Entanglement-assisted concatenated quantum codes.

Entanglement-assisted concatenated quantum codes (EACQCs), constructed by concatenating two quantum codes, are proposed. These EACQCs show significant advantages over standard concatenated quantum codes (CQCs). First, we prove that, unlike standard CQCs, EACQCs can beat the nondegenerate Hamming bound for entanglement-assisted quantum error-correction codes (EAQECCs). Second, we construct families of EACQCs with parameters better than the best-known standard quantum error-correction codes (QECCs) and EAQECCs. Moreover, these EACQCs require very few Einstein-Podolsky-Rosen (EPR) pairs to begin with. Finally, it is shown that EACQCs make entanglement-assisted quantum communication possible, even if the ebits are noisy. Furthermore, EACQCs can outperform CQCs in entanglement fidelity over depolarizing channels if the ebits are less noisy than the qubits. We show that the error-probability threshold of EACQCs is larger than that of CQCs when the error rate of ebits is sufficiently lower than that of qubits. Specifically, we derive a high threshold of 47% when the error probability of the preshared entanglement is 1% to that of qubits.

Quantum error-correction codes (QECCs) are necessary to realize quantum communications and to make fault-tolerant quantum computers (1, 2). The stabilizer formalism provides a useful way to construct QECCs from classical codes, but certain orthogonality constraints are required (3). The entanglement-assisted (EA) QECC (EAQECC) (4–6) generalizes the stabilizer code. By presharing some entangled states between the sender (Alice) and the receiver (Bob), EAQECCs can be constructed from any classical linear codes without the orthogonality constraints. Therefore, the construction could be greatly simplified. As an important physical resource, entanglement can boost the classical information capacity of quantum channels (7–12). Recently, it has been shown that EAQECCs can violate the nondegenerate quantum Hamming bound (13) or the quantum Singleton bound (14).

Compared to standard QECCs, EAQECCs must establish some amount of entanglement before transmission. This preshared entanglement is the price to be paid for enhanced communication capability. In a sense, we need to consider the net transmission of EAQECCs—i.e., the number of qubits transmitted minus that of ebits preshared. Further, it is difficult to preserve too many noiseless ebits in EAQECCs at present. Thus, we have to use as few ebits as possible to conduct the communication—e.g., one or two ebits are preferable (15–18). In addition, EAQECCs with positive net transmission and little entanglement can lead to catalytic quantum codes (4, 6), which are applicable to fault-tolerant quantum computation (FTQC). In ref. 4, a table of best-known EAQECCs of length up to 10 was established through computer search or algebraic methods. Several EAQECCs in ref. 4 have larger minimum distances than the best-known standard QECCs of the same length and net transmission. However, for larger code lengths, the efficient construction of EAQECCs with better parameters than standard QECCs is still unknown.

In classical coding theory, concatenated codes (CCs), originally proposed by Forney in the 1960s (19), provide a useful way of constructing long codes from short ones. CCs can achieve very large coding gains with reasonable encoding and decoding complexity (20). Moreover, CCs can have large minimum distances since the distances of the component codes are multiplied. As a result, CCs have been widely used in many digital communication systems—e.g., the NASA standard for the Voyager program (21) and the compact disc (20). Similarly, in QECCs, the concatenated quantum codes (CQCs), introduced by Knill and Laflamme in 1996 (22), are also effective for constructing good quantum codes. In particular, it has been shown that CQCs are of great importance in realizing FTQC (23–25).

Moreover, there exists a specific phenomenon in QECCs, called “error degeneracy,” which distinguishes quantum codes from classical ones in essence. It is widely believed that degenerate codes can correct more quantum errors than nondegenerate ones. Indeed, there are some open problems concerning whether degenerate codes can violate the nondegenerate quantum Hamming bound (26) or can improve the quantum-channel capacity (27, 28). Many CQCs have been shown to be degenerate, even if the component codes are nondegenerate—e.g., Shor’s code and the CQC (23, 29). If we introduce extra entanglement to CQCs, it is possible to improve the error-degeneracy performance of CQCs.

