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arXiv:2104.04365 · IEEE Journal on Selected Areas in Information Theory

Short Codes for Quantum Channels With One Prevalent Pauli Error Type

Marco Chiani, Lorenzo Valentini

Published Jul 29, 2020• 20 citations
arXivHTMLPDF

Abstract

One of the main problems in quantum information systems is the presence of errors due to noise, and for this reason quantum error-correcting codes (QECCs) play a key role. While most of the known codes are designed for correcting generic errors, i.e., errors represented by arbitrary combinations of Pauli X, Y and Z operators, in this paper we investigate the design of stabilizer QECC able to correct a given number egof generic Pauli errors, plus eZPauli errors of a specified type, e.g., Z errors. These codes can be of interest when the quantum channel is asymmetric in that some types of error occur more frequently than others. We first derive a generalized quantum Hamming bound for such codes, then propose a design methodology based on syndrome assignments. For example, we found a [[9, 1]] quantum error-correcting code able to correct up to one generic qubit error plus one Z error in arbitrary positions. This, according to the generalized quantum Hamming bound, is the shortest code with the specified error correction capability. Finally, we evaluate analytically the performance of the new codes over asymmetric channels.

DOI
10.1109/jsait.2020.3012827

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One of the main problems in quantum information systems is the presence of errors due to noise, and for this reason quantum error-correcting codes (QECCs) play a key role. While most of the known codes are designed for correcting generic errors, i.e., errors represented by arbitrary combinations of Pauli X, Y and Z operators, in this paper we investigate the design of stabilizer QECC able to correct a given number egof generic Pauli errors, plus eZPauli errors of a specified type, e.g., Z errors. These codes can be of interest when the quantum channel is asymmetric in that some types of error occur more frequently than others. We first derive a generalized quantum Hamming bound for such codes, then propose a design methodology based on syndrome assignments. For example, we found a [[9, 1]] quantum error-correcting code able to correct up to one generic qubit error plus one Z error in arbitrary positions. This, according to the generalized quantum Hamming bound, is the shortest code with the specified error correction capability. Finally, we evaluate analytically the performance of the new codes over asymmetric channels.
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The employment of quantum error correction codes (QECCs) within quantum computers potentially offers a reliability improvement for both quantum computation and communications tasks. However, the laws of quantum mechanics prevent us from directly invoking the mature family of classical error correction codes in the quantum domain. In order to circumvent the associated problems, the notion of quantum stabilizer codes (QSCs) was proposed in conjunction with syndrome-based decoding in the quantum regime. However, most of the powerful QSC schemes require long codewords for achieving a high performance, which potentially imposes additional challenges for implementation concerning their implementation, since their decoding may require longer than the quantum circuit's coherence time. Hence at the time of writing, QSCs exhibiting short to moderate codeword lengths are preferable. We commence by describing the pivotal problem encountered by classical error-correction codes, which also emerges when designing the QSCs, namely the intrinsic trade-off between the minimum distance versus coding rate. The complete formulation of this particular trade-off does not exist, but several lower and upper bounds can be found in the literature. It has been shown that a substantial gap can be observed between the upper and lower bound of the minimum distance, given the codeword length and the quantum coding rate. Hence, we propose an appealingly simple and invertible analytical approximation, for characterizing the trade-off between the quantum coding rate and the minimum distance of QSCs as well as their corresponding quantum bit error rate (QBER) performance upper-bound. For example, for a half-rate QSC having a codeword length of n = 128, the minimum distance is bounded by 11 < d < 22, while our approximation yields a minimum distance of d = 17 for the above-mentioned code. Next, we link this parametric study of the minimum distance versus quantum coding rate to the popular QSCs, namely to the family of quantum topological error correction codes (QTECCs). In order to construct the classical-to-quantum isomorphism, we conceive and investigate the family of classical topological error correction codes (TECCs), assuming that the bits of a codeword can be arranged in a lattice structure. We then present the classical-to-quantum isomorphism to pave the way for constructing their dual pairs in the quantum domain, which are the QTECCs. Finally, we characterize the performance of QTECCs in the face of the quantum depolarizing channel in terms of both their QBER and fidelity. Specifically, we demonstrate that for quantum coding rate rQ ~ 0, the threshold probability of the QBER below which the colour, rotated-surface, surface, and toric codes become capable of improving the uncoded QBER are given by 1.8 ×10-2, 1.3 ×10-2, 6.3 ×10-2 and 6.8 ×10-2, respectively. Furthermore, we also demonstrate that we can achieve beneficial fidelity improvements above the minimum fidelity of 0.94, 0.97 and 0.99 by employing the rQ = 1/7 colour code, the rQ = 1/9 rotated-surface code, and the rQ = 1/13 surface code, respectively. However, QSCs require additional quantum gates for their employment. Incorporating more quantum gates for performing error correction potentially introduces further sources of quantum decoherence into quantum computers. In this scenario, the primary challenge is to find the sufficient condition required by each of the quantum gates for beneficially employing QECCs in order to yield reliability improvements, given that the quantum gates utilized by the QECCs also introduce quantum decoherence. In this treatise, we approach this problem by firstly presenting the general framework of protecting quantum gates by the amalgamation of the transversal configuration of quantum gates and QSCs, which can be viewed as syndrome-based QECCs. Secondly, we provide examples of the advocated framework by invoking QTECCs for protecting both transversal Hadamard gates and controlled (CNOT) gates. Both our simulation and analytical results explicitly show that by utilizing QTECCs, the fidelity of the quantum gates can be beneficially improved, provided that quantum gates satisfying a certain