Having surveyed the necessary background in Chapter 1, this chapter begins developing the theory of quantum error correcting codes and fault-tolerant quantum computing. Section 2.1 introduces the dynamical framework of quantum operations. In this approach a quantum computer is treated as an open quantum system and a superoperator describes the effective interaction it has with its environment. The operator-sum representation for a quantum operation is derived, as are some of its essential properties. This section also shows how the noise model introduced in Section 1.4.1 can be represented as a quantum operation, and briefly describes other important noise models. Sec tion 2.2 introduces a number of fundamental notions associated with quantum error correcting codes, and the seven-qubit Calderbank-Shor-Steane code is used in Section 2.3 to show how these notions manifest in a specific quantum error correcting code.
Having surveyed the necessary background in Chapter 1, this chapter begins developing the theory of quantum error correcting codes and fault-tolerant quantum computing. Section 2.1 introduces the dynamical framework of quantum operations. In this approach a quantum computer is treated as an open quantum system and a superoperator describes the effective interaction it has with its environment. The operator-sum representation for a quantum operation is derived, as are some of its essential properties. This section also shows how the noise model introduced in Section 1.4.1 can be represented as a quantum operation, and briefly describes other important noise models. Sec tion 2.2 introduces a number of fundamental notions associated with quantum error correcting codes, and the seven-qubit Calderbank-Shor-Steane code is used in Section 2.3 to show how these notions manifest in a specific quantum error correcting code.
"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."
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.
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.
This thesis is concerned with certain theoretical problems that arise naturally in the context of fault-tolerant quantum computation. Fault-tolerance can be defined as the art of building reliable devices from unreliable components, and is of particular importance for quantum computers that aim to precisely control the dynamics of extremely sensitive quantum systems. A model of quantum computation is a specification of the basic building blocks by which a quantum computation is implemented. The best known model is the quantum circuit model, where computations are implemented by means of unitary quantum gates that are applied to two-level quantum systems known as qubits. In a physical implementation of a quantum circuit, the gates and qubits will inevitably be affected by noise. Fault-tolerant quantum circuits are designed to be resilient against the effects of this noise, provided that it is not too strong. Fault-tolerance in the quantum circuit model is well developed thanks to the theory of quantum error-correcting codes. These codes allow for the correction of small numbers of errors introduced by a variety of noise processes. In a fault-tolerant quantum circuit, qubits are replaced with encoded qubits, and quantum gates with encoded gates that are immediately followed by special quantum circuits for error correction. Provided the rate at which errors occur is below a constant threshold value, the accumulation of errors can be checked so that the correct output of the computation correctly determined. The threshold acts as both a measure of how good the design of a quantum circuit is, and also as a target for experimenters aiming to implement quantum circuits. Much current research in quantum computation is aimed at designing quantum circuits that increase the noise threshold, hopefully to the point where it becomes within the reach of experimenters. A fault-tolerant encoded quantum gate must limit the propagation of errors so that the code’s corrective capabilities are not overwhelmed. It is not so easy to design encoded gates that satisfy this property, and to date only a finite handful of such gates are known. In a faulttolerant quantum circuit, all quantum gates must be decomposed or compiled in terms of those that may be implemented fault-tolerantly. In the first part of this thesis, we present two results that may be applied to this problem of gate compilation. The first is a generic method based on the Solovay-Kitaev theorem that may be applied to all quantum gates, but is most effective for those that act on single qubits. We present the Solovay-Kitaev theorem in its simples known form as an algorithm, together with novel constructions that can be used to implement it. Following this we give a two specialized methods for two-qubit gate decomposition, based on the Cartan decomposition of the Lie group SU(4). The cluster-state model of computation is an alternative to the quantum circuit model, and makes use of quantum measurements and highly entangled cluster-states to implement a quantum computation. The fault-tolerant techniques developed for quantum circuits are not immediately applicable in this model, so in order for it to be a realistic candidate for performing computations we prove that such techniques are possible. In the second part of the thesis we prove that constant fault-tolerance thresholds may be achieved in the cluster-state model, and in particular in an adaption to an optical implementation. Following this we design a complete error correction scheme for optical cluster-state computation, and numerically determine the threshold of this model in the face of the dominant noise models likely to affect such an implementation.
Fault-tolerant state preparation is essential for reliable quantum error correction, particularly in Steane-type error correction, which relies on robust ancilla states for syndrome readout. One method of fault-tolerant state preparation is to initialize multiple ancilla states and check them against each other to detect problematic errors. In the worst case, the number of states required for successful initialization grows polynomially with the code distance, but it has been shown that this can be reduced to constant ancilla overhead-in the best case, only four states are required. However, existing techniques for finding low-overhead initialization schemes are limited to codes with large symmetry groups, such as the Golay code. In this work, we propose a general, automated synthesis methodology for Steane-type fault-tolerant state preparation circuits that applies to arbitrary Calderbank-Shor-Steane (CSS) codes and does not rely on code symmetries. We apply the proposed methods to various CSS codes up to a distance of seven and simulate the successful fault-tolerant initialization of logical basis states under circuit-level depolarizing noise. The circuits synthesized using the proposed methodology provide an important step towards experimental realizations of high-fidelity ancilla states for near-term demonstration of fault-tolerant quantum computation.
Abstract This chapter treats the subject of quantum error correction and fault-tolerant quantum computation. In particular, it presents the various representations of open-system quantum dynamics, insisting on the indistinguishability between different unravelings. Quantum error correction with discrete gates reacting to discrete error syndromes obtained from projective measurement can be seen as a form of quantum feedback that stabilizes a whole manifold of quantum states rather than a single one. The chapter gives a useful general introduction to the theory of stabilizers, which is key to the understanding of error correcting codes. It explains in full detail the principles behind the size of the overhead needed to make the entire quantum processor fault-tolerant, not just its memory.