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.
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.
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)].
Abstract Quantum computing1—the processing of information according to the fundamental laws of physics—offers a means to solve efficiently a small but significant set of classically intractable problems. Quantum computers are based on the controlled manipulation of entangled quantum states, which are extremely sensitive to noise and imprecision; active correction of errors must therefore be implemented without causing loss of coherence. Quantum error-correction theory2, 3, 4, 5, 6, 7, 8, 9 has made great progress in this regard, by predicting error-correcting ‘codeword’ quantum states. But the coding is inefficient and requires many quantum bits10, 11, 12, which results in physically unwieldy fault-tolerant quantum circuits10, 11, 12, 13, 14, 15, 16, 17, 18. Here I report a general technique for circumventing the trade-off between the achieved noise tolerance and the scale-up in computer size that is required to realize the error correction. I adapt the recovery operation (the process by which noise is suppressed through error detection and correction) to simultaneously correct errors and perform a useful measurement that drives the computation. The result is that a quantum computer need be only an order of magnitude larger than the logic device contained within it. For example, the physical scale-up factor10, 11 required to factorize a thousand-digit number is reduced from 1, 500 to 22, while preserving the original tolerated gate error rate (10−5) and memory noise per bit (10−7). The difficulty of realizing a useful quantum computer is therefore significantly reduced.
Accurate control and addressability of quantum devices will come with the promise of improvement in a wide variety of theoretical and applied fields, such as chemistry, condensed matter physics, theoretical computer science, foundational physics, communications, metrology and others. Decoherence of quantum states and the loss of quantum systems have adverse effects and deter a satisfactory usage of quantum devices. This is the main problem to be overcome, which is the goal of quantum fault tolerance. In this thesis we present a series of works that contribute to some of the fields mentioned above, in the direction of fighting decoherence and loss. These works fall in two categories: on one hand, we looked at computer architectures which can be used to combat errors, using techniques of quantum error correcting codes. In a first project we found decoherence and loss probability thresholds below which quantum computing is provably possible. We assumed a very particular error model tailored specially to quantum dots as single photon sources and linear optics. Subsequently we looked at the problem of loss, both of heralded and unheralded, and devised some ways to fight it. The framework under which this work was done was used to develop theory which is currently being tested in a quantum optics experimental group and will be reported in an article later this year. On the other hand, we studied how the error probability can be reduced at the physical level, thanks exclusively to the properties of the system in which information is stored, as opposed to making use of quantum codes. We looked at a particular superconducting circuit, which is potentially very well protected against some types of decoherence. In particular, we observed that the interaction with the environment become weaker for certain values of the circuit external parameters.
Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. \n \nQuantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. \n \nThis thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. \n \nThree techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.
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.