In this article, we generalize the idea of concatenation to EAQECCs and propose EACQCs. We show that EACQCs can beat the nondegenerate quantum Hamming bound, while standard CQCs cannot. Several families of degenerate EACQCs that can surpass the nondegenerate Hamming bound for EAQECCs are constructed. The same conclusion could be reached for asymmetric error models, in which the phase-flip errors (Z errors) happen more frequently than the bit-flip errors (X errors) (30, 31). Furthermore, we derive a number of EACQCs with better parameters than the best-known QECCs and EAQECCs. In particular, we see that many EACQCs have positive net transmission, and each of them consumes only one or two ebits. Thus, they give rise to catalytic EACQCs with little entanglement and better parameters than the best-known QECCs. Further, we show that the EACQC scheme makes EA quantum communication possible, even if the ebits are noisy. We compute the entanglement fidelity (EF) of the [[15,1,9;10]] EACQC by using Bowen’s [[3,1,3;2]] EAQECC (32) or the [[3,1,3;2]] EA repetition code (4, 6) as the inner code. The outer code is the standard [[5,1,3]] stabilizer code. We show that the [[15,1,9;10]] EACQC performs much better than the [[25,1,9]] CQC over depolarizing channels if the ebits suffer a lower error rate than the qubits. Moreover, we compute the error-probability threshold of EACQCs, and we show that EACQCs have much higher thresholds than CQCs when the error rate of ebits is sufficiently lower than that of qubits.

EA Stabilizer Formalism

Let ( is an integer) and denote by GF(q) the extension field of the binary field GF(2). Let be the field of complex numbers, and let be the q -dimensional Hilbert space, where n is a positive integer. Define two error operators on by and , where , and “tr” denotes the trace operator from GF(q) to GF(2). For a vector , denote by and . Let and let be the group generated by Ξ. For the operator , the weight of e is defined by . The definition of quantum stabilizer codes is given below.

A stabilizer code Q is a qdimensional () subspace of V such that where T is a subgroup of . has minimum distance d if it can detect all errors of weight up to d –1. Further, Q is called nondegenerate if every stabilizer in T has weight larger than or equal to d; otherwise, it is called degenerate.

A CQC is derived from an inner code and an outer code. In general, the component codes of CQCs can be chosen as stabilizer codes or nonstabilizer codes. In this article, it suffices to consider only the case of stabilizer codes. Let the inner and outer codes be and , respectively. Then, we can derive a CQC (33) with parameters

An EAQECC with parameters can encode k qudits into n qudits by consuming c pairs of maximally entangled states between Alice and Bob. It should be noted that EAQECCs can be constructed from arbitrary classical linear codes directly. The Calderbank–Shor–Steane framework (3, 34) provides a useful way to construct both QECCs and EAQECCs from classical linear codes.

Let and be two linear codes over GF(q). Denote the parity-check matrices of C1 and C2 by H1 and H2, respectively. There exists an EAQECC with parameters where , and is the transpose of H2.

EAQECCs can also be constructed by using the Hermitian construction (3, 4, 35) as follows.

Let be a linear code over . Denote the parity-check matrix of C by H. There exists an EAQECC with parameters where , and is the conjugate transpose of H over .

Results

We organize the main results of our study in the following order. First, we present the construction of EACQCs from two component quantum codes. Second, we construct several families of EACQCs violating the nondegenerate Hamming bound for EAQECCs. Third, we derive a number of EACQCs with better parameters than the best-known QECCs and EAQECCs. Finally, we show that EACQCs can correct errors in the ebits. It is shown that EACQCs can outperform CQCs in EF and have higher error-probability thresholds than CQCs.

EACQCs.

We generalize CQCs to EACQCs by concatenating two quantum codes, which can be chosen as either standard QECCs or EAQECCs. In this article, sometimes we represent an QECC as an EAQECC so that we can unify the representation of QECCs and EAQECCs. Let the inner code be , which requires c1 ebits. Denote by the net transmission of Q. Let the outer code be , which can either be binary or nonbinary depending on . Q uses c2 edits, or, equivalently, ebits. Denote by the net transmission of Q. Notice that, for classical linear codes and quantum codes over the binary field GF(2), we usually neglect the index in the code parameters if there is no ambiguity.

