlequbit error patterns, we can evaluate the syndrome string and the associated error recovery operator, as shown in Table III. Observe that the first three elements of the syndrome string are exclusively used for identifying errors, which are obtained from basis measurements. By contrast, the last three elements are used for identifying errors, which are obtained from basis measurements. Finally, the errors can be identified based on the combination of and .
Error pattern  Syndrome  Error recovery  Error pattern  Syndrome  Error recovery 

First, let us investigate the performance of the proposed scheme in the face of bitflip and phaseflip quantum channels. For quantum bitflip channels, we have observed a total of possible error patterns inflicted by the quantum bitflip channels and calculated their associated syndrome strings. Based on these syndrome strings, we apply the error recovery operator given in Table III. First, it is clear that the proposed scheme can always correct any single qubit error pattern. Additionally, the proposed scheme is also capable of correcting 28 error patterns exhibiting three errors, seven error patterns exhibiting four errors as well as 21 error patterns exhibiting five errors. Therefore, the success probability of the proposed errorcorrection scheme of Fig. 14 in the face of quantum bitflip channels is given by
(32) 
Similarly, we can make the same observation for the proposed scheme in the face of quantum phaseflip channels. We found that the proposed scheme provides us with an identical errorcorrection behaviour for both bitflip and phaseflip quantum channels. In fact, the success probability of the errorcorrection scheme in Fig. 14 in the face of quantum phaseflip channels is given by Eq. (32) upon substituting with :
(33) 
Let us now extend the above calculation to the quantum depolarizing channel, where we have a total of error patterns represented by the total number of combinations of bitflip errors, of phaseflip errors, as well as of the simultaneous bitflip and phaseflip errors. Based on the observation made for quantum bitflip channels, we know that the errorcorrection scheme of Fig. 14 is capable of correcting a total of error patterns (including ) in the face of quantum bitflip channels. Similarly, we also have a total of correctable error patterns in the presence of quantum phaseflip channels. Therefore, we observe a total of correctable error patterns in the face of quantum depolarizing channels, since we also consider the combination of and errors, namely the errors.
After scrutinizing all error patterns, we obtain the Pauli weight distribution of the error patterns in the face of quantum depolarizing channels as follows: one error pattern is the allidentity operator (weight = ); 21 error patterns having weight = 1; 42 error patterns having weight = 2; 252 error patterns having weight = 3; 609 error patterns having weight = 4; 1281 having weight = 5; 1428 error patterns having weight = 6; 462 error patterns having weight = 7. This distribution is identical to that of a QSCbased scheme utilizing the stabilizer operators of the Steane code. Given that we have , the success probability of the proposed errorcorrection scheme of Fig. 14 in the face of quantum depolarizing channels is given by
(34) 
Let us now compare the QBER of the proposed scheme to those of the QSCbased scheme. Indeed for a fair comparison, we do not consider the recurrencebased scheme, since it is an errordetection scheme, not an errorcorrection one. For the QSCbased scheme, we utilized the stabilizer operators of Steane code [35, 17]. The resultant QBER is depicted in Fig. 15(a), which is identical to the QBER of the QSCbased scheme utilizing the stabilizer operators of the Steane code.
Both our proposed scheme and the QSCbased scheme provide a constant level of yield, namely and , respectively, since they perform errorcorrection, instead of errordetection. Specifically, the proposed errorcorrection scheme provides better yield than the QSCbased scheme, while exhibiting the same QBER. The yield of the proposed errorcorrection scheme is evaluated according to Def. 5, i.e. , while the yield of the QSCbased scheme is equal to its quantum coding rate given by . Since and , it is clear that . Since both the proposed and the QSCbased schemes provide an identical QBER, while our proposed scheme offers higher yield, we can infer that the proposed scheme will also exhibit a higher goodput, as reported in Fig. 15(b).
As for their complexities, our proposed scheme requires a total of 22 CNOT gates as seen in Fig 14. By contrast, the QSCbased scheme requires a total of 71 CNOT gates, namely 48 CNOT gates for stabilizer measurements, 22 CNOT gates for the quantum inverse encoder, and one CNOT gate for quantum teleportation. To elaborate a little further on the complexity of the quantum circuit required for performing QED followed by quantum teleportation, please refer to Fig. 16, which portrays the QSCbased scheme using the stabilizer operators of the Steane code followed by quantum teleportation.
