The goal of this paper is to review the theoretical basis for achieving a faithful quantum information transmission and processing in the presence of noise. Initially encoding and decoding, implementing gates and quantum error correction will be considered error free. Finally we will relax this non realistic assumption, introducing the quantum fault-tolerant concept. The existence of an error threshold permits to conclude that there is no physical law preventing a quantum computer from being built. An error model based on the depolarizing channel will be able to provide a simple estimation of the storage or memory computation error threshold: < 5.2 10-5. The encoding is made by means of the [[7,1,3]] Calderbank-Shor-Steane quantum code and the Shor's fault-tolerant method to measure the stabilizer's generators is used.
The goal of this paper is to review the theoretical basis for achieving a faithful quantum information transmission and processing in the presence of noise. Initially encoding and decoding, implementing gates and quantum error correction will be considered error free. Finally we will relax this non realistic assumption, introducing the quantum fault-tolerant concept. The existence of an error threshold permits to conclude that there is no physical law preventing a quantum computer from being built. An error model based on the depolarizing channel will be able to provide a simple estimation of the storage or memory computation error threshold: < 5.2 10-5. The encoding is made by means of the [[7,1,3]] Calderbank-Shor-Steane quantum code and the Shor's fault-tolerant method to measure the stabilizer's generators is used.
"The goal of this paper is to review the theoretical basis for achieving a faithful quantum information transmission and processing in the presence of noise. Initially, encoding and decoding, implementing gates and quantum error correction will be considered error-free. Finally, we shall relax this non-realistic assumption, introducing the quantum fault-tolerant concept. The existence of an error threshold permits us to conclude that there is no physical law preventing a quantum computer from being built. An error model based on the depolarising channel will be able to provide a simple estimate of the storage or memory computation error threshold: 5.2 10. The encoding is made by means of the [[7,1,3]] Calderbank-Shor-Steane quantum code, and Shor´s fault-tolerant method is used to measure the stabiliser´s generators."
In its short history, the field of quantum computing has produced a num ber of truly startling revelations about the computational power inherent in quantum systems. These discoveries generated a great deal of excitement, though this excitement was tempered by the realization that the quantum correlations responsible for this power are extremely fragile. It became clear that to properly assess the performance of a quantum computer, the effects of noise and imperfect quantum gates would have to be included in the analysis. It was thought by many that the virtues of a quantum computer would not survive the presence of these effects. Remarkably, in three years (1995-1997), theoretical tools were developed that stood this expectation on its head. Be ginning with the groundbreaking work of P. W. Shor and A. M. Steane, a verygeneral framework for constructing quantum error correcting codes was estab lished, and protocols were developed that allowed quantum computations to be done in a fault-tolerant manner. This line of development culminated in a demonstration of the accuracy threshold theorem. This theorem showed that fault-tolerant protocols for quantum computation, in conjunction with concatenated quantum error correcting codes, would allow a quantum compu tation of arbitrary duration to be done with arbitrarily small error probability, provided all quantum gates used in the computation had error probabilities that fell below a value known as the accuracy threshold Pa. This threshold was calculated for a number of simple error models yielding values in the range 10-6 < p a < io -3 . The good news is that Pa ^ 0 so that reliable quantum computing in the presence of noise and imperfect quantum gates is possible. The bad news is that Pa is small so that building quantum gates that satisfy the tolerances of the threshold theorem will be technically challenging.
Designing encoding and decoding circuits to reliably send messages over many uses of a noisy channel is a central problem in communication theory. When studying the optimal transmission rates achievable with asymptotically vanishing error it is usually assumed that these circuits can be implemented using noise-free gates. While this assumption is satisfied for classical machines in many scenarios, it is not expected to be satisfied in the near term future for quantum machines where decoherence leads to faults in the quantum gates. As a result, fundamental questions regarding the practical relevance of quantum channel coding remain open. By combining techniques from fault-tolerant quantum computation with techniques from quantum communication, we initiate the study of these questions. We introduce fault-tolerant versions of quantum capacities quantifying the optimal communication rates achievable with asymptotically vanishing total error when the encoding and decoding circuits are affected by gate errors with small probability. Our main results are threshold theorems for the classical and quantum capacity: For every quantum channelTand everyϵ>0there exists a thresholdp(ϵ,T)for the gate error probability below which rates larger than$C-\epsilon arefault−tolerantlyachievablewithvanishingoverallcommunicationerror,whereC$denotes the usual capacity. Our results are not only relevant in communication over large distances, but also on-chip, where distant parts of a quantum computer might need to communicate under higher levels of noise than affecting the local gates.
