The experiments described in this thesis form an investigation into the path towards establishing the requirements of quantum computing in a linear optical system. Our qubits are polarisation encoded photons for which the basic operations of quantum computing, single qubit rotations, are a well understood problem. The difficulty lies in the interaction of photons. To achieve these we use measurement induced non-linearities. The first experiment in this thesis describes the thorough characterisation of a controlled-sign gate based on such non-linearities. The photons are provided as pairs generated through parametric down-conversion, and as such share correlations unlikely to carry over into large scale implementations of the future. En route to such larger circuits, a characterisation of the actions of the controlled-sign gate is conducted, when the input qubits have been generated independently from each other, revealing a large drop in process fidelity. To explore the cause of this degradation of the gate performance a thorough and highly accurate model of the gate is derived including the realistic description of faulty circuitry, photon loss and multi-photon emission by the source. By simulating the effects of the various noise sources individually, the heretofore largely ignored multi-photon emission is identified as the prime cause of the degraded gate performance, causing a drop in fidelity nearly three times as large as any other error source. I further draw the first comparison between the performance of an experimental gate to the error probabilities per gate derived as thresholds for fault-tolerant quantum computing. In the absence of a single vigourous threshold value, I compare the gate performance to the models that yielded the highest threshold to date as an upper bound and to the threshold of the Gremlin-model, which allows for the most general errors. Unsurprisingly this comparison reveals that the implemented gate is clearly insufficient, however just remedying the multi-photon emission error will allow this architecture to move to within striking distance of