Direct Quantum Communications in the Presence of Realistic Noisy Entanglement
Daryus Chandra, Angela Sara Cacciapuoti, Marcello Caleffi, Lajos Hanzo
NNoiseless Direct Quantum Communicationsin the Face of Noisy Entanglement
Daryus Chandra, Angela Sara Cacciapuoti,
Senior Member, IEEE ,Marcello Caleffi,
Senior Member, IEEE , Lajos Hanzo,
Fellow, IEEE
Abstract
The availability of pre-shared entanglement among remote quantum nodes is required forfacilitating quantum communications within the Quantum Internet. However, the generation and thedistribution of the entanglement is inherently contaminated by quantum decoherence. Conventionally,the quantum decoherence in quantum communications is mitigated by performing the consecutivesteps of quantum entanglement distillation followed by quantum teleportation. However, thisconventional approach imposes a long delay. To circumvent this impediment, we propose a noiselessquantum communication scheme in the face of realistic noisy pre-shared entanglement, whicheliminates the sequential steps imposing delay in the standard approach. More precisely, ourproposed scheme can be viewed as a direct quantum communication scheme capable of flawlessoperation in the face of realistic noisy pre-shared entanglement. Our performance analysis showsthat the proposed scheme offers improved qubit error ratio, yield, and goodput over the existingstate-of-the-art quantum entanglement distillation based schemes, despite requiring fewer quantumgates.
Index Terms
Quantum communication, quantum error-correction codes, quantum entanglement
I. I
NTRODUCTION
Enabling quantum communications among quantum devices within the Quantum Internet [1]–[3] will ultimately lead to various groundbreaking applications. These radically new ap-plications do not necessarily have classical counterparts [4] and they are not limited tothe already well-known secure classical communications, blind computation, distributedquantum computing, and quantum secret sharing [2], [5]–[8]. Naturally, the reliable transfer ofquantum 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
D. Chandra, A.S. Cacciapuoti and M. Caleffi are with DIETI, University of Naples Federico II, Naples, 80125, Italy.L. Hanzo is with School of ECS, University of Southampton, Southampton, SO17 1BJ, UK.
December 23, 2020 DRAFT a r X i v : . [ qu a n t - ph ] D ec the communication channels can be mitigated using error-control codes [13]. The key ideaof error-control codes is to attach appropriately designed redundancy to the informationbits by an encoding process, which is utilized by the decoder to correct a certain numberof errors. However, observing and/or copying quantum information is not allowed in thequantum domain due to the no-cloning theorem and the quantum measurement postulate. Thismotivates the carefully constructed design of quantum error-correction codes (QECCs) [14]–[17].QECCs constitute potent error mitigation techniques required for tackling the deleteriouseffect of quantum decoherence. Similar to the classical error-correction codes, QECCs rely onattaching redundant qubits to the logical qubits to provide additional information that can beexploited for quantum error-correction during the decoding step [18]. Interestingly, the wholeencoding and decoding process can be completed without actually observing the physicalqubits and thus, preserving the integrity of the quantum information conveyed by the physicalqubits. In the quantum domain, the redundant qubits can be in form of auxiliary qubitsinitialized to the | (cid:105) or | + (cid:105) states, or in the form of pre-shared maximally-entangled quantumstates, which are assumed to be noise-free. For a two-qubit system, the maximally-entangledquantum states are represented by the Einstein-Podolsky-Rosen (EPR) pairs. The state-of-the-art studies typically assume that the EPR pairs are pre-shared among quantum devices withinthe quantum networks before any quantum communication protocol is initiated. Hence, theEPR pairs can be considered as the primary resource within the Quantum Internet [10].Having pre-shared entanglement offers several beneficial features for QECCs. Firstly, it canbe used for conveniently transforming some powerful classical error-correction codes that donot satisfy the symplectic criterion into their quantum counterparts [19]–[21]. Secondly, theycan also be used for increasing the error-correction capability of quantum stabilizer codes(QSCs) [22]. Indeed, there are several types of QECCs in the literature that exploit pre-shared entanglement, such as entanglement-assisted QSCs [23], entanglement-aided canonicalcodes [24], as well as teleportation-based QECCs [25]. However, in all the above-mentionedschemes, the pre-shared entanglement is considered to be noise-free.In a scenario having realistic noisy pre-shared entanglement, QECCs are invoked forquantum entanglement distillation (QED) [17], [26]–[29], which is followed by quantumteleportation [30] for transferring the quantum information. QED can be viewed as a specificapplication of QECCs, where several copies of noisy pre-shared EPR pairs are discarded to A pair of classical error-correction codes having parity-check matrices H x and H z can be transformed to a quantumerror-correction code if they satisfy H x H Tz + H z H Tx = 0 mod 2 . December 23, 2020 DRAFT obtain fewer but less noisy EPR pairs. In this approach, QED and quantum teleportation haveto be performed subsequently, which typically imposes excessive practical delay. Additionally,state-of-the-art QED schemes will always have some residual quantum noise, unless infinitelymany noisy pre-shared EPR pairs are discarded during QED. Unfortunately, this residualquantum noise is carried over to the logical qubits during the quantum teleportation processand hence it affects the integrity of the quantum information.Having said that, in this treatise, we propose a novel solution for achieving noiselessquantum communication, despite using noisy pre-shared entanglement. Firstly, we eliminatethe idealized simplifying assumption of having noise-free pre-shared EPR pairs. Secondly, wedevise a scheme for avoiding the undesired delay imposed by the consecutive steps of QEDand quantum teleportation. More explicitly, our proposed scheme can be viewed as a directquantum communication scheme, which exploits the quantum noise experienced by the pre-shared EPR pairs for improving the reliability of quantum communications. As it will becomemore evident later in this treatise, our proposal may be deemed philosophically reminiscentof training-based equalization techniques in classical communications, which rely on pilotsequences for estimating the channel and then eliminating its impairments. Table I boldlyand explicitly contrasts our proposed scheme to the existing techniques of amalgamatingpre-shared entanglement and QECCs. In Table I, we explicitly indicate that our proposalrelies on realistic noisy pre-shared entanglement. Naturally, our proposal is also suitable forthe scenario of noise-free pre-shared entanglement, similarly to the EA-QECC schemes. InSection VI, we formally show that our proposed scheme outperforms the state-of-the-art. Ournovel contributions can be summarized as follows:1) We propose a new scheme for achieving noiseless quantum communications despiterelying on noisy pre-shared entanglement.2) We carry out the performance analysis of the proposed scheme for both error-detectionand error-correction based schemes in the face of quantum depolarizing, bit-flipand phase-flip quantum channels. The results show that the proposed scheme offerscompetitive performance in terms of its qubit error ratio, yield, and goodput despiterequiring fewer quantum gates than the existing state-of-the-art quantum entanglementdistillation schemes.TABLE I: Comparison of our proposed scheme with the existing approaches.