We prove the following result about EACQCs.

Let be the inner code, and let be the outer code. There exists an EACQC with parameterswhere is the number of ebits. The net transmission is .

Based on the idea of code concatenation, we simply concatenate the inner code Q with the outer code Q to derive the EACQC (19, 22, 33). First, we encode the information state by using the outer code Q, i.e.,where there are Einstein–Podolsky–Rosen (EPR) pairs, , preshared between Alice and Bob during the outer encoding, and is the identity operator on Bob’s halves of ebits during the outer encoding. The outer encoding operation U is applied to the qubits on Alice’s side.

Suppose that we can represent bywhere should satisfy the normalization condition. We separate each basis state in into n2 subblocks, i.e., for . For each subblock , we do the inner encoding as follows:where are EPR pairs preshared between Alice and Bob during each inner encoding. The encoding operation is applied to the qubits in Alice’s side, while Bob’s halves of ebits do not need to be encoded during each inner encoding. It is easy to see that the number of ebits used during the whole inner encoding is . The encoding process of EACQCs is given in Fig. 1.

Fig. 1.

The encoding circuit of EACQCs. The information state is first encoded with the outer encoder U by presharing EPR pairs between Alice and Bob. For the output of U, each subblock is encoded with the inner encoder U by presharing c1 EPR pairs between Alice and Bob.

The numbers of ebits used during the outer and the inner encoding are equal to and , respectively. Therefore, the total number of ebits is equal to . It is easy to see that the dimension of the EACQC is equal to . Similar to the principle of code concatenation in refs. 19, 22, and 33, the minimum distance of is at least . Therefore, we can obtain an EAQECC with parameters □

It is easy to see that if the inner and outer codes are both standard QECCs, then the EACQC is a standard CQC. Moreover, we can use different inner codes in EACQCs. Let () be n2 inner codes. For simplicity, we let , and let . Let the outer code be . Then, we can derive an EACQC with parameterswhere . The net transmission is .

EACQCs Beating the Nondegenerate Quantum Hamming Bound.

First, let us review the nondegenerate Hamming bound for EAQECCs (36).

For a binary nondegenerate EAQECC, it must satisfy

Taking the limit as , this yields the asymptotic bound on the rate k/n:where , and is the binary entropy function.

To the best of our knowledge, no degenerate CQCs have been discovered that violate the nondegenerate quantum Hamming bound (36). However, the situation is quite different in the EA case. We can easily construct several families of degenerate EACQCs that violate the Hamming bound in Lemma 3. We summarize these EACQCs as follows.

There exist the following four families of EACQCs with parameters

① where is odd.

② where is even.

③ where is odd.

④ where is even.

EACQCs in ①−④ can beat the nondegenerate quantum Hamming bound for EAQECCs.

The proof is given in .□

We give an explicit example to illustrate the construction of EACQCs. Let be the inner code, and let be the outer code in ref. 4. Then, we can derive an EACQC with parameters by Theorem 1. This code can beat the nondegenerate Hamming bound for EAQECCs in Eq. . Notice that and are both nondegenerate codes (3, 4), while is degenerate. Also notice that Q and Q cannot beat the nondegenerate Hamming bound in Eq. , but their EACQC can do so. If we encode one of the qubits of the outer encoding by using the EAQECC, then we derive a EACQC. This code can also beat the nondegenerate Hamming bound for EAQECCs.

For asymmetric channel models, we present a construction of EACQCs that can beat the nondegenerate Hamming bound for asymmetric EAQECCs. Let d and d be two positive integers. From ref. 37, an asymmetric EAQECC can detect any X error of weight up to and any Z error of weight up to simultaneously. The number of edits is c. One can further obtain nondegenerate Hamming bounds for asymmetric EAQECCs (36, 37).

