In recent years, significant progress has been made in the research on errors in quantum deletion/insertion channels along with the development of error-correcting codes. However, the current channel model may not be applicable in specific scenarios, such as when using coherent-state qubits as funda-mental qubits. Here, we materialize the conventional error model of synchronization errors to accommodate coherent-state qubits and their superposition states. Additionally, we quantitatively assess the discrepancies between the conventional error model and our more realistic approach, underscoring the limitations of the former in practical applications.
In recent years, significant progress has been made in the research on errors in quantum deletion/insertion channels along with the development of error-correcting codes. However, the current channel model may not be applicable in specific scenarios, such as when using coherent-state qubits as funda-mental qubits. Here, we materialize the conventional error model of synchronization errors to accommodate coherent-state qubits and their superposition states. Additionally, we quantitatively assess the discrepancies between the conventional error model and our more realistic approach, underscoring the limitations of the former in practical applications.
Quantum synchronisation errors are a class of errors that change the number of qubits in a quantum system. The classical error correction of synchronisation errors has been well-studied, including an insertion-deletion equivalence more than a half-century ago, but little progress has been made towards the quantum counterpart since the birth of quantum error correction. We address the longstanding problem of a quantum insertion-deletion equivalence on permutation-invariant codes, detailing the conditions under which such codes are t-insertion error-correctable. We extend these conditions to quantum insertion-deletion errors, formulating a more restrictive set of conditions under which permutation-invariant codes are (t,s)-insdel error-correctable. For completeness, we extend our approach to qudit errors on qudit permutation-invariant codes. Our work resolves many of the outstanding questions regarding the quantum error correction of synchronisation errors.
We introduce a model of quantum insertion-deletion (insdel) channels. Insdel channels are meant to represent, for example, synchronization errors arising in data transmission. In the classical setting, they represent a strict generalization of the better-understood corruption error channels, and until recently, had mostly resisted effort toward a similar understanding as their corruption counterparts. They have received considerable attention in recent years. Very recently, Haeupler and Shahrasbi developed a framework, using what they call synchronisation strings, that allows one to turn insdel-type errors into corruption-type errors. These can then be handled by the use of standard error-correcting codes. We show that their framework can be extended to the quantum setting, providing a way to turn quantum insdel errors into quantum corruption errors, which can be handled with standard quantum error-correcting codes.
This paper discusses harnessing of pure states in the study of quantum deletion. Quantum deletion is one of the recently prominent aspects in the error model of quantum error correction codes, and pure states represent a class of quantum states widely used in the study of other error models. This study reveals that, under specific encoding and decoding operations, the possibility of deletion error-correction for pure states extends to deletion error-correction for mixed states. Through this paper, it is hoped that the understanding of quantum deletion errors deepens, contributing to the advancement of research in quantum deletion error-correction codes.
A universal, scalable quantum computer will require the use of quantum error correction in order to achieve fault tolerance. The assessment and comparison of error-correcting strategies is performed by classical simulation. However, due to the prohibitive exponential scaling of general quantum circuits, simulations are restrained to specific subsets of quantum operations. This creates a gap between accuracy and efficiency which is particularly problematic when modeling noise, because most realistic noise models are not efficiently simulable on a classical computer. We have introduced extensions to the Pauli channel, the traditional error channel employed to model noise in simulations of quantum circuits. These expanded error channels are still computationally tractable to simulate, but result in more accurate approximations to realistic error channels at the single qubit level. Using the Steane [[7,1,3]] code, we have also investigated the behavior of these expanded channels at the logical error-corrected level. We have found that it depends strongly on whether the error is incoherent or coherent. In general, the Pauli channel will be an excellent approximation to incoherent channels, but an unsatisfactory one for coherent channels, especially because it severely underestimates the magnitude of the error. Finally, we also studied the honesty and accuracy of the expanded channels at the logical level. Our results suggest that these measures can be employed to generate lower and upper bounds to a quantum code's threshold under the influence of a specific error channel.
We reexamine a non-Gaussian quantum error-correction code designed to protect optical coherent-state qubits against errors due to an amplitude-damping channel. We improve on a previous result [R. Wickert, N. K. Bernardes, and P. van Loock, Phys. Rev. A 81, 062344 (2010)] by providing a tighter upper bound on the performance attained when considering realistic assumptions, which constrain the operation of the gates employed in the scheme. The quantitative characterization is performed through measures of fidelity and concurrence, the latter obtained by employing the code as an entanglement distillation protocol. We find that, when running the code in fully deterministic error-correction mode, direct transmission can only be beaten for certain combinations of channel and input state parameters. In contrast, in error-detection mode, the usage of higher repetition encodings remains beneficial throughout, however, at the expense of diminishing success probabilities.
Quantum computing has demonstrated the potential to transform our world, with applications ranging from drug discovery to logistics. One of the largest problems standing in the way of practical quantum computers is the fragility of the quantum states that power them. This fragility inevitably leads to computational errors that require detection and correction. This review leverages peer-reviewed foundational papers published about quantum mechanics, quantum computing and error correction with emphasis on the most recently published research. It a ims to provide an overview of Quantum Error Correction (QEC) for those who have a grasp on the fundamentals of quantum computing and want to explore the gaps in knowledge, recent developments, and future research directions in QEC. Key challenges such as quantum noise, code distance, error threshold, and real-time decoders are presented. Core principles, and the current state of the art in QEC, explaining the strengths and weaknesses of each method, are discussed. The findings of published research in literature are showcased, encompassing the most recent developments. Noting the fascinating progress made in this time frame, it can be concluded that, with cautious optimism, we can expect the creation of QEC schemes capable of facilitating practical quantum computers in the future.