Scheme Noisy Entanglement Direct TransferEntanglement-assisted quantum error-correction codes No
Yes
Quantum entanglement distillation + Quantum teleportation
Yes
NoProposed
Yes Yes
December 23, 2020 DRAFT
EntanglementGenerator andDistributor | Φ + i B | Φ + i AB Receiver (B) | ψ i| ψ i | ψ i | e ψ i| Φ + i A | f Φ + i A Transmitter (A)
Encoder DecoderClassical Channel
Quantum Channel
Fig. 1: The quantum communication model considered for our proposed scheme.3) In case of noise-free pre-shared entanglement, the proposed scheme outperforms eventhe existing entanglement-assisted quantum stabilizer codes.The rest of the treatise is organized as follows. In Section II, we commence by presentingthe quantum communication model. In Section III, we detail the explicit formulation ofour proposed scheme for direct noiseless quantum communication over noisy pre-sharedentanglement. In Section IV, we exemplify our scheme proposed for error-detection, whilein Section V, we conceive its counterpart for error-correction. In Section VI, we showthe suitability of our proposal for quantum computing applications. Finally, we concludein Section VII by also discussing some future research directions.II. S
YSTEM M ODEL
As discussed in [10], both entanglement generation and distribution are key for the QuantumInternet. The specific “location” of the device implementing these functionalities – a.k.a.the entanglement generator and distributor – varies among the different schemes andsolutions [10]. However, there is a general agreement in the literature that the employmentof the so-called “ at both end-points ” scheme is vital for the Quantum Internet by enablingon-demand communication capabilities at the quantum nodes. According to the “ at bothend-points ” scheme, the entanglement generator and distributor is embedded within both thetransmitter and the receiver [10]. In this light, we consider the quantum communication modeldepicted in Fig. 1. The model includes a transmitter ( A ) , a receiver ( B ) , the entanglementgenerator and distributor, a noisy quantum channel and a classical channel. Without loss ingenerality, in the figure we only highlighted the entanglement generator and distributor usedat the receiver, since it is exploited by the proposed scheme. The quantum communicationsession commences with the generation of the EPR pairs, whose quantum state is | Φ ± (cid:105) AB = 1 √ | (cid:105) ± | (cid:105) ) AB , | Ψ ± (cid:105) AB = 1 √ | (cid:105) ± | (cid:105) ) AB . (1) December 23, 2020 DRAFT
Quantum Channel
EntanglementDistiller
QuantumTeleporter
Transmitter ( A ) | ψ i ... | f Φ + i A | f Φ + i A ... | f Φ + i A | Φ + i A | Φ + i A | Φ + i A | Φ + i B ... | Φ + i B | Φ + i B Post-Processor
Classical Channel | ψ i Receiver ( B ) | Φ + i A Entanglement
Entanglement | Φ + i B Generator and
Distiller
Distributor | Φ + i AB Fig. 2: The conventional quantum communication model relying on consecutive steps ofquantum entanglement distillation and quantum teleportation.In the rest of this treatise, we assume that the pre-shared EPR pairs are initialized to thequantum state of | Φ + (cid:105) AB , where the subscript AB indicates that the first qubit of each EPRpair is held by A and the second qubit is held by B . In Fig. 1, the entanglement generator islocated at B . Hence, the first qubit of the EPR pairs | Φ + (cid:105) A has to be sent by B through thequantum channel, while the second qubit of the EPR pairs | Φ + (cid:105) B is available immediately at B . After A obtains the first qubit of the EPR pairs | Φ + (cid:105) A , it can be exploited for transmittingthe quantum information embedded within the logical quantum qubit | ψ (cid:105) . In addition to thepre-shared EPR pairs, A and B are also connected via a classical communication channel,which is considered to be noise-free .The main goal of the quantum communication model of Fig. 1 is to faithfully transfer thequantum state | ψ (cid:105) from A to B assisted by the pre-shared EPR pairs and also by classicalcommunications. To achieve this goal, A may exploit the noisy pre-shared EPR pairs | Φ + (cid:105) A for appropriately encoding the logical qubits | ψ (cid:105) into | ψ (cid:105) , which is sent to B . In additionto the received encoded quantum state | ˜ ψ (cid:105) , B also obtains the classical bits gleaned fromthe measurement of the EPR-pair members | Φ + (cid:105) A at A . Finally, B performs a decodingprocedure to reconstruct the original quantum state | ψ (cid:105) of the logical qubits by utilizing thequbits of the EPR-pair members | Φ + (cid:105) B at B .Before outlining the mathematical details of our proposal, let us briefly discuss the con-ventional quantum communication model that relies on the consecutive steps of QED andquantum teleportation, as portrayed in Fig. 2. Similar to the model of Fig. 1, the “ at both end-points ” scheme is adopted. In contrast to our proposal, QED-based schemes rely on numerouscopies of EPR pairs, first generated and then distributed. At the transmitter A and receiver B ,the entanglement distillers discard the several copies of the noisy EPR pairs to obtain fewerbut less-noisy EPR pairs. Once the QED step is completed, quantum teleportation is usedfor faithfully transferring the quantum state | ψ (cid:105) of the logical qubits from A to B . Since the This assumption is not restrictive since we focus our attention on the quantum noise only. In case of a realistic noisyclassical channel, the well-known classical error-mitigation techniques can be implemented.
December 23, 2020 DRAFT
QED and the quantum teleportation are performed subsequently, the conventional quantumcommunication model seen in Fig. 2 imposes a longer communication delay than the mdoel ofFig. 1. Additionally, the state-of-the-art QED schemes always have some residual quantumnoise unless infinitely many noisy pre-shared EPR pairs are discarded during QED. Thisresidual quantum noise is carried over to the logical qubits during the quantum teleportationstep and it affects the integrity of the quantum information.
A. Error Model
In this treatise, we consider one of the most general quantum channel model, namely thequantum depolarizing channel N ( · ) , a type of quantum Pauli channel. For a single-qubitsystem, the quantum depolarizing channel is described by N ( ρ ) = (1 − p ) ρ + p XρX + Y ρY + ZρZ ) , (2)where { I, X, Y, Z } are the Pauli matrices and ρ denotes the density matrix of the inputquantum state. The Kraus operators of N ( · ) are given by N = √ − pI , N = (cid:112) p X , N = (cid:112) p Y , N = (cid:112) p Z [11], where p is the depolarizing probability of N ( · ) .We also consider two other types of quantum Pauli channel, i.e. the bit-flip and the phase-flipquantum channels. The other types of quantum Pauli channels are unitarily equivalent to bit-flip and phase-flip quantum channels. Hence, the ensuing analysis can be readily extended byconsidering suitable pre-processing and post-processing operations [31]. The quantum bit-flipchannel is defined as N ( ρ ) = (1 − p x ) ρ + p x XρX, (3)with the Kraus operators given by N = √ − p x I and N = √ p x X , where p x is the bit-flip probability. By contrast, the quantum phase-flip channel, which is characterized by thephase-flip probability p z and by the Kraus operators N = √ − p z I and N = √ p z Z , isdefined as N ( ρ ) = (1 − p z ) ρ + p z ZρZ. (4)III. Q
UANTUM C OMMUNICATION WITH N OISY P RE -S HARED E NTANGLEMENT
In this section, we present the general concept of our proposed scheme for performing botherror-detection and error-correction. The schematic of the proposed scheme is depicted inFig 3. Its operation commences by preparing the initialized quantum state as follows: | ψ p (cid:105) = | ψ (cid:105) k ⊗ | Φ + (cid:105) n − kAB , (5)where | ψ (cid:105) k represents the quantum state of k logical qubits, while | Φ + (cid:105) n − kAB represents ( n − k ) pairs of pre-shared EPR pairs | Φ + (cid:105) between A and B . The subscripts A and B indicate that December 23, 2020 DRAFT | ψ i k | e ψ i k NV A M V † B M R
Modulo-2Addition s A,n − k s B,n − k s A,n − k | Φ + i n − kB | f Φ + i n − kA | b ψ i k | ψ i k Fig. 3: The scheme proposed for performing noiseless quantum communication using noisypre-shared EPR pairs.half of the EPR pairs are held by A and the other half by B .As we elucidated in Section II, the generation and the distribution of the EPR pairs to A is contaminated by the quantum noise imposed by the quantum channels. Let us denote the ( n − k ) -tuple Pauli operator inflicted by the quantum channel as P n − k . Then, we have | (cid:102) Φ + (cid:105) n − kA = P n − k | Φ + (cid:105) n − kA . (6)The quantum state | ψ (cid:105) k of the logical qubits is encoded by a quantum encoder V A , wherewe exploit the noisy EPR-pair members at A | (cid:102) Φ + (cid:105) n − kA . The encoded state | ψ (cid:105) k of the logicalqubits is then sent through the quantum channel N ( · ) . Let us denote the k -tuple Pauli operatorinflicted by the quantum channel as P k . Then, we have | (cid:101) ψ (cid:105) k = P k | ψ (cid:105) k . (7)At the receiver side, the quantum decoder V † B of Fig. 3 decodes the corrupted quantum state | (cid:101) ψ (cid:105) k with the aid of the ( n − k ) EPR-pair members | Φ + (cid:105) n − kB at B . To design the quantumencoder V A and the quantum decoder V † B , we impose the reversible property on the initializedquantum state in Eq. (5), which is formulated as V † B V A ( | ψ (cid:105) k ⊗ | Φ + (cid:105) n − kAB ) = | ψ (cid:105) k ⊗ | Φ + (cid:105) n − kAB . (8) Remark.