Vi Discussion: A Quantum Computing Perspective
In the previous sections, we have shown the advantages of our proposal in quantum communication applications. In this section, we demonstrate that the proposed scheme can also be adopted for quantum computing applications. In quantum computing applications, the quantum information is usually protected with the aid of noisefree auxiliary qubits, which may also take form of preshared entanglement [23, 24, 19, 20, 21, 25]. A prime example is constituted by the family of entanglementassisted quantum stabilizer codes (EAQSCs). Compared to the conventional QSCs, which are unassisted by noisefree preshared entanglement, EAQSCs offer an errorcorrection capability improvement. This is reminiscent of having an additional errorfree side channel between the transmitter and the receiver in the classical domain. The argument that we can always have noisefree preshared entanglement relies on the assumption that EPR pairs can be created abundantly and quantum entanglement distillation can be applied to them. The concept of EAQSCs is favourable in the realms of quantum computation, since the EAQSCs can be readily amalgamated both with transversal implementation of quantum gates [36, 37] as well as with magic state distillation [38] for creating a universal set of faulttolerant quantum gates. In the following, we propose an errorcorrection scheme that outperforms the stateoftheart EAQSC.
Any EAQSC can be defined as , where is the number of physical qubits, is the number of logical qubits, is the minimum distance of the code, and is the number of noisefree preshared entangled qubits. The errordetection and errorcorrection capability of any EAQSC can be determined by its minimum distance . An EAQSC exhibiting a minimum distance is capable of detecting quantum errors or correcting quantum errors. Based on the quantum Singleton bound of EAQSCs [23], there exists a EAQSC capable of correcting a singlequbit error , which encodes one logical qubit into three physical qubits with the aid of two noisefree preshared entangled qubits . This specific code is denoted by . In the following, we will show that by utilizing two noisefree preshared EPR pairs, instead of errorcorrection, we can achieve error elimination, implying that in this specific context, we can always obtain a noisefree logical qubit.
Let us now discuss our proposed scheme portrayed in Fig. 17(a), which is rearranged into Fig. 17(b) for facilitating our analysis. The quantum channel in Fig. 17(a) and 17(b) represents a quantum channel contaminating the logical qubit. According to Fig. 17(b), the quantum encoder
is represented by the following unitary matrix:
(35) 
while the quantum decoder is described by the following unitary matrix:
(36) 
It can be readily verified that the reversible property is satisfied, i.e. we have .
Upon denoting the density matrix of the initial global quantum state of by , the proposed scheme can be formulated with the aid of the following supermap:
(37) 
where is the Kraus operator describing the quantum channel, while and represent the unitary matrices of Eq. (35) and (36). Therefore, Eq. (37) can be rewritten as:
(38) 
After the decoding operation, we perform the measurement of the EPR pairs. Observe that we can apply basis measurement to the first EPR pair and basis measurement to the second EPR pair for determining the type of Pauli error experienced by the logical qubit . To elaborate a little further, we design a scheme so that requiring a joint measurement of the EPR pairs can be avoided to reduce the complexity of the quantum encoder and decoder. We combine the classical bits of and of Fig. 17(b) to determine the error recovery operator .
Error recovery  
0  0  
1  0  
1  1  
0  1 
To expound a little further, let us denote the syndrome string as , where is obtained from the measurement of the first EPR pair in basis and is gleaned from the measurement of the second EPR pair in basis. The error recovery operator associated with the syndrome value is portrayed in Table IV. Finally, it may be inferred from Eq. (38), that after the error recovery operator of Fig. 17(a), we always obtain the legitimate quantum state of the logical qubit. Hence, we have demonstrated that with the aid of two noisefree EPR pairs, instead of correcting a singlequbit achievable by an EAQSC, we can always recover a noisefree logical qubit. Observe that when we replace the quantum channel by realistic noisy quantum Pauli gates, we can modify the LUT of Table IV to benefit from the noisefree operation of the quantum Pauli gates.
Remark.
This is a significant improvement compared to the existing EAQSC , which is only capable of correcting a singlequbit error by utilizing two noisefree EPR pairs. Furthermore, the 5qubit code of [16] is also only capable of correcting a singlequbit error. Hence, our scheme represents an improvement also with respect to [16].
