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 [[9,1,3]] code and the [[25,1,9]] 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.