We note that in conventional QECCs, the reversible property of a noise-free scenariocan always be guaranteed, since the quantum encoder V and decoder V † act on the samephysical qubits. By contrast, in our scheme, the quantum encoder V A only processes thelogical qubits | ψ (cid:105) k and the EPR-pair members at A , whilst the quantum decoder V † B onlyprocesses the logical qubits | (cid:101) ψ (cid:105) k received via the noisy quantum channel N ( · ) and the EPR-pair members at B .By denoting the density matrix of | ψ p (cid:105) = | ψ (cid:105) ⊗ k | Φ + (cid:105) n − kAB as ρ , it is possible to reformulate We note that in Fig. 3 there is a little notation-abuse, since we use the symbols V A and V † B to denote the encoding anddecoding performed on the qubits available at A and B , respectively. Instead, in Eq. (8), V A and V † B denote the encoderand decoder acting on the global quantum state | ψ (cid:105) k ⊗ | Φ + (cid:105) n − kAB . However, this notation abuse can be tolerated since V A and V † B in Eq. (8) leave the qubits unavailable at A and B , respectively, unchanged. December 23, 2020 DRAFT the proposed general scheme of Fig. 3 as the following supermap S : S ( V A , V † B , N , ρ ) = (cid:88) i,j ( V B N j V A N i ) ρ ( V B N j V A N i ) † . (9)In Eq. (9), we take into account the effects of the quantum noise inflicted by the quantumchannels utilized for both the distribution of the EPR-pair members at A and for thetransmission of the encoded state of the logical qubits. Furthermore, in Eq. (9), N i , N j represent the Kraus operators of the quantum channels , while V A and V B are the matrixrepresentations of the quantum encoder and decoder, respectively.The scheme proposed in Fig. 3 is completed by local measurements M on the EPR pairswhose outcomes control the operator R depending the particular error-control strategy imple-mented. Specifically, to perform the associated error-control procedure, local measurementsof the EPR pairs are performed for obtaining the classical bits s A,n − k and s B,n − k . Sinceno joint measurements are applied to the EPR pairs for the sake of reducing the number ofquantum channels utilization, a syndrome-like quantity may be constructed from the modulo-2addition of the classical measurement results as follows: s n − k = s A,n − k ⊕ s B,n − k . (10)It is important to note that both A and B have chosen the appropriate pre-determinedmeasurement basis M for each of the EPR pairs.In the case of the proposed error-detection schemes, the operator R of Fig. 3 acts as adiscard-retain unit based on the syndrome of Eq. (10). More specifically, if the syndromevalues of Eq. (10) indicate the presence of errors, i.e. the syndrome values are not zeros ( s n − k (cid:54) = 0) , the operator R will decide to discard the logical qubits | ψ (cid:105) k , otherwise it willretain the logical qubits. By contrast, in the case of the proposed error-correction schemes, theoperator R represents an error-recovery procedure based on maximum-likelihood decodingrelying on the syndrome values of Eq. (10). Specifically, the error-recovery procedure canbe formally expressed as (cid:98) L k ( s n − k ) = arg max L k P ( L k | s n − k ) , (11)where P ( L k | s n − k ) denotes the probability of experiencing the logical error L k imposed onthe logical qubits | ψ (cid:105) k , given that we obtain the syndrome values s n − k . To be more precise and with a little notation-abuse, N i , N j 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 thesupermap of Eq. (9), acting on the global state | ψ (cid:105) ⊗ k | Φ + (cid:105) n − kAB . When n − k EPR pairs are considered, the local measurements of the EPR pairs produce n − k ) outcomes. To denotethe associated vectors, we utilize the notation s . December 23, 2020 DRAFT
IV. E
RROR -D ETECTION S CHEME
In this section, we consider the error-detection of either a single logical qubit or of twological qubits and carry out its performance analysis. We rely on Definition 1 and 2 forcharacterizing the performance of the proposed error-detection schemes.
Definition 1.
The success probability p s of the proposed error-detection schemes is definedas the conditional probability of obtaining the legitimate quantum state ρ of the logical qubits,given that we obtain the all-zero syndrome values s n − k = 0 : p s = p ( ρ | s n − k = 0) = p ( ρ ∩ s n − k = 0) p ( s n − k = 0) . (12)The relationship between the qubit error ratio (QBER) and the success probability p s cansimply be defined as QBER = 1 − p s . Definition 2.
The yield Y of the proposed error-detection schemes is defined as the ratio of k logical qubits retained after the detection using ( n − k ) noisy pre-shared EPR pairs, i.e.: Y = (cid:18) kn − k (cid:19) p ( s n − k = 0) . (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 acommon metric utilized for normalizing the performance of classical coded communicationsystems with respect to the associated coding rate. The notion of goodput in the quantumdomain is clarified in [32], where it is used for comparing the performance of various QECCsexhibiting different quantum coding rates and for determining their performance discrepancieswith respect to the quantum capacity also known as the quantum hashing bound. We underlinethat yield and goodput are not the same metric, although they are intimately linked. Morespecifically, the goodput G is defined as the product of the success probability of a givenQECC by its quantum coding rate [32], i.e. G = (cid:0) kn (cid:1) p s = (cid:0) kn (cid:1) (1 − QBER ) . Therefore, thegoodput of our proposed scheme may be reformulated as in Definition 3. Definition 3.
The goodput of the proposed error-detection schemes is defined as the productof the success probability by the ratio of k logical qubits to the ( n − k ) noisy pre-sharedEPR pairs: G = (cid:18) kn − k (cid:19) p s = (cid:18) kn − k (cid:19) (1 − QBER ) . (14)By comparing Def. 2 and Def. 3 we can observe the intrinsic relationship between the yieldand the goodput. December 23, 2020 DRAFT0 M Z N M Z N Communication V † B | f Φ + i A | Φ + i B V A | Φ + i AB R | ψ i | e ψ i Classical
Fig. 4: The quantum circuit conceived for performing a single-qubit error-detection using asingle noisy EPR pair.
A. Error-Detection for A Single Logical Qubit
Let us consider the proposed single qubit error-detection scheme depicted in Fig. 4, whichutilizes only a single noisy EPR pair. Specifically, the encoding and decoding circuit ofFig. 3 is detailed in Fig. 4. We design the quantum encoder and decoder for ensuring that thereversible condition of Eq. (8) is satisfied. The quantum encoder V A and quantum decoder V † B of Fig. 4 can be represented using unitary matrices as follows: V A = | (cid:105)(cid:104) | ⊗ I ⊗ I + | (cid:105)(cid:104) | ⊗ X ⊗ I, V B = | (cid:105)(cid:104) | ⊗ I ⊗ I + | (cid:105)(cid:104) | ⊗ I ⊗ X. (15)By scrutinizing Eq. (15), it is readily seen that the reversible property is indeed satisfied, i.e. V † B V A ( | ψ (cid:105) ⊗ | Φ + (cid:105) AB ) = | ψ (cid:105) ⊗ | Φ + (cid:105) AB . The performance of the scheme proposed in Fig. 4is characterized by Proposition 1. Proposition 1.
The success probability of the error-detection scheme depicted in Fig. 4 inthe face of quantum depolarizing channels relying on a single noisy EPR pair is given by: p s = 1 − p − O ( p ) , (16)while the yield is given by: Y = 1 − p p . (17) Proof:
Please refer to Appendix A.We compare our proposed scheme to the state-of-the-art QED followed by quantumteleportation schemes of Fig. 2. Specifically, we compare the scheme proposed in Fig. 4to the single-round recurrence-based QED of [27] and to the quantum stabilizer code (QSC)-based QED of [17], [28], [29] having the stabilizer operator of S = ZZ . We assume that thequantum teleportation step is noise-free and therefore the QBER of the benchmark schemesis directly determined by the QBER of the associated QED scheme. Note that both thebenchmark schemes require two noisy pre-shared EPR pairs, while our proposed schemeonly needs a single noisy pre-shared EPR pair.The QBER is portrayed in Fig. 5, where we label the performance of the scheme presented inFig. 4 as ‘Proposed 1’. We observe that the QBER of the scheme presented in Fig. 4 matches December 23, 2020 DRAFT1
Fig. 5: The QBER of our proposed error-detection schemes compared to the existing schemesfor mitigating the effect of quantum depolarizing channels. The uncoded QBER curve is givenby QBER = p . The inset is the QBER in the log-log scale. (a) Yield (b) Goodput Fig. 6: The (a) yield and the (b) goodput of the proposed error-detection schemes comparedto the existing schemes for mitigating the effect of quantum depolarizing channels. The insetsare the yield and the goodput in the logarithmic x axis.that of the recurrence-based and QSC-based schemes without requiring the additional quantumteleportation step, which also relies on the idealized assumption of being noise-free for bothbenchmarks. Furthermore, we observe that all the schemes considered are only capable ofdetecting a single X error. This will be confirmed later in Fig. 7(a), where we report theQBER in the presence of quantum bit-flip channels.In Fig. 6(a), we report the performance of our proposed scheme in terms of its yield. Weobserve that our proposed scheme provides a significantly better yield than the benchmarkschemes. Two noisy pre-shared EPR pairs are used for obtaining a single less noisy pre-sharedEPR pair for both the recurrence-based and the QSC-based schemes. This means that duringthe process one of the noisy pre-shared EPR pairs is discarded. By contrast, our protocol December 23, 2020 DRAFT2 only needs a single noisy pre-shared EPR pair for achieving the same QBER performance.Finally, the goodput of our proposed error-detection scheme is presented in Fig. 6(b), whichconfirms again the intrinsic relationship between the yield and the goodput. Specifically, ourproposal that provides a better yield, gives us also an improved goodput. Apart from itsbenefit of having a higher yield and goodput, our proposed scheme also offers a pair ofadditional advantages: • It does not suffer from long communication delay, since it does not require theconsecutive steps of performing QED followed by quantum teleportation. • It requires fewer controlled-NOT (CNOT) quantum gates. Quantitatively, the proposedscheme of Fig. 4 requires a total of only two CNOT gates. By contrast, the recurrence-based scheme requires a total of three CNOT gates: two for a single-round recurrenceand one for quantum teleportation. As for the QSC-based scheme, we need a total ofseven CNOT gates: four for the measurement of stabilizer operators, two for the quantuminverse encoder, and one for quantum teleportation. The number of CNOT gates providesa reasonable estimate of how severe of the quantum error proliferation is expected tobe when the realistic quantum encoder V A and the decoder V † B are error-infested asdemonstrated in [33], [34]. In this case, the overall proliferation of quantum errors isheavily dependent on the number of two-qubit quantum gates. In our future work, wewill carry out the performance analysis of our scheme in the presence of noisy quantumgates. Remark.
