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arXiv:2312.08322 · Physical Review A

Concatenating quantum error-correcting codes with decoherence-free subspaces and vice versa

Nihar Ranjan Dash, Sanjoy Dutta, R. Srikanth, Subhashish Banerjee

Published Jun 10, 2024• 6 citations
arXivHTMLPDF

Abstract

Quantum error-correcting codes (QECCs) and decoherence-free subspace (DFS) codes provide active and passive means, respectively, to address certain types of errors that arise during quantum computation. The latter technique is suitable to correct correlated errors with certain symmetries and the former to correct independent errors. The concatenation of a QECC and a DFS code results in a degenerate code that splits into actively and passively correcting parts, with the degeneracy impacting either part, leading to degenerate errors as well as degenerate stabilizer operators. The concatenation of the two types of code can aid universal fault-tolerant quantum computation when a mix of correlated and independent errors is encountered. In particular, we show that for sufficiently strongly correlated errors, the concatenation with the DFS as the inner code provides better entanglement fidelity, whereas for sufficiently independent errors, the concatenation with the QECC as the inner code is preferable. As illustrative examples, we examine in detail the concatenation of a two-qubit DFS code and a three-qubit repetition code or five-qubit Knill-Laflamme code, under independent and correlated errors.

DOI
10.1103/physreva.109.062411

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Quantum error-correcting codes (QECCs) and decoherence-free subspace (DFS) codes provide active and passive means, respectively, to address certain types of errors that arise during quantum computation. The latter technique is suitable to correct correlated errors with certain symmetries and the former to correct independent errors. The concatenation of a QECC and a DFS code results in a degenerate code that splits into actively and passively correcting parts, with the degeneracy impacting either part, leading to degenerate errors as well as degenerate stabilizer operators. The concatenation of the two types of code can aid universal fault-tolerant quantum computation when a mix of correlated and independent errors is encountered. In particular, we show that for sufficiently strongly correlated errors, the concatenation with the DFS as the inner code provides better entanglement fidelity, whereas for sufficiently independent errors, the concatenation with the QECC as the inner code is preferable. As illustrative examples, we examine in detail the concatenation of a two-qubit DFS code and a three-qubit repetition code or five-qubit Knill-Laflamme code, under independent and correlated errors.
The exchange interaction between identical qubits in a quantum-information processor gives rise to unitary two-qubit errors. It is shown here that decoherence-free subspaces (DFSs) for collective decoherence undergo Pauli errors under exchange, which, however, do not take the decoherence-free states outside of the DFS. In order to protect DFSs against these errors it is sufficient to employ a recently proposed concatenated DFS quantum-error-correcting code scheme [D. A. Lidar, D. Bacon, and K.B. Whaley, Phys. Rev. Lett. 82, 4556 (1999)].
In this work, the efficient quantum error-correction protocol against the general independent noise is constructed with the three-qubit codes. The rules of concatenation are summarized according to the error-correcting capability of the codes. The codes not only play the role of correcting errors, but the role of polarizing the effective channel. For any independent noise, the most suitable error-correction protocol can be constructed based on the rules of concatenation. The most significant aspect of using the concatenated three-qubit codes is to realize quantum error-correction with the non-ideal gate operations, because the error-correction schemes of the three-qubit codes are simple and modular. For example, for the amplitude damping noise with the initial channel fidelity 0.9, the effective channel fidelity can reach 0.94731 (or 0.961634 with the ideal quantum gate operations) when using the 4 levels concatenated quantum error-correction with the three-qubit codes. The protocol with the three-qubit codes needs 81 qubits and 366 quantum gate operations when the accuracy rate is 0.9995. Meanwhile, when using the 3 levels concatenated quantum error-correction with the five-qubit code, the effective channel fidelity can only reach 0.922798 (or 0.975488 with the ideal quantum gate operations). The protocol with the five-qubit code needs 125 qubits and 1147 quantum gate operations when the accuracy rate is 0.9995. The physical resources costed is multiple of the physical resources costed by realizing the three-qubit quantum error-correction, the protocol has a higher fault tolerance threshold, and no increase in complexity. So, we believe it will be helpful for realizing quantum error-correction in the physical system.
Decoherence-free subspaces (DFSs) shield quantum information from errors induced by the interaction with an uncontrollable environment. Here we study a model of correlated errors forming an Abelian subgroup (stabilizer) of the Pauli group (the group of tensor products of Pauli matrices). Unlike previous studies of DFSs, this type of error does not involve any spatial symmetry assumptions on the system-environment interaction. We solve the problem of universal, fault-tolerant quantum computation on the associated class of DFSs. We do so by introducing a hybrid DFS quantum error-correcting-code approach, where errors that arise due to departure of the codewords from the DFS are corrected actively.
Concatenating quantum error correction codes scales error correction capability by driving logical error rates down double-exponentially across levels. However, the noise structure shifts under concatenation, making it hard to choose an optimal code sequence. We automate this choice by estimating the effective noise channel after each level and selecting the next code accordingly. In particular, we use learning-based methods to tailor small, non-additive encoders when the noise exhibits sufficient structure, then switch to standard codes once the noise is nearly uniform. In simulations, this level-wise adaptation achieves a target logical error rate with far fewer qubits than concatenating stabilizer codes alone--reducing qubit counts by up to two orders of magnitude for strongly structured noise. Therefore, this hybrid, learning-based strategy offers a promising tool for early fault-tolerant quantum computing.
We consider quantum error correction against correlated noise using simple and concatenated Calderbank-Shor-Steane codes as well as nnn-qubit repetition codes. We characterize the performance of various codes by means of the fidelity following the error correction of a single logical qubit in a quantum register. For concatenated codes we find a threshold in the single-qubit error rate below which the encoded qubit is perfectly protected. The threshold depends on the correlation strength of the noise and goes to zero for perfect correlation. Finally, we concatenate the traditional error correcting codes with a decoherence free subspace and evaluate the performance over the whole range from uncorrelated noise to perfectly correlated noise.
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.