Vii Conclusions and Future Research
In this treatise, we have conceived direct noiseless quantum communication using noisy preshared EPR pairs. Conventionally, noiseless quantum communication tends to rely on the consecutive steps of QED followed by quantum teleportation. One of the salient benefits that we can offer is the elimination of the long communication delay imposed by the aforementioned consecutive steps, despite relying on noisy preshared EPR pairs. Additionally, our proposed schemes offer better QBER than the recurrencebased schemes and provide identical QBER to the QSCbased schemes. Moreover, compared to the QSCbased schemes, our proposal attains better yield, despite requiring fewer CNOT gates. We have also compared our proposed scheme to EAQSC, which requires noisefree preshared EPR pairs. EAQSCs require joint eigenvalue measurements relying on all the qubits gleaned from the EPR pairs for performing errorcorrection. Despite relying only on the local measurements of the EPR pairs and classical communications, we can always obtain a noisefree logical qubit. In our future research, we are interested in finding a systematic way of constructing the quantum encoder and decoder pair. In fact, we found that an arbitrary quantum encoder and decoder pair cannot always satisfy the reversible property of Eq. (
8). Therefore, the sufficient and necessary conditions of generating the quantum encoder and decoder pair should be found. Furthermore, since our proposed scheme performs identically to the QSCbased schemes, it remains to be shown whether a wider range of QSCs can be directly embedded into our scheme.Appendix A: Proof of Proposition 1
By exploiting the error model of Section IIA and by utilizing the expressions of Eq. (15), Eq. (9) can be reformulated as
(39) 
where is the density matrix of the logical qubit and we assume that the quantum depolarizing channels experienced by and exhibit an identical depolarizing probability . After the decoding, a measurement in the basis of the EPR pair shared between and is performed. Every time we find a disagreement in the classical measurement results from the EPR pair , the associated logical qubit is discarded, otherwise, it is retained. We note that the syndrome value of is obtained if the EPR pair is in the quantum state or , while the EPR pair in the state or gives us a syndrome value of . Hence, the probability of retaining the logical qubit is equal to the probability of obtaining the syndrome value . Based on these considerations and by accounting for Eq. (Appendix A: Proof of Proposition 1), we can evaluate the probability of obtaining the syndrome value as , which is obtained from the following error operators . Then, based on this set of error operators, we can determine the probability of obtaining the syndrome value of and obtain the legitimate quantum state of the logical qubit as , which is obtained from the following error operators . Finally, the success probability of the scheme presented in Fig. 4 can be determined according to Def. 12 as follows:
(40) 
which gives us an approximately linear performance improvement over the uncoded QBER as a function of . By accounting for Def. 2, the yield is and the proof follows.
Appendix B: Proof of Corollary 1
The supermap of Eq. (9) of the errordetection scheme proposed in Fig. 4 operating in the face of quantum bitflip channels is given by . Therefore, the probability of measuring is given by . Hence, the probability of obtaining the legitimate quantum state while measuring is given by The success probability of arriving at the legitimate quantum state given that the syndrome value is becomes , which gives us a quadratic performance improvement over uncoded QBER as a function of . The yield can be directly determined according to Def. 2 and the proof follows.
Appendix C: Proof of Corollary 2
The supermap of Eq. (9) based on the errordetection scheme proposed in Fig. 4 operating in the face of quantum phaseflip channels is given by . Therefore, the probability of measuring is given by , since the measurement is carried out in the basis. Hence, the probability of measuring and obtaining the legitimate quantum state is . The yield can be directly determined according to Def. 2 and the proof follows.
Appendix D: Proof of Proposition 2
After applying the first CNOT of the decoder of Fig. 9, we can determine the probability of obtaining the syndrome value from the first EPR pair as . If , we discard the logical qubit. If , the firsttwo quantum depolarizing channels are reduced into a single depolarizing channel having Kraus operators: , , , and . By applying the second CNOT of the decoder of Fig. 9, we can determine the probability of obtaining the syndrome value from the second EPR pair as . The probability of arriving at the legitimate quantum state and measuring the syndrome value is . By using Def. 12 and 2, the proof follows.
Appendix E: Proof of Proposition 3
Similar to the proof of Prop. 1, By exploiting the error model of Section IIA and relying on the quantum encoder and decoder of Fig. 10, the supermap of Eq. (9) can be readily obtained. After the decoding operation, the first EPR pair is measured in the basis while the second one in the basis. Let us distinguish the components of the syndrome string in Eq. (10) according to the basis used for the measurement. Specifically, let us denote the syndrome component obtained when the second EPR pair is measured in the basis by , while the syndrome component obtained when the first EPR pair is measured in the basis by . The overall syndrome string is . Since no error operators exhibiting even numbers of errors and even numbers of errors can be detected, which gives us the syndrome vector of , the probability we retain the logical qubits is equal to the sum of the probabilities of all these possible error patterns resulting in the syndrome vector of . After observing error patterns and by accounting for Eq. (9), we have
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