By invoking the simple scheme presented in Fig 4, we can attain both an improvedyield and a reduced delay, despite relying on a reduced number of CNOT gates compared tothe benchmarks, which is achieved without degrading the QBER.In the following, we extend the above performance analysis of the proposed error-detectionscheme to both bit-flip and phase-flip quantum channels. Specifically, in Corollary 1, weevaluate the performance in the presence of quantum bit-flip channels, while in Corollary 2,we assume the presence of quantum phase-flip channels.
Corollary 1.
The success probability of the error-detection scheme depicted in Fig. 4 in theface of quantum bit-flip channels by relying on a single noisy EPR pair is given by: p s = 1 − p x − O ( p x ) , (18)while the yield is formulated as Y = 1 − p x + 2 p x . (19) December 23, 2020 DRAFT3 (a) Bit-flip (b) Phase-flip
Fig. 7: The QBER of the proposed error-detection scheme compared to the existing protocolsfor mitigating the effect of (a) bit-flip and (b) phase-flip quantum channels. The insets arethe QBER in the log-log scale. (a) Bit-flip (b) Phase-flip
Fig. 8: The yield of the proposed error-detection scheme compared to the existing schemesfor mitigating the effect of (a) bit-flip and (b) phase-flip quantum channels. The insets arethe yield in the logarithmic x axis. Proof:
Please refer to Appendix B.
Corollary 2.
The success probability of the error-detection scheme depicted in Fig. 4 in theface of quantum phase-flip channels by relying on a single noisy EPR pair is given by: p s = 1 − p z + 2 p z , (20)while the yield is formulated as Y = 1 . (21) Proof:
Please refer to Appendix C.Observe from Fig. 7(a) and 7(b) that similar trends are also valid for bit-flip and phase-flip
December 23, 2020 DRAFT4
N RNN | ψ i ClassicalCommunication V A V † B | Φ + i AB M X | f Φ + i A M Z M Z M X | e ψ i | Φ + i B Fig. 9: The quantum circuit for performing a single-qubit error-detection using two noisypre-shared EPR pairs.quantum channels. Furthermore, our proposal significantly outperforms the state-of-the-art inboth 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 schemesby using the same number of noisy pre-shared EPR-pairs. More specifically, we assumehaving two noisy pre-shared EPR-pairs for all the schemes considered. Specifically, wemodify the scheme proposed in Fig. 4 as seen in Fig. 9, where the first EPR pair ismeasured in the Z basis ( M Z = {| (cid:105)(cid:104) | , | (cid:105)(cid:104) |} ) , while the second pair in the X basis ( M X = {| + (cid:105)(cid:104) + | , |−(cid:105)(cid:104)−|} ) . Let us distinguish the components of the syndrome vector inEq. (10) according to the observation basis used for the measurement. Specifically, let usdenote the syndrome component obtained when the first EPR pair is measured in the Z basisby s Z = s A ⊕ s B and that obtained when the second EPR pair is measured in the X basis by s X = s A ⊕ s B . The operator R acts as follows: if s Z = 0 , the measurement of the second EPRpair is performed to obtain s X . Otherwise, the logical qubit is discarded immediately, sincethere is no need to measure the syndrome value s X , if the syndrome value s Z already indicatesthat the logical qubit is corrupted. The aforementioned decision strategy is summarized as alook-up table (LUT) in Table II(a). The performance of the error-detection scheme depictedin Fig. 9 is quantified in terms of its QBER and yield presented in Proposition 2. Proposition 2.
The success probability of the proposed error-detection scheme of Fig. 9operating in the face of quantum depolarizing channels by utilizing two noisy EPR pairs is: p s = 1 − p − O ( p ) , (22)TABLE II: Syndrome values and associated decision R for the error-detection schemes. (a) Scheme in Fig. 9. s Z s X Decision R (b) Scheme in Fig. 10. s X s Z Decision R December 23, 2020 DRAFT5
NN RRM X M Z M X M Z NN V A V † B | f Φ + i A | Φ + i B ClassicalCommunication | e ψ i| ψ i | Φ + i AB Fig. 10: The quantum circuit designed for the proposed error-detection for two logical qubits,which utilizes two noisy pre-shared EPR pairs.while the yield is expressed as Y = 12 (cid:18) − p p − p (cid:19) . (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 theerror-detection scheme proposed in Fig. 9 outperforms all the benchmark schemes, while inFig. 6(b), the proposed error-detection 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 providebetter QBER. For the sake of conciseness, we refrain from reporting the performance resultsfor the proposed scheme of Fig. 9 in the presence of bit-flip and phase-flip quantum channels.
Remark.
By maintaining the same maximal yield and goodput as the state-of-the-art schemes,our proposed scheme provides better error-detection performance.
B. Error-Detection for Two Logical Qubits
Let us now shift our focus to the scheme presented in Fig. 10, where we use two noisyEPR pairs for constructing an error-detection scheme for two logical qubits. Similar to theprevious subsection, we commence by determining the performance of our scheme in theface of quantum depolarizing channels. Then, we evaluate the performance of the proposedscheme in mitigating the X errors imposed by quantum bit-flip channels and the Z errorsinflicted by quantum phase-flip channels. As mentioned in Section II, the quantum encoder V A and decoder V † B are designed for satisfying the reversible property. The resultants quantumencoder V A and decoder V † B are seen in Fig. 10. Additionally, the decision block R ofFig. 10 is represented by the LUT of Table II(b). We summarize our performance results inProposition 3. Proposition 3.
The success probability of the proposed error-detection scheme of Fig. 9
December 23, 2020 DRAFT6 (a) QBER (b) Yield
Fig. 11: The (a) QBER and the (b) yield of the proposed error-detection scheme in Fig. 10compared to the existing schemes for mitigating the effect of quantum depolarizing channels.The uncoded QBER curve is given by QBER = 1 − (1 − p ) = 2 p − p . The insets are theQBER and the yield in the logarithmic x axis.operating in the face of quantum depolarizing channels is given by: p s = 1 − p − O ( p ) , (24)while the yield is expressed as Y = 1 − p + 8 p − p p . (25) Proof:
Please refer to Appendix E.To benchmark the performance of the proposed scheme, we have chosen the following QEDschemes. Firstly, for the recurrence-based method, we carry out two single-round distillationsto obtain two less noisy EPR pairs from four noisy EPR pairs. Secondly, for the QSC-based scheme, we choose the stabilizer operators of S = XXXX and S = ZZZZ toapply error-detection to a set of four noisy EPR pairs. The uncoded QBER is given byQBER = 1 − (1 − p ) = 2 p − p , which means that any error experienced by any logicalqubit within the two qubits is considered as an error. The resultant QBER is portrayed inFig. 11(a), while the yield is quantified in Fig. 11(b).In Fig. 11(a), our proposed scheme can be seen to outperform the recurrence-based schemefor p < . , while exhibiting an identical QBER to the QSC-based scheme. However, observein Fig. 11(b) that our proposed scheme attains better yield than both the recurrence-basedand the QSC-based schemes. Additionally, as shown in Fig. 10, the total number of CNOTgates required by the entire error-detection scheme is eight. Compared to the QSC-basedscheme, which required a total of 25 CNOT gates, namely 16 CNOT gates for the stabilizermeasurements, eight CNOT gates for the quantum inverse encoder, and one CNOT gate for December 23, 2020 DRAFT7 quantum teleportation. Hence, our proposed scheme requires significantly fewer CNOT gateswhile offering an identical QBER and better yield.
Remark.
While providing an identical QBER performance to the QSC-based schemes, ourerror-detection scheme always provides better yield and requires fewer CNOT gates.Next, we apply the above performance analysis of the proposed error-detection scheme tothe bit-flip and phase-flip quantum channels. Specifically, in Corollary 3, we evaluate theperformance of our proposal for quantum bit-flip channels, while in Corollary 4, for quantumphase-flip channels.
Corollary 3.
The success probability of the proposed error-detection scheme of Fig. 9operating in the face of quantum bit-flip channels is formulated as p s = 1 − p x − O ( p x ) , (26)while the yield is given by Y = 1 − p x + 12 p x − p x + 8 p x . (27) Proof:
Please refer to Appendix F.
Corollary 4.
