We examine the efficiency of pure, nondegenerate quantum-error correction-codes for Pauli channels. Specifically, we investigate if correction of multiple errors in a block is more efficient than using a code that only corrects one error per block. Block coding with multiple-error correction cannot increase the efficiency when the qubit error-probability is below a certain value and the code size fixed. More surprisingly, existing multiple-error correction codes with a code length equal or less than 256 qubits have lower efficiency than the optimal single-error correcting codes for any value of the qubit error-probability. We also investigate how efficient various proposed nondegenerate single-error correcting codes are compared to the limit set by the code redundancy and by the necessary conditions for hypothetically existing nondegenerate codes. We find that existing codes are close to optimal.
We examine the efficiency of pure, nondegenerate quantum-error correction-codes for Pauli channels. Specifically, we investigate if correction of multiple errors in a block is more efficient than using a code that only corrects one error per block. Block coding with multiple-error correction cannot increase the efficiency when the qubit error-probability is below a certain value and the code size fixed. More surprisingly, existing multiple-error correction codes with a code length equal or less than 256 qubits have lower efficiency than the optimal single-error correcting codes for any value of the qubit error-probability. We also investigate how efficient various proposed nondegenerate single-error correcting codes are compared to the limit set by the code redundancy and by the necessary conditions for hypothetically existing nondegenerate codes. We find that existing codes are close to optimal.
A striking feature of quantum error correcting codes is that they can sometimes be used to correct more errors than they can uniquely identify. Such degenerate codes have long been known, but have remained poorly understood. We provide a heuristic for designing degenerate quantum codes for high noise rates, which is applied to generate codes that can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code. The gap between nondegenerate and degenerate code performance is quite large, in contrast to the tiny magnitude of the only previous demonstration of this effect. We also identify a channel for which none of our codes outperform the best nondegenerate code and show that it is nevertheless quite unlike any channel for which nondegenerate codes are known to be optimal.
Quantum error correction is the art of protecting quantum states from the detrimental influence from the environment. To master this art, one must understand how the system interacts with the environment and gives rise to a full set of quantum phenomena, many of which have no correspondence in classical information theory. Such phenomena include decoherence, an effect that in general destroys superpositions of pure states as a consequence of entanglement with the environment. But decoherence can also be understood as “information leakage”, i.e., when knowledge of an encoded code block is transferred to the environment. In this event, the block’s information or entanglement content is typically lost.In a typical scenario, however, not all types of destructive events are likely to occur, but only those allowed by the information carrier, the type of interaction with the environment, and how the environment “picks up” information of the error events. These characteristics can be incorporated into a code, i.e., a channel-adapted quantum error-correcting code.Often, it is assumed that the environment’s ability to distinguish between error events is small, and I will denote such environments “memory-less”. But this assumption is not always valid, since the ability to distinguish error events is related to the temperature of the environment, and in the particular case of information coded onto photons, kBTR «ℏω typically holds, and one must then assume that the environment has a “memory”. In the thesis I describe a short quantum error-correction code adapted for photons interacting with a “cold” reservoir, i.e., a reservoir which continuously probes what error occurred in the coded state.I also study other types of environments, and show how to distill meaningful figures of merit from codes adapted for these channels, as it turns out that resource-based figures reflecting both information and entanglement can be calculated exactly for a well-studied class of channels: the Pauli channels. Starting from these resource-based figures, I establish the notion of efficiency and quality and show that there will be a trade-off between efficiency and quality for short codes. Finally I show how to incorporate, into these calculations, the choices one has to make when handling quantum states that have been detected as incorrect, but where no prospect of correcting them exists, i.e., so-called detection errors.
This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution. The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively. The next four chapters present achievable rates for quantum codes in various scenarios. The final chapter is dedicated to an upper bound on the quantum channel capacity.
Chapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists. These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is, impossible.
Chapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes. By studying the scaling of the optimal block length as a function of the channel's parameters, we develop a heuristic for designing even better codes.
Chapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called twisted states.
In Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible.
Finally, Chapter 7 presents a new upper bound on the quantum channel that is both additive and convex, and which can be interpreted as the of the channel for communication given access to side channels from a class of zero cloning channels. This clone assisted capacity is equal to the unassisted for channels that are degradable, which we use to find new upper bounds on the of a depolarizing channel.
We consider the problem of optimally decoding a quantum error correction code---that is, to find the optimal recovery procedure given the outcomes of partial ``check'' measurements on the system. In general, this problem is NP hard. However, we demonstrate that for concatenated block codes, the optimal decoding can be efficiently computed using a message-passing algorithm. We compare the performance of the message-passing algorithm to that of the widespread blockwise hard decoding technique. Our Monte Carlo results using the five-qubit and Steane's code on a depolarizing channel demonstrate significant advantages of the message-passing algorithms in two respects: (i) Optimal decoding increases by as much as 94 the error threshold below which the error correction procedure can be used to reliably send information over a noisy channel; and (ii) for noise levels below these thresholds, the probability of error after optimal decoding is suppressed at a significantly higher rate, leading to a substantial reduction of the error correction overhead.
Motivated by the fast block Jacket transform based on Pauli matrices,this paper showed how block Jacket matrix could simplify the coding theory of quantum error-correction.Quantum Jacket codes(QJCs) did not require the dual-containing constraint necessary for standard quantum error-correction codes(QECCs),thus allowing to construct quantum codes of large block length with efficiency.These QJCs were generated from some Abelian subgroups of Pauli group that were composed of different rows of the yielded block Jacket matrix,and had advantages of being fast constructed with asymptotically good behaviors.
This paper presents a novel approach to quantum error correction (QEC) utilizing redundancy coding, focusing on enhanced efficiency in quantum information transmission. The core claim revolves around designing multi-particle error-correcting codes to minimize the resource consumption associated with encoding and decoding. The proposed mechanism involves encoding information across multiple qubits, leveraging error detection and correction capabilities inherent in these codes. Crucially, the strategy optimizes the distribution of redundant qubits, significantly reducing unnecessary information redundancy. The research addresses the limitations of existing QEC schemes, which often rely on excessive redundancy, impacting transmission efficiency. This work demonstrates a pathway towards more practical and resource-conscious QEC, representing a key step toward reliable long-distance quantum communication. The primary goal is to reduce the overhead associated with error correction, making quantum information transfer more viable. This is achieved through a refined approach to redundancy, concentrating correction resources where they are most needed.