minimum depolarization fidelity threshold (Fth) are available. For instance, for protecting transversal Hadamard gates, the minimum fidelity values required for each of the gates in order to attain fidelity improvements are 99.74%, 99.73%, 99.87%, and 99.86%, when they are protected by colour, rotated-surface, surface, and toric codes, respectively. Unfortunately, these specific Fth values can only be obtained for a very large number of physical qubits (n → ∞), when the quantum coding rate of the QTECCs approaches zero (rQ → 0). Finally, in order to conceive QSCs exhibiting a high quantum coding rate, we modify the construction of QTECCs for conceiving a low-complexity concatenated quantum turbocode (QTC). The above-mentioned high quantum coding rate is obtained by combining the quantum-domain version of short-block codes (SBCs) also known as single parity-check (SPC) codes as the outer codes and quantum unity-rate codes (QURCs) as the inner codes. Despite its design simplicity, the proposed QTC yields a near-hashing-bound error correction performance. For instance, compared to the best half-rate QTC known in the literature, namely the quantum irregular convolutional codes (QIrCCs) combined with the QURC scheme, which operates at the distance of D = 0.037 from the quantum hashing bound, our novel QSBC-QURC scheme can operate at the lower distance of D = 0:029. It is also worth mentioning that this is the first instantiation of QTCs capable of adjusting the quantum encoders according to the quantum coding rate required for mitigating the Pauli errors imposed by the time-variant depolarizing probabilities of the quantum channel.
Quantum technology offers transformative potential for future applications, including secure communications and high-speed computing. However, quantum systems are highly susceptible to decoherence caused by environment interactions, making error correction essential for reliable operation. This paper proposes a new 7-qubit quantum error correction code constructed from classical Hamming codes through a distinct derivation of the parity matrix. Unlike the well-known Steane code, our scheme employs a different parity matrix within the stabilizer framework, resulting in a novel code structure. We detail the step-by-step construction of codewords, syndrome extraction using stabilizer measurements, and the corresponding quantum circuits. Performance evaluation is conducted through simulations under depolarizing channels across error probabilities from to 0, using Quantum Word Error Rate (QWER) as the metric. Simulation results confirm that the proposed stabilizers can uniquely identify all single-qubit error patterns of Pauli X, Y, and Z. Moreover, the scheme achieves a significant error reduction of approximately 28 times (about 96%) compared to uncoded transmission. These findings demonstrate that deriving quantum codes from modified classical matrices offers an effective strategy for enhancing the reliability of future quantum communication systems.
We discuss a method to adapt the codeword stabilized (CWS) quantum code framework to the problem of finding asymmetric quantum codes. We focus on the corresponding Pauli error models for amplitude damping noise and phase damping noise. In particular, we look at codes for Pauli error models that correct one or two amplitude damping errors. Applying local Clifford operations on graph states, we are able to exhaustively search for all possible codes up to length 999. With a similar method, we also look at codes for the Pauli error model that detect a single amplitude error and detect multiple phase damping errors. Many new codes with good parameters are found, including nonadditive codes and degenerate codes.
The error threshold of a one-parameter family of quantum channels is defined as the largest noise level such that the quantum capacity of the channel remains positive. This in turn guarantees the existence of a quantum error correction code for noise modeled by that channel. Discretizing the single-qubit errors leads to the important family of Pauli quantum channels; curiously, multipartite entangled states can increase the threshold of these channels beyond the so-called hashing bound, an effect termed superadditivity of coherent information. In this work, we divide the simplex of Pauli channels into one-parameter families and compute numerical lower bounds on their error thresholds. We find substantial increases of error thresholds relative to the hashing bound for large regions in the Pauli simplex corresponding to biased noise, which is a realistic noise model in promising quantum computing architectures. The error thresholds are computed on the family of graph states, a special type of stabilizer state. In order to determine the coherent information of a graph state, we devise an algorithm that exploits the symmetries of the underlying graph, resulting in a substantial computational speed-up. This algorithm uses tools from computational group theory and allows us to consider symmetric graph states on a large number of vertices. Our algorithm works particularly well for repetition codes and concatenated repetition codes (or cat codes), for which our results provide the first comprehensive study of superadditivity for arbitrary Pauli channels. In addition, we identify a novel family of quantum codes based on tree graphs. The error thresholds of these tree graph states outperform repetition and cat codes in large regions of the Pauli simplex, and hence form a new code family with desirable error correction properties.
Entanglement-assisted quantum error correction codes (EAQECCs) play an important role in quantum communications with noise. Such a scheme can use arbitrary classical linear code to transmit qubits over noisy quantum channels by consuming some ebits between the sender (Alice) and the receiver (Bob). It is usually assumed that the preshared ebits of Bob are error free. However, noise on these ebits is unavoidable in many cases. In this work, we evaluate the performance of EAQECCs with noisy ebits over asymmetric quantum channels and quantum channels with memory by computing the exact entanglement fidelity of several EAQECCs. We consider asymmetric errors in both qubits and ebits and show that the performance of EAQECCs in entanglement fidelity gets improved for qubits and ebits over asymmetric channels. In quantum memory channels, we compute the entanglement fidelity of several EAQECCs over Markovian quantum memory channels and show that the performance of EAQECCs is lowered down by the channel memory. Furthermore, we show that the performance of EAQECCs is diverse when the error probabilities of qubits and ebits are different. In both asymmetric and memory quantum channels, we show that the performance of EAQECCs is improved largely when the error probability of ebits is reasonably smaller than that of qubits.