The success probability p s of the proposed error-detection scheme of Fig. 10operating in the face of quantum phase-flip channels is expressed as p s = 1 − p z − O ( p z ) , (28)while the yield is given by: Y = 1 − p z + 12 p z − p z + 8 p z . (29) Proof:
Please refer to Appendix G.Observe from Fig. 12(a) that the QBER of the recurrence-based scheme is better than thatof our proposed error-detection scheme as well as of the QSC-based scheme. This is dueto the fact that the recurrence-based scheme is specifically tailored for detecting X errors.Hence, it mitigates more effectively the X errors imposed by the quantum bit-flip channels.By contrast, we can observe from Fig. 12(b) that the recurrence-based scheme degrades theQBER in the face of the quantum phase-flip channels, instead of improving it, since therecurrence-based scheme is not designed for detecting the Z errors. In Fig. 13(a), we portraythe yield of the proposed error-detection scheme against both the recurrence-based and theQSC-based scheme for quantum bit-flip channels. Observe that our proposed scheme providesa better maximal yield. The same comment can be made for the yield obtained for quantum December 23, 2020 DRAFT8 (a) Bit-flip (b) Phase-flip
Fig. 12: The QBER of our proposed error-detection scheme of Fig. 10 compared to theexisting schemes for mitigating the effect of (a) bit-flip and (b) phase-flip quantum channels.The insets are the QBER in the log-log scale. (a) Bit-flip (b) Phase-flip
Fig. 13: The yield of our proposed error-detection scheme of Fig. 10 compared to the existingschemes for mitigating the effect of (a) bit-flip and (b) phase-flip quantum channels. The insetsare the yield in the logarithmic x axis.phase-flip channels as presented in Fig. 13(b).V. E RROR -C ORRECTION S CHEME
Error-detection schemes provide dynamic yields, since they rely on a retain-and-discard actionof the operator R , while error-correction schemes provide a constant yield, since they attemptto recover the legitimate quantum state of the logical qubits from the received encoded state.Therefore, a modification of Def. 1 and 2 is required in order to accurately evaluate theperformance of the proposed error-correction scheme. Definition 4.
The success probability p s of the proposed error-correction scheme is definedas the sum of the conditional probabilities p ( (cid:98) L k = L k | s n − k ) , i.e. the sum of the probabilitiesthat the error-recovery operator R successfully applies (cid:98) L k = L k based on the syndrome value December 23, 2020 DRAFT9 | ψ i M Z M Z M X M X M X M Z M Z M Z M Z M X M X M X N M Z RNNNNNN | Φ + i B | ψ iV † B V A | f Φ + i A | ψ i Fig. 14: The quantum encoder V A and the quantum decoder V † B for performing the proposederror-correction scheme. s n − k : p s = (cid:88) L k p ( (cid:98) L k = L k | s n − k ) , (30)where the relationship between p s and the QBER can be expressed as QBER = 1 − p s . Definition 5.
The yield of the proposed error-correction scheme is defined as : Y = kn − k , (31)while its goodput is similarly defined in Def. 3.Let us now consider the quantum encoder V A and decoder V † B of Fig. 14. To investigate itserror-correction performance, we have to check first that the scheme of Fig. 14 is capableof discriminating all the single-qubit error patterns based on the measured syndrome values.In Fig. 14, we can observe that the overall scheme requires six noisy pre-shared EPR pairs,which means that we have a six-bit syndrome string denoted by s = s s s s s s , where theindices i ∈ { , , , , , } represent the EPR pair starting from the top. Therefore, for eachof the single-qubit error patterns, we can evaluate the syndrome string and the associatederror recovery operator, as shown in Table IV. Observe that the first three elements of thesyndrome string s Z = s s s are exclusively used for identifying X errors, which are obtainedfrom Z basis measurements. By contrast, the last three elements s X = s s s are used foridentifying Z errors, which are obtained from X basis measurements. Finally, the Y errorscan be identified based on the combination of s Z and s X .First, let us investigate the performance of the proposed scheme in the face of bit-flip andphase-flip quantum channels. For quantum bit-flip channels, we have observed a total of = 128 possible error patterns inflicted by the quantum bit-flip channels and calculated their We note that while Def. 4 is the standard definition utilized in the QECC literature, Def. 5 is not standard. Indeed, inthe available QSC-based QED literature, the yield is evaluated as the quantum coding rate. In our proposal, we cannot usesuch a definition since the yield does not equal to quantum coding rate. Hence, we evaluate the yield as the ratio betweenthe number of logical qubits and the number of utilized pre-shared EPR pairs.
December 23, 2020 DRAFT0 associated syndrome strings. Based on these syndrome strings, we apply the error recoveryoperator R given in Table IV. First, it is clear that the proposed scheme can always correctany single qubit error pattern. Additionally, the proposed scheme is also capable of correcting28 error patterns exhibiting three X errors, seven error patterns exhibiting four X errors aswell as 21 error patterns exhibiting five X errors. Therefore, the success probability of theproposed error-correction scheme of Fig. 14 in the face of quantum bit-flip channels is givenby p s = 1 − p x + 98 p x − p x + 252 p x − p x + 48 p x . (32)Similarly, we can make the same observation for the proposed scheme in the face of quantumphase-flip channels. We found that the proposed scheme provides us with an identical error-correction behaviour for both bit-flip and phase-flip quantum channels. In fact, the successprobability of the error-correction scheme in Fig. 14 in the face of quantum phase-flipchannels is given by Eq. (32) upon substituting p x with p z : p s = 1 − p z + 98 p z − p z + 252 p z − p z + 48 p z . (33)Let us now extend the above calculation to the quantum depolarizing channel, where we havea total of = 16 , error patterns represented by the total number of combinations ofbit-flip X errors, of phase-flip Z errors, as well as of the simultaneous bit-flip and phase-flip Y errors. Based on the observation made for quantum bit-flip channels, we know that theerror-correction scheme of Fig. 14 is capable of correcting a total of error patterns (including IIIIIII ) in the face of quantum bit-flip channels. Similarly, we alsohave a total of correctable error patterns in the presence of quantum phase-flip channels.Therefore, we observe a total of ×
64 = 4 , correctable error patterns in the face ofquantum depolarizing channels, since we also consider the combination of X and Z errors,namely the Y errors.After scrutinizing all , error patterns, we obtain the Pauli weight distribution of theTABLE IV: Syndrome values and the associated error-recovery operator R of the error-correction scheme presented in Fig. 14. Error pattern Syndrome Error recovery Error pattern Syndrome Error recovery P k , P n − k s R P k , P n − k s R XIIIIII (1 1 0 0 0 0)
X ZIIIIII (0 0 0 1 1 0)
ZIXIIIII (1 0 0 0 0 0)
I IZIIIII (0 0 0 1 0 1)
ZIIXIIII (0 1 0 0 0 0)
I IIZIIII (0 0 0 0 1 1)
ZIIIXIII (0 0 1 0 0 0)
I IIIZIII (0 0 0 1 1 1)
IIIIIXII (0 1 1 0 0 0)
X IIIIZII (0 0 0 1 0 0)
IIIIIIXI (1 0 1 0 0 0)
X IIIIIZI (0 0 0 0 1 0)
IIIIIIIX (1 1 1 0 0 0)
I IIIIIIZ (0 0 0 0 0 1) I December 23, 2020 DRAFT1 (a) QBER (b) Goodput
Fig. 15: The (a) QBER and the (b) goodput of our error-correction scheme proposed inFig. 14 compared to the existing QSC-based scheme for mitigating the effect of quantumdepolarizing channels. The insets are the QBER and the goodput in the logarithmic x axis.error patterns in the face of quantum depolarizing channels as follows: one error pattern isthe all-identity operator (weight = ); 21 error patterns having weight = 1; 42 error patternshaving 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 havingweight = 7. This distribution is identical to that of a QSC-based scheme utilizing the stabilizeroperators of the Steane code. Given that we have p x = p z = p y = p , the success probability ofthe proposed error-correction scheme of Fig. 14 in the face of quantum depolarizing channelsis given by p s = 1 − p p − p
