The one-way quantum computing model introduced by Raussendorf and Briegel [Phys. Rev. Lett. 86, 5188 (2001)] shows that it is possible to quantum compute using only a fixed entangled resource known as a cluster state, and adaptive single-qubit measurements. This model is the basis for several practical proposals for quantum computation, including a promising proposal for optical quantum computation based on cluster states [M. A. Nielsen, Phys. Rev. Lett. (to be published), quant-ph/0402005]. A significant open question is whether such proposals are scalable in the presence of physically realistic noise. In this paper we prove two threshold theorems which show that scalable fault-tolerant quantum computation may be achieved in implementations based on cluster states, provided the noise in the implementations is below some constant threshold value. Our first threshold theorem applies to a class of implementations in which entangling gates are applied deterministically, but with a small amount of noise. We expect this threshold to be applicable in a wide variety of physical systems. Our second threshold theorem is specifically adapted to proposals such as the optical cluster-state proposal, in which nondeterministic entangling gates are used. A critical technical component of our proofs is two powerful theorems which relate the properties of noisy unitary operations restricted to act on a subspace of state space to extensions of those operations acting on the entire state space. We expect these theorems to have a variety of applications in other areas of quantum-information science.
The one-way quantum computing model introduced by Raussendorf and Briegel [Phys. Rev. Lett. 86, 5188 (2001)] shows that it is possible to quantum compute using only a fixed entangled resource known as a cluster state, and adaptive single-qubit measurements. This model is the basis for several practical proposals for quantum computation, including a promising proposal for optical quantum computation based on cluster states [M. A. Nielsen, Phys. Rev. Lett. (to be published), quant-ph/0402005]. A significant open question is whether such proposals are scalable in the presence of physically realistic noise. In this paper we prove two threshold theorems which show that scalable fault-tolerant quantum computation may be achieved in implementations based on cluster states, provided the noise in the implementations is below some constant threshold value. Our first threshold theorem applies to a class of implementations in which entangling gates are applied deterministically, but with a small amount of noise. We expect this threshold to be applicable in a wide variety of physical systems. Our second threshold theorem is specifically adapted to proposals such as the optical cluster-state proposal, in which nondeterministic entangling gates are used. A critical technical component of our proofs is two powerful theorems which relate the properties of noisy unitary operations restricted to act on a subspace of state space to extensions of those operations acting on the entire state space. We expect these theorems to have a variety of applications in other areas of quantum-information science.
In this paper we do a detailed numerical investigation of the fault-tolerant threshold for optical cluster-state quantum computation. Our noise model allows both photon loss and depolarizing noise, as a general proxy for all types of local noise other than photon loss noise. We obtain a threshold region of allowed pairs of values for the two types of noise. Roughly speaking, our results show that scalable optical quantum computing is possible in the combined presence of both noise types, provided that the loss probability is less than $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and the depolarization probability is less than ${10}^{\ensuremath{-}4}$. Our fault-tolerant protocol involves a number of innovations, including a method for syndrome extraction known as telecorrection, whereby repeated syndrome measurements are guaranteed to agree. This paper is an extended version of Dawson et al. [Phys. Rev. Lett. 96, 020501 (2006)].
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.
We propose a measurement-based model for fault-tolerant quantum computation that can be realized with one-dimensional cluster states and fusion measurements only; basic resources that are readily available with scalable photonic hardware. Our simulations demonstrate high thresholds compared with other measurement-based models realized with basic entangled resources and 2-qubit fusion measurements. Its high tolerance to noise indicates that our practical construction offers a promising route to scalable quantum computing with quantum emitters and linear-optical elements.