A binary nondegenerate asymmetric EAQECC must satisfy

Let be a binary asymmetric EAQECC derived from the repetition code, where is an integer. We use Q as the inner code. Let be the outer code, where for even , or d2 = n2 for odd . We concatenate Q with Q according to Fig. 1. Then, we have the following result about asymmetric EACQCs.

There exists a family of asymmetric EACQCs with parameterswhere is an integer, for even , or for odd .

For any integer and any odd in Corollary 1 can beat the nondegenerate Hamming bound for asymmetric EAQECCs in Lemma 4. For any integer and any even can also beat the nondegenerate Hamming bound. Let and . We can derive an asymmetric EACQC with parameters .

EACQCs Beating Existing QECCs and EAQECCs.

Similar to classical coding theory, constructing quantum codes with parameters better than the best-known results is a central topic in quantum coding theory. It is even more attractive since degenerate quantum codes have significant potential to outperform any nondegenerate quantum code. Indeed, a number of the best-known QECCs in ref. 29 have been shown to be degenerate.

As argued in ref. 4, we say that an EAQECC is better than a QECC if the net transmission is larger than k. Ref. 29 collects a list of classical linear codes and QECCs with the best parameters currently known. According to the construction of EAQECCs in Lemma 2, the quaternary codes in ref. 29 correspond to the best-known nondegenerate EAQECCs. In general, it is not difficult to construct nondegenerate EAQECCs with positive net transmissions better than the best-known QECCs based on ref. 29. However, how to construct degenerate EAQECCs with positive net transmissions that can beat the best-known nondegenerate EAQECCs is largely unknown. This addresses the important question of whether degeneracy can improve on the coding limit in EAQECCs.

We give two explicit constructions to show that EACQCs can beat the best-known QECCs and EAQECCs. According to refs. 38 and 39, there exists a cyclic maximum-distance-separable (MDS) code with parameters . From Lemma 2, we can derive an EA quantum MDS (EAQMDS) code with parameters . Let be the inner code, and let be the outer code. Then, we can derive an EACQC with parameters . Compared with the best-known QECC in ref. 29, the EACQC has a larger minimum distance, while maintaining the same length and net transmission. also has a larger minimum distance than the best-known nondegenerate EAQECC from ref. 29 of the same length and net transmission.

Let be an EAQMDS code constructed from a cyclic MDS code [65,33,33] in ref. 38, and let . Then, we can derive an EACQC with parameters by using and as the outer and inner codes, respectively. This EACQC is better than the asymptotic Gilbert–Varshamov bound for EAQECCs in ref. 36. In , we list more constructions of EACQCs with parameters better than the best-known QECCs and EAQECCs.

In practice, we prefer to use as few ebits as possible to do the EA communication since storing a large number of noiseless ebits is quite difficult. Let be the inner code, and let be the outer code; then, we can derive a [[15,2,6;1]] EACQC. This code has larger minimum distance than the best-known standard [[15,1,5]] QECC in ref. 29. By using the MAGMA software (40), we know that there exists a nondegenerate [[15, 8,6;7]] EAQECC. This code has the same minimum distance and net transmission as the EACQC. However, the EACQC consumes only one ebit and, thus, is more practical. In , we list a number of EACQCs with parameters better than the best-known QECCs and EAQECCs, and each EACQC consumes only one ebit.

In ref. 41, several families of q-ary EAQMDS codes with distances larger than q + 1 and consuming very few edits were constructed. We use EAQMDS codes in ref. 41 as the outer codes to construct EACQCs that consume very few ebits. We give an example to illustrate the construction. Let be the inner code, and let a EAQMDS code in ref. 41 be the outer code. Then, we can derive an EACQC with parameters . This code has a larger minimum distance than the best-known [[68,6,14]] QECC in ref. 29 of the same length and net transmission. It also has a larger minimum distance than the best-known nondegenerate EAQECC in ref. 29 of the same length and net transmission. In , we list a number of EACQCs with better parameters than the best-known QECCs and EAQECCs in ref. 29, and each code consumes only a few ebits.

Thresholds of EACQCs with Noisy Ebits.