27 + 6160 p − p
243 + 4824 p . (34)Let us now compare the QBER of the proposed scheme to those of the QSC-based scheme.Indeed for a fair comparison, we do not consider the recurrence-based scheme, since itis an error-detection scheme, not an error-correction one. For the QSC-based scheme, weutilized the stabilizer operators of Steane code [17], [35]. The resultant QBER is depicted inFig. 15(a), which is identical to the QBER of the QSC-based scheme utilizing the stabilizeroperators of the Steane code.Both our proposed scheme and the QSC-based scheme provide a constant level of yield,namely Y = and Y = , respectively, since they perform error-correction, instead oferror-detection. Specifically, the proposed error-correction scheme provides better yield thanthe QSC-based scheme, while exhibiting the same QBER. The yield of the proposed error-correction scheme is evaluated according to Def. 5, i.e. Y proposed = kn − k , while the yield of theQSC-based scheme is equal to its quantum coding rate given by Y stabilizer = kn . Since n, k > and n > k , it is clear that Y proposed > Y stabilizer . Since both the proposed and the QSC-based December 23, 2020 DRAFT2
NNNNNNN M Z HHH HHH R HHH H M Z M Z X Z M Z HHH HHH M Z R HHH M Z | i| Φ + i AB | i| i| i| i| i| i| i| i| i| i| i ClassicalClassical CommunicationCommunication | ψ i | ψ i Fig. 16: The quantum circuit for performing QSC-based scheme using the stabilizer operatorsof Steane code followed by quantum teleportation.schemes provide an identical QBER, while our proposed scheme offers higher yield, we caninfer 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 inFig 14. By contrast, the QSC-based scheme requires a total of 71 CNOT gates, namely 48CNOT 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 complexityof the quantum circuit required for performing QED followed by quantum teleportation,please refer to Fig. 16, which portrays the QSC-based scheme using the stabilizer operatorsof the Steane code followed by quantum teleportation.VI. D
ISCUSSION : A Q
UANTUM C OMPUTING P ERSPECTIVE
In the previous sections, we have shown the advantages of our proposal in quantumcommunication applications. In this section, we demonstrate that the proposed scheme canalso be adopted for quantum computing applications. In quantum computing applications,the quantum information is usually protected with the aid of noise-free auxiliary qubits,which may also take form of pre-shared entanglement [19]–[21], [23]–[25]. A primeexample is constituted by the family of entanglement-assisted quantum stabilizer codes (EA-QSCs). Compared to the conventional QSCs, which are unassisted by noise-free pre-sharedentanglement, EA-QSCs offer an error-correction capability improvement. This is reminiscentof having an additional error-free side channel between the transmitter and the receiver in theclassical domain. The argument that we can always have noise-free pre-shared entanglementrelies on the assumption that EPR pairs can be created abundantly and quantum entanglementdistillation can be applied to them. The concept of EA-QSCs is favourable in the realms of
December 23, 2020 DRAFT3
Noise−free pre−sharing M X V A N M Z M X RM Z V † B ClassicalCommunication | ψ i | ψ i| Φ + i| Φ + i (a) | ψ i N M Z M Z M X M X | Φ + i| Φ + i R | ψ i| ψ i| ψ i p | e ψ i | b ψ i (b) Fig. 17: (a) The quantum circuit of the proposed scheme utilizing two noise-free pre-sharedEPR pairs. (b) The rearranged quantum circuit of (a) for analysis.quantum computation, since the EA-QSCs can be readily amalgamated both with transversalimplementation of quantum gates [36], [37] as well as with magic state distillation [38] forcreating a universal set of fault-tolerant quantum gates. In the following, we propose anerror-correction scheme that outperforms the state-of-the-art EA-QSC.Any EA-QSC can be defined as C [ n, k, d, c ] , where n is the number of physical qubits, k is thenumber of logical qubits, d is the minimum distance of the code, and c is the number of noise-free pre-shared entangled qubits. The error-detection and error-correction capability of anyEA-QSC can be determined by its minimum distance d . An EA-QSC exhibiting a minimumdistance d is capable of detecting ( d − quantum errors or correcting t = (cid:98) ( d − / (cid:99) quantum errors. Based on the quantum Singleton bound of EA-QSCs [23], there exists aEA-QSC capable of correcting a single-qubit error ( d = 3) , which encodes one logicalqubit ( k = 1) into three physical qubits ( n = 3) with the aid of two noise-free pre-sharedentangled qubits ( c = 2) . This specific code is denoted by C [ n, k, d, c ] = C = [3 , , , . Inthe following, we will show that by utilizing two noise-free pre-shared EPR pairs, instead oferror-correction, we can achieve error elimination, implying that in this specific context, wecan always obtain a noise-free logical qubit.Let us now discuss our proposed scheme portrayed in Fig. 17(a), which is rearranged intoFig. 17(b) for facilitating our analysis. The quantum channel N ( · ) in Fig. 17(a) and 17(b)represents a quantum channel contaminating the logical qubit. According to Fig. 17(b), thequantum encoder V A is represented by the following unitary matrix: V A = ( | (cid:105)(cid:104) | ⊗ I ⊗ I ⊗ I ⊗ I + | (cid:105)(cid:104) | ⊗ X ⊗ I ⊗ I ⊗ I )( I ⊗ I ⊗ I ⊗ | (cid:105)(cid:104) | ⊗ I + X ⊗ I ⊗ I ⊗ | (cid:105)(cid:104) | ⊗ I ) , (35)while the quantum decoder V † B is described by the following unitary matrix: V B = ( I ⊗ I ⊗ I ⊗ I ⊗ | (cid:105)(cid:104) | + X ⊗ I ⊗ I ⊗ I ⊗ | (cid:105)(cid:104) | )( I ⊗ I ⊗ I ⊗ | (cid:105)(cid:104) | ⊗ I + X ⊗ I ⊗ I ⊗ | (cid:105)(cid:104) | ⊗ I ) . (36) December 23, 2020 DRAFT4
It can be readily verified that the reversible property is satisfied, i.e. we have V † B V A ( | ψ (cid:105) ⊗ | Φ + (cid:105) AB ) = | ψ (cid:105) ⊗ | Φ + (cid:105) AB .Upon denoting the density matrix of the initial global quantum state of | ψ (cid:105) ⊗ | Φ + (cid:105) AB by ρ ,the proposed scheme can be formulated with the aid of the following supermap: S ( V A , N , V † B , ρ ) = (cid:88) i ( V B N i V A ) ρ ( V B N i V A ) † , (37)where N i is the Kraus operator describing the quantum channel, while V A and V B representthe unitary matrices of Eq. (35) and (36). Therefore, Eq. (37) can be rewritten as: S ( ρ ) = (1 − p ) ρ ⊗ | Φ + (cid:105)(cid:104) Φ + | ⊗ | Φ + (cid:105)(cid:104) Φ + | + p XρX ) ⊗ | Ψ + (cid:105)(cid:104) Ψ + | ⊗ | Φ + (cid:105)(cid:104) Φ + | + p Y ρY ) ⊗ | Ψ + (cid:105)(cid:104) Ψ + | ⊗ | Φ − (cid:105)(cid:104) Φ − | + p ZρZ ) ⊗ | Φ + (cid:105)(cid:104) Φ + | ⊗ | Φ − (cid:105)(cid:104) Φ − | . (38)After the decoding operation, we perform the measurement of the EPR pairs. Observe thatwe can apply Z basis measurement to the first EPR pair and X basis measurement to thesecond EPR pair for determining the type of Pauli error experienced by the logical qubit | ψ (cid:105) .To elaborate a little further, we design a scheme so that requiring a joint measurement of theEPR pairs can be avoided to reduce the complexity of the quantum encoder and decoder. Wecombine the classical bits of A and B of Fig. 17(b) to determine the error recovery operator R .To expound a little further, let us denote the syndrome string as s = s Z s X , where s Z isobtained from the measurement of the first EPR pair in Z basis and s X is gleaned fromthe measurement of the second EPR pair in X basis. The error recovery operator associatedwith the syndrome value s = s Z s X is portrayed in Table V. Finally, it may be inferredfrom Eq. (38), that after the error recovery operator R of Fig. 17(a), we always obtain thelegitimate quantum state ρ of the logical qubit. Hence, we have demonstrated that with theaid of two noise-free EPR pairs, instead of correcting a single-qubit achievable by an EA-QSC, we can always recover a noise-free logical qubit. Observe that when we replace thequantum channel N ( · ) by realistic noisy quantum Pauli gates, we can modify the LUT ofTable V to benefit from the noise-free operation of the quantum Pauli gates. Remark.