This thesis is concerned with fault-tolerant quantum information processing using quantum error-correcting codes. It contains two major pieces of work. The first is a study of coherent noise in the context of stabilizer error-correcting codes. The second is a proposed scheme for a universal set of fault-tolerant logical gates in a particular code family built out of the 3D toric code. Chapter 1 provides an introduction to quantum computation and fault tolerance. Many basic concepts in error-correcting codes are defined. Special attention is paid to the set of code properties that are most likely to determine how easily a given fault-tolerant scheme might be implemented on a physical device. These include the fault-tolerant noise threshold and the overhead. In Chapters 2 and 3 we study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In Chapter 2 we introduce coherent noise and incoherent noise and a number of methods that are useful for the study of coherent noise. We study the repetition code as a basic example, and we also study a correlated noise model. In Chapter 3 we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment. Chapter 4 is connected to another aspect of fault tolerance, fault-tolerant logical gates. The toric code is a promising candidate for fault-tolerant quantum computation because of its high threshold and low-weight stabilizers. A universal gate set in the toric code generally requires magic state distillation, which can incur a significant qubit overhead. In this work we construct an error-correcting code in three dimensions based on the toric code that features a fault-tolerant T gate with no magic state distillation required. We further describe a subsystem version of our code which supports a universal set of fault-tolerant gates. This code can be converted into the stabilizer version using gauge-fixing. We also argue that our code can be converted to a (2+1)-D protocol, where a 2D lattice undergoes a measurement-based protocol over time. In this way, a fault-tolerant logical T gate can be realized in a 2D toric code structure.
The states needed in quantum computation are extremely affected by decoherence. Several methods have been proposed to control error spreading. They use two main tools: fault-tolerant constructions and concatenated quantum error correcting codes. In this work, we estimate the threshold conditions necessary to make a long enough quantum computation. The [[7,1,3]] CSS quantum code together with the Shor method to measure the error syndrome is used. No concatenation is included. The decoherence is introduced by means of the depolarizing channel error model, obtaining several thresholds from the numerical simulation. Regarding the maintenance of a qubit stabilized in the memory, the error probability must be smaller than 2.9 × 10-5. In order to implement a one or two-qubit encoded gate in an effective fault-tolerant way, it is possible to choose an adequate non-encoded noisy gate if the memory error probability is smaller than 1.3 × 10-5. In addition, fulfilling this last condition permits us to assume a more efficient behavior compared to the equivalent non-encoded process.
Quantum error correction (QEC) represents the cornerstone technology for realizing fault-tolerant quantum computing, addressing the fundamental challenge of quantum state decoherence in noisy intermediate-scale quantum (NISQ) devices. This comprehensive review examines recent advances in QEC implementations from 2020-2024, spanning theoretical foundations to practical hardware demonstrations. We analyze the evolution from pioneering stabilizer codes to modern topological approaches, with particular emphasis on surface codes and low-density parity-check (LDPC) codes that have demonstrated below-threshold error correction in superconducting and trapped-ion systems. The paper provides a comparative analysis of classical and machine learning-based decoder algorithms, evaluating their performance through complexity analysis and threshold comparisons in real-time error correction scenarios. Through examination of breakthrough experiments by Google, IBM, Quantinuum, and emerging platforms, we assess the current state of fault-tolerant quantum computing and identify critical engineering challenges including correlated noise models, cryogenic control systems, and scalable logical gate implementations. Our quantitative analysis reveals that while significant progress has been made toward practical fault-tolerance thresholds, achieving the resource efficiency and reliability required for large-scale quantum computation remains challenging. We provide detailed resource estimates for key applications including Shor's algorithm and optimization problems, along with analysis of industry roadmaps toward thousand-logical-qubit systems. This review serves as a comprehensive guide for researchers and engineers working toward the next generation of fault-tolerant quantum computers, with particular focus on information-theoretic perspectives relevant to quantum information entropy and syndrome processing.