In this section, we evaluate the performance of EACQCs with noisy ebits. We compute the EF and the error-probability threshold of EACQCs and compare them to standard CQCs. For a quantum channel, the use of a QECC should improve the EF when the error probability is below a specific value, which we call the “threshold.” In practical applications, QECCs with sufficiently high thresholds are needed. We will show that EACQCs can outperform CQCs in EF if the ebits are less noisy than the qubits. Further, we will show that the threshold of EACQCs is much higher than that of CQCs when the error probability of ebits is sufficiently lower than that of qubits.

During the process of EA quantum communication, the preshared ebits of Bob need to be stored faultlessly, and EAQECCs can only correct errors on the transmitted qubits. However, noise in Bob’s ebits may be inevitable in practical applications (6, 42), and maintaining a large number of noiseless ebits is extremely difficult. In this section, we use EACQCs to correct errors in ebits. In the EACQC scheme, suppose that we use an EAQECC Q as the inner code and use a standard stabilizer code Q as the outer code. We show that the outer code Q cannot only correct errors on the physical qubits, but also can correct errors on the ebits. We construct two EACQCs and show that they can outperform CQCs in EF when the error probability of ebits is lower than that of qubits. We construct a EACQC by using the stabilizer code as the outer code and Bowen’s EAQECC (32) as the inner code. Alternately, we can use the stabilizer code as the outer code and the EA repetition code as the inner code to construct another EACQC with the same parameters. Recall that the standard CQC is the concatenation of the stabilizer code with itself. It is known that Bowen’s EAQECC is equivalent to the stabilizer code, and they have the same stabilizers. Thus, the EACQC is equivalent to the CQC. Then, the EACQC has the same error-correction ability as the CQC. Nevertheless, we show that EACQCs can outperform CQCs in EF if the error probability of ebits is lower than that of qubits.

The detailed EF computation of the two EACQCs and the CQC was put in . The EFs of the two EACQCs and the CQC are plotted in Fig. 2. We compare the EF of EACQCs with that of the CQC. If p = p, the EF of the EACQC is equal to that of the CQC. When , the EF of the and the EACQCs can outperform that of the [[25,1,9]] CQC (Fig. 2). As p becomes even lower—e.g., —the EF of and performs much better than that of the CQC (Fig. 2 ). Moreover, performs better than when p = p (Fig. 2). While performs much better than and the [[25,1,9]] CQC (Fig. 2 ).

Fig. 2.

The EF of EACQCs and CQCs for p = p (A), (B), (C), and (D).

We compare the error-probability threshold of the two EACQCs with that of the CQC. For the [[5,1,3]] stabilizer code and the CQC, the thresholds are p > 0.09 and p > 0.18, respectively. Thus, the CQC scheme can improve the error-probability threshold. For the EACQCs, when , the thresholds of and are p > 0.25 and p > 0.14, respectively. While p becomes sufficiently lower—e.g., —the thresholds of and are p > 0.41 and p > 0.47, respectively. Therefore, the EACQC scheme can greatly improve the error-probability threshold when the error probability of ebits is much lower than that of qubits.

Discussion

In this article, we have proposed the construction of EACQCs by concatenating an inner code with an outer code. We not only have generalized the idea of concatenation to EAQECCs, but also have shown that EACQCs can outperform many existent results. We have further shown that EACQCs can beat the nondegenerate Hamming bound for EAQECCs, while standard CQCs cannot do so. We have derived many EACQCs with larger minimum distances than the best-known QECCs and EAQECCs in ref. 29 of the same length and net transmission. In addition, we have constructed several catalytic EACQCs with little entanglement and better parameters than the best-known QECCs and EAQECCs. We have also constructed a family of asymmetric EACQCs that can beat the nondegenerate Hamming bound for asymmetric EAQECCs. Finally, we have computed the EF of two EACQCs and compared them with the CQC. We have shown that EACQCs can outperform CQCs in EF when the ebits are less noisy than qubits. In particular, we have shown that EACQCs have much higher error thresholds than CQCs when the error probability of ebits is sufficiently lower than that of qubits. These properties of EACQCs make them very competitive with standard CQCs for both quantum communication and FTQC.