This is a significant improvement compared to the existing EA-QSC C [ n, k, d, c ] = TABLE V: Syndrome values and associated error recovery R for the scheme in Fig. 17(a). s Z s X Error recovery R I X Y Z December 23, 2020 DRAFT5 C [3 , , , , which is only capable of correcting a single-qubit error by utilizing two noise-freeEPR pairs. Furthermore, the 5-qubit code C [ n, k, d ] = C [5 , , of [16] is also only capableof correcting a single-qubit error. Hence, our scheme represents an improvement also withrespect to [16]. VII. C ONCLUSIONS AND F UTURE R ESEARCH
In this treatise, we have conceived direct noiseless quantum communication using noisypre-shared EPR pairs. Conventionally, noiseless quantum communication tends to relyon the consecutive steps of QED followed by quantum teleportation. One of the salientbenefits that we can offer is the elimination of the long communication delay imposedby the aforementioned consecutive steps, despite relying on noisy pre-shared EPR pairs.Additionally, our proposed schemes offer better QBER than the recurrence-based schemes andprovide identical QBER to the QSC-based schemes. Moreover, compared to the QSC-basedschemes, our proposal attains better yield, despite requiring fewer CNOT gates. We have alsocompared our proposed scheme to EA-QSC, which requires noise-free pre-shared EPR pairs.EA-QSCs require joint eigenvalue measurements relying on all the qubits gleaned from theEPR pairs for performing error-correction. Despite relying only on the local measurementsof the EPR pairs and classical communications, we can always obtain a noise-free logicalqubit. In our future research, we are interested in finding a systematic way of constructingthe quantum encoder and decoder pair. In fact, we found that an arbitrary quantum encoderand decoder pair cannot always satisfy the reversible property of Eq. (8). Therefore, thesufficient and necessary conditions of generating the quantum encoder and decoder pairshould be found. Furthermore, since our proposed scheme performs identically to the QSC-based schemes, it remains to be shown whether a wider range of QSCs can be directlyembedded into our scheme.A
PPENDIX
A: P
ROOF OF P ROPOSITION S ( ρ ) = (cid:104) (1 − p ) ρ + p XρX + p Y ρY + p − p ) ZρZ (cid:105) ⊗ | Φ + (cid:105)(cid:104) Φ + | + (cid:104) p ρ p XρX p Y ρY p (1 − p ) ZρZ (cid:105) ⊗ | Φ − (cid:105)(cid:104) Φ − | + (cid:104) p ρ p XρX p Y ρY p (1 − p ) ZρZ (cid:105) ⊗ | Ψ − (cid:105)(cid:104) Ψ − | + (cid:104) p − p ) ρ + p − p ) XρX + p − p ) Y ρY + p ZρZ (cid:105) ⊗ | Ψ + (cid:105)(cid:104) Ψ + | , (39) December 23, 2020 DRAFT6 where ρ is the density matrix of the logical qubit and we assume that the quantum depolarizingchannels experienced by | ψ (cid:105) and | Φ + (cid:105) A exhibit an identical depolarizing probability p . Afterthe decoding, a measurement in the Z basis of the EPR pair shared between A and B isperformed. Every time we find a disagreement in the classical measurement results fromthe EPR pair ( s = s A ⊕ s B = 1) , the associated logical qubit is discarded, otherwise, itis retained. We note that the syndrome value of s = 0 is obtained if the EPR pair is inthe quantum state | Φ + (cid:105) or | Φ − (cid:105) , while the EPR pair in the state | Ψ + (cid:105) or | Ψ − (cid:105) gives us asyndrome value of s = 1 . Hence, the probability of retaining the logical qubit is equal tothe probability of obtaining the syndrome value s = 0 . Based on these considerations andby accounting for Eq. (39), we can evaluate the probability of obtaining the syndrome value s = 0 as p ( s = 0) = 1 − p + p , which is obtained from the following error operators P ∈ { II, IZ, XX, XY, Y X, Y Y, ZI, ZZ } . Then, based on this set of error operators, we candetermine the probability of obtaining the syndrome value of s = 0 and obtain the legitimatequantum state of the logical qubit ρ as p ( ρ ∩ ( s = 0)) = 1 − p + p , which is obtained fromthe following error operators P ∈ { II, ZZ } . Finally, the success probability of the schemepresented in Fig. 4 can be determined according to Def. 1 as follows: p s = p ( ρ | ( s = 0)) = 1 − p + p − p + p = 1 − p − O ( p ) , (40)which gives us an approximately linear performance improvement over the uncoded QBERas a function of p . By accounting for Def. 2, the yield is Y = p ( s = 0) and the proof follows.A PPENDIX
B: P
ROOF OF C OROLLARY S ( ρ ) = [(1 − p x ) ρ + p x XρX ] ⊗ | Φ + (cid:105)(cid:104) Φ + | +[ p x (1 − p x ) ρ + p x (1 − p x ) XρX ] ⊗ | Ψ + (cid:105)(cid:104) Ψ + | . Therefore, the probability of measuring s = 0 is given by p ( s = 0) = 1 − p x + 2 p x . Hence, the probability of obtaining the legitimatequantum state ρ while measuring s = 0 is given by p ( ρ ∩ s = 0) = 1 − p x + p x . The successprobability p s of arriving at the legitimate quantum state ρ given that the syndrome valueis s = 0 becomes p s = p ( ρ | s = 0) = − p x + p x − p x +2 p x , which gives us a quadratic performanceimprovement over uncoded QBER as a function of p . The yield can be directly determinedaccording to Def. 2 and the proof follows.A PPENDIX
C: P
ROOF OF C OROLLARY S ( ρ ) = [(1 − p z ) ρ + p z (1 − p z ) ZρZ ] ⊗| Φ + (cid:105)(cid:104) Φ + | + [ p z ρ + p z (1 − p z ) ZρZ ] ⊗ | Φ − (cid:105)(cid:104) Φ − | . Therefore, the probability of measuring December 23, 2020 DRAFT7 s = 0 is given by p ( s = 0) = 1 , since the measurement is carried out in the Z basis.Hence, the probability of measuring s = 0 and obtaining the legitimate quantum state ρ is p ( ρ ∩ s = 0) = 1 − p z + 2 p z . The yield can be directly determined according to Def. 2 andthe proof follows. A PPENDIX
D: P
ROOF OF P ROPOSITION V † B of Fig. 9, we can determine the probabilityof obtaining the syndrome value s Z = 0 from the first EPR pair as p ( s Z = 0) = 1 − p + p . If s Z = 1 , we discard the logical qubit. If s Z = 0 , the first-two quantum depolarizing channelsare reduced into a single depolarizing channel having Kraus operators: N = (cid:114) − p + p − p + p I , N = (cid:114) p − p + p X , N = (cid:114) p − p + p Y , and N = (cid:114) p − p − p + p Z . By applying the secondCNOT of the decoder V † B of Fig. 9, we can determine the probability of obtaining thesyndrome value s X = 0 from the second EPR pair as p ( s X = 0) = − p + p − p − p + p . Theprobability of arriving at the legitimate quantum state ρ and measuring the syndrome value s X = 0 is p ( ρ ∩ ( s X = 0)) = − p + p − p − p + p . By using Def. 1 and 2, the proof follows.A PPENDIX
E: P
ROOF OF P ROPOSITION X basis whilethe second one in the Z basis. Let us distinguish the components of the syndrome stringin Eq. (10) according to the basis used for the measurement. Specifically, let us denote thesyndrome component obtained when the second EPR pair is measured in the Z basis by s Z = s A ⊕ s B , while the syndrome component obtained when the first EPR pair is measuredin the X basis by s X = s A ⊕ s B . The overall syndrome string is s = s X s Z . Since no erroroperators exhibiting even numbers of X errors and even numbers of Z errors can be detected,which gives us the syndrome vector of s = 00 , the probability we retain the logical qubitsis equal to the sum of the probabilities of all these possible error patterns resulting in thesyndrome vector of s = 00 . After observing = 256 error patterns and by accounting forEq. (9), we have p ( s = 00) = 1 − p + 8 p − p + p , which is obtained from oneall-identity error pattern, 18 error patterns having a weight of two, 24 error patterns having aweight of three, and 21 error patterns having a weight of four, which generate the syndromevector s = 00 . However, from all of those error patterns, only four of the error patterns thatare actually associated with the legitimate quantum state ρ of the logical qubits, which are P ∈ { IIII, XXXX, Y Y Y Y, ZZZZ } . Hence, the probability of obtaining the legitimate December 23, 2020 DRAFT8 S ( ρ ) = (1 − p x ) ρ ⊗ ρ Φ + ⊗ ρ Φ + + p x (1 − p x ) ( IX ) ρ ( IX ) † ⊗ ρ Φ + ⊗ ρ Φ + + p x (1 − p x ) ( XI ) ρ ( XI ) † ⊗ ρ Φ + ⊗ ρ Φ + + p x (1 − p x ) ( XX ) ρ ( XX ) † ⊗ ρ Φ + ⊗ ρ Φ + + p x ρ ⊗ ρ Ψ + ⊗ ρ Φ + + p x (1 − p x ) ( IX ) ρ ( IX ) † ⊗ ρ Ψ + ⊗ ρ Φ + + p x (1 − p x ) ( XI ) ρ ( XI ) † ⊗ ρ Ψ + ⊗ ρ Φ + + p x (1 − p x ) ( XX ) ρ ( XX ) † ⊗ ρ Ψ + ⊗ ρ Ψ + + p x (1 − p x ) ( XX ) ρ ( XX ) † ⊗ ρ Ψ + ⊗ ρ Φ + + p x (1 − p x ) ρ ⊗ ρ Φ + ⊗ ρ Ψ + + p x (1 − p x ) ( IX ) ρ ( IX ) † ⊗ ρ Φ + ⊗ ρ Ψ + + p x (1 − p x ) ( XI ) ρ ( XI ) † ⊗ ρ Φ + ⊗ ρ Ψ + + p x (1 − p x )( XX ) ρ ( XX ) † ⊗ ρ Φ + ⊗ ρ Ψ + + p x (1 − p x ) ρ ⊗ ρ Ψ + ⊗ ρ Ψ + + p x (1 − p x )( IX ) ρ ( IX ) † ⊗ ρ Ψ + ⊗ ρ Ψ + + p x (1 − p x )( XI ) ρ ( XI ) † ⊗ ρ Ψ + ⊗ ρ Ψ + . (41)quantum state ρ while measuring s = 00 is p ( ρ ∩ ( s = 00)) = 1 − p + 6 p − p + p . Byusing Def. 1 and 2, the proof follows.A PPENDIX