Quantum information processing is the coherent manipulation of a quantum state for the purpose of performing an information processing task. It is known that certain tasks can in principle be performed much more efficiently using quantum information processing rather than ordinary classical processing. Examples include factoring, secure key distribution, and the simulation of quantum systems. However, significant theoretical and technological obstacles stand in the road to achieving these promised benefits. It is exceedingly difficult to construct devices that are able to achieve an extensive level of control of a quantum state while keeping that state well isolated from the effects of noise. Although there is considerable experimental effort underway to build devices that are less noisy and more controllable, there is also a significant theoretical program aimed at finding schemes that make such physical limitations have less effect on the overall reliability of a device. The crowning achievement is the theory of fault-tolerant quantum error-correction, which shows that a noisy quantum device can be efficiently made to behave as though it were noise free, so long as the amount of noise present is below the “noise threshold”. A central result of this thesis is the calculation of the noise threshold for optical cluster-state quantum computing. Optical cluster-state quantum computing is one of the most promising proposals for the physical implementation of a quantum computer (i.e., a generic quantum information processing device). Previous studies of the value of the threshold, that considered other physical implementations, do not apply to optical quantum computing due to the unusual features of the optical proposal such as nondeterministic gates and photon loss. We present the first detailed analysis of the value of the noise threshold for this proposal. Our analysis involves a number of innovations, including a method for error-correction known as telecorrection, whereby repeated error-syndrome measurements are guaranteed to agree due to the use of teleportation during the correction process. This thesis also considers how to overcome limits to the amount of physical control able to be applied to a quantum device. We ask, if a quantum device can only control limited parts of its quantum state, can that device still be used to achieve a useful information processing task? We consider simple networks of interacting quantum spins where only a few of the spins are controllable, and consider the problem of using this system for high-fidelity quantum communication (i.e., using the system as a simple “quantum wire”). Without control, such systems generally yield a very poor communication fidelity. We show that a very simple scheme that involves controlling a small number of the spins can be used to greatly increase the fidelity across the entire network. The scheme is designed using techniques of state encoding. The thesis also considers the problem of engineering the interactions in a quantum device so that the ground state is a quantum error-correcting code. Such a problem is an important part of the proposal for naturally fault-tolerant systems, which have the ability to resist noise whilst using little or no external control. Our results prove that a certain important class of quantum error-correcting codes, the so called nondegenerate codes, cannot be the eigenstate of any physically-plausible quantum system. This result places significant restrictions on the design of naturally fault-tolerant devices, and sheds light on why current proposals for natural fault tolerance use codes that are degenerate.
In this thesis, we describe the one-way quantum computer [Formula: see text], a scheme of universal quantum computation that consists entirely of one-qubit measurements on a highly entangled multiparticle state, i.e. the cluster state. We prove the universality of the [Formula: see text], describe the underlying computational model and demonstrate that the [Formula: see text] can be operated fault-tolerantly. In Sec. 2, we show that the [Formula: see text] can be regarded as a simulator of quantum logic networks. In this way, we prove the universality and establish the link to the network model — the common model of quantum computation. We also indicate that the description of the [Formula: see text] as a network simulator is not adequate in every respect. In Sec. 3, we derive the computational model underlying the [Formula: see text], which is very different from the quantum logic network model. The [Formula: see text] has no quantum input, no quantum output and no quantum register, and the unitary gates from some universal set are not the elementary building blocks of [Formula: see text] quantum algorithms. Further, all information that is processed with the [Formula: see text] is the outcomes of one-qubit measurements and thus processing of information exists only at the classical level. The [Formula: see text] is nevertheless quantum-mechanical, as it uses a highly entangled cluster state as the central physical resource. In Sec. 4, we show that there exist nonzero error thresholds for fault-tolerant quantum computation with the [Formula: see text]. Further, we outline the concept of checksums in the context of the [Formula: see text], which may become an element in future practical and adequate methods for fault-tolerant [Formula: see text] computation.
This thesis contains the results of research projects broadly related to the pursuit of universal fault-tolerant quantum computation. We are interested in questions such as which resources are required to achieve a quantum computational advantage, how to utilize such resources in practice, and how to characterize and control noise throughout a computation. \n \nIn part 1, we cover results pertaining to magic state distillation, one of the leading approaches to implementing universal fault-tolerant quantum computation. Early work in this area focused on determining the region of distillable states for qubit protocols, yet comparatively little is known about which states can be distilled and with what distillable region for d>2. In the first project, we focus on d=3 and present new four-qutrit distillation schemes that improve upon the known distillable region, and achieve distillation tight to the boundary of undistillable states for some classes of states. As a consequence of recent results, this implies that there is a family of quantum states that enable universality if and only if they exhibit contextuality with respect to stabilizer measurements. We also identify a new routine whose fixed point is a magic state with maximal sum-negativity i.e., it is maximally non-stabilizer in a specific sense. In the second project, we return to d=2 and present a number of interesting new distillation routines based on small codes. Many of these distill noisy states right up to the boundary of the known undististillable region, while some distill toward non-stabilizer states that have not previously been considered. \n \nIn part 2, we address the issue of noise characterization. The characterization of noise in a quantum system serves as a foundation towards meeting two important goals. Firstly, we may wish to know about specific sources of noise in order to adapt or correct errors as much as possible. Secondly, we would like a way to verify that noise is below some threshold value in order to meet the criteria of threshold theorems. We consider the case where noise is approximately known to coincide with the generalized damping channel (encompassing the common intrinsic processes of amplitude damping and dephasing), but may contain additional unknown noise sources. We provide methods to accurately obtain the noise parameters, and compare with the results of a randomized benchmarking experiment. Using this information, we show how to make meaningful statements about fault-tolerance threshold theorems by considering the diamond distance.