F: P
ROOF OF C OROLLARY V A and V † B in Fig. 10 as well as on the Kraus operators of quantum bit-flip channel,we specify Eq. (9) as in Eq. (41) given at the top of this page, where ρ Φ + (cid:52) = | Φ + (cid:105) (cid:104) Φ + | and ρ Ψ + (cid:52) = | Ψ + (cid:105) (cid:104) Ψ + | . In our scheme proposed in Fig. 10, we measure the first EPR pair inthe X basis and the second EPR pair in the Z basis. For the sake of X error-detection, wedecide to discard the logical qubits if we measure ρ Ψ + for the second EPR pair, which canbe inferred from classical measurement results s Z = s A ⊕ s B = 1 . By contrast, we retainthe logical qubits if we measure ρ Φ + , which implies s Z = s A ⊕ s B = 0 . Therefore, fromEq. (41), the probability of getting the syndrome value of s Z = 0 from the second EPR pairis p ( s Z = 0) = 1 − p x + 12 p x − p x + 8 p x . It is important to observe that any error operatorexhibiting odd numbers of X errors can be detected and the logical qubits are discarded.By contrast, any error operator exhibiting even numbers of X errors cannot be detected andhence the logical qubits are retained. There are two possible error patterns that preserve thelegitimate encoded state of the logical qubits, i.e. P ∈ { IIII, XXXX } . Then, we have p ( ρ ∩ ( s = 0)) = 1 − p x + 6 p x − p x + 2 p x . By using Def. 1 and 2, the proof follows.A PPENDIX
G: P
ROOF OF C OROLLARY | Φ − (cid:105)(cid:104) Φ − | on the first EPR pair, i.e. when s X = s A ⊕ s B = 1 . By contrast, we retain the logicalqubits if we measure | Φ + (cid:105)(cid:104) Φ + | , i.e. when s X = s A ⊕ s B = 0 . Hence, we obtain p ( s X = 0) from the first EPR pair upon measurement is p ( s X = 0) = 1 − p z + 12 p z − p z + 8 p z .Similarly, any error operators exhibiting even numbers of Z errors cannot be detected andhence the logical qubits are retained. There are two possible error patterns that preserve the December 23, 2020 DRAFT9 legitimate encoded state of the logical qubits, i.e. P ∈ { IIII, ZZZZ } . Therefore, we obtain p ( ρ ∩ ( s X = 0)) = 1 − p z + 6 p z − p z + 2 p z . By using Def. 1 and 2, the proof follows.R EFERENCES [1] H. J. Kimble, “The quantum Internet,”
Nature , vol. 453, no. 7198, pp. 1023–1030, 2008.[2] M. Caleffi, A. S. Cacciapuoti, and G. Bianchi, “Quantum Internet: From communication to distributed computing!,”in
Proc. of the 5th ACM International Conference on Nanoscale Computing and Communication , pp. 1–4, 2018.[3] S. Wehner, D. Elkouss, and R. Hanson, “Quantum Internet: A vision for the road ahead,”
Science , vol. 362, no. 6412,2018.[4] C. Wang, A. Rahman, and R. Li, “Applications and use cases for the quantum Internet,” Internet Draft draft-wang-qirg-quantum-internet-use-cases-04, Internet Engineering Task Force, Mar. 2020. Work in Progress.[5] M. Razavi, “Multiple-access quantum key distribution networks,”
IEEE Transactions on Communications , vol. 60,no. 10, pp. 3071–3079, 2012.[6] D. Cuomo, M. Caleffi, and A. S. Cacciapuoti, “Towards a distributed quantum computing ecosystem,”
IET QuantumCommunication (Invited Paper) , vol. 1, no. 1, pp. 3–8, 2020.[7] M. Caleffi, D. Chandra, D. Cuomo, S. Hassanpour, and A. S. Cacciapuoti, “The rise of the quantum Internet,”
Computer , vol. 53, no. 6, pp. 67–72, 2020.[8] Z. Sun, L. Song, Q. Huang, L. Yin, G. Long, J. Lu, and L. Hanzo, “Toward practical quantum secure directcommunication: A quantum-memory-free protocol and code design,”
IEEE Transactions on Communications , vol. 68,no. 9, pp. 5778–5792, 2020.[9] A. S. Cacciapuoti, M. Caleffi, F. Tafuri, F. S. Cataliotti, S. Gherardini, and G. Bianchi, “Quantum Internet: Networkingchallenges in distributed quantum computing,”
IEEE Network , vol. 34, no. 1, pp. 137–143, 2019.[10] A. S. Cacciapuoti, M. Caleffi, R. Van Meter, and L. Hanzo, “When entanglement meets classical communications:Quantum teleportation for the quantum Internet,”
IEEE Transactions on Communications (Invited Paper) , vol. 68,no. 6, pp. 3808–3833, 2020.[11] M. A. Nielsen and I. L. Chuang,
Quantum computation and quantum information . Cambridge University Press, 2000.[12] G. Cariolaro and G. Pierobon, “Performance of quantum data transmission systems in the presence of thermal noise,”
IEEE Transactions on Communications , vol. 58, no. 2, pp. 623–630, 2010.[13] C. E. Shannon, “A mathematical theory of communication,”
The Bell System Technical Journal , vol. 27, no. 3,pp. 379–423, 1948.[14] P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,”
Phys. Rev. A , vol. 52, no. 4, 1995.[15] A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,”
Phys. Rev. A , vol. 54, no. 2, 1996.[16] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, “Perfect quantum error correcting code,”
Phys. Rev. Lett. , vol. 77,no. 1, 1996.[17] D. A. Lidar and T. A. Brun,
Quantum error correction . Cambridge University Press, 2013.[18] Z. Babar, D. Chandra, H. V. Nguyen, P. Botsinis, D. Alanis, S. X. Ng, and L. Hanzo, “Duality of quantum and classicalerror correction codes: Design principles and examples,”
IEEE Communications Surveys & Tutorials , vol. 21, no. 1,pp. 970–1010, 2018.[19] Y. Fujiwara, D. Clark, P. Vandendriessche, M. De Boeck, and V. D. Tonchev, “Entanglement-assisted quantum low-density parity-check codes,”
Phys. Rev. A , vol. 82, no. 4, 2010.[20] M. M. Wilde and J. M. Renes, “Quantum polar codes for arbitrary channels,” in
Proc. of the IEEE InternationalSymposium on Information Theory (ISIT) , pp. 334–338, IEEE, 2012.[21] M. M. Wilde, M.-H. Hsieh, and Z. Babar, “Entanglement-assisted quantum turbo codes,”
IEEE Transactions onInformation Theory , vol. 60, no. 2, pp. 1203–1222, 2013.
December 23, 2020 DRAFT0 [22] D. Chandra, Z. Babar, H. V. Nguyen, D. Alanis, P. Botsinis, S. X. Ng, and L. Hanzo, “Quantum coding bounds anda closed-form approximation of the minimum distance versus quantum coding rate,”
IEEE Access , vol. 5, pp. 11557–11581, 2017.[23] T. A. Brun, I. Devetak, and M.-H. Hsieh, “Correcting quantum errors with entanglement,”
Science , vol. 314, no. 5798,pp. 436–439, 2006.[24] I. Devetak, T. A. Brun, and M.-H. Hsieh, “Entanglement-assisted quantum error-correcting codes,” in
New Trends inMathematical Physics: Selected Contributions of the 15th International Congress on Mathematical Physics , pp. 161–172, Springer, 2009.[25] M. Grassl, “Entanglement-assisted quantum communication beating the quantum Singleton bound,” in
Proc. of the16th Asian Quantum Information Science Conference (AQIS) , pp. 20–21, 2016.[26] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by localoperations,”
Phys. Rev. A , vol. 53, no. 4, 1996.[27] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisyentanglement and faithful teleportation via noisy channels,”
Phys. Rev. Lett. , vol. 76, no. 5, p. 722, 1996.[28] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum errorcorrection,”
Phys. Rev. A , vol. 54, no. 5, 1996.[29] R. Matsumoto, “Conversion of a general quantum stabilizer code to an entanglement distillation protocol,”
Journal ofPhysics A: Mathematical and General , vol. 36, no. 29, 2003.[30] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantumstate via dual classical and Einstein-Podolsky-Rosen channels,”
Phys. Rev. Lett. , vol. 70, no. 13, 1993.[31] M. Caleffi and A. S. Cacciapuoti, “Quantum switch for the quantum Internet: Noiseless communications through noisychannels,”
IEEE Journal on Selected Areas in Communications , vol. 38, no. 3, pp. 575–588, 2020.[32] D. Chandra, Z. Babar, S. X. Ng, and L. Hanzo, “Near-hashing-bound multiple-rate quantum turbo short-block codes,”
IEEE Access , vol. 7, pp. 52712–52730, 2019.[33] D. Chandra, Z. Babar, H. V. Nguyen, D. Alanis, P. Botsinis, S. X. Ng, and L. Hanzo, “Quantum topological errorcorrection codes are capable of improving the performance of Clifford gates,”
IEEE Access , vol. 7, pp. 121501–121529,2019.[34] R. Cane, D. Chandra, S. X. Ng, and L. Hanzo, “Mitigation of decoherence-induced quantum-bit errors and quantum-gate errors using Steane’s code,”
IEEE Access , vol. 8, pp. 83693–83709, 2020.[35] A. Steane, “Multiple-particle interference and quantum error correction,”
Proc. of the Royal Society of London. SeriesA: Mathematical, Physical and Engineering Sciences , vol. 452, no. 1954, pp. 2551–2577, 1996.[36] J. Preskill, “Reliable quantum computers,”
Proc. of the Royal Society of London. Series A: Mathematical, Physicaland Engineering Sciences , vol. 454, no. 1969, pp. 385–410, 1998.[37] E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation,”
Nature ,vol. 549, no. 7671, pp. 172–179, 2017.[38] S. Bravyi and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas,”
Phys. Rev. A ,vol. 71, no. 2, 2005.,vol. 71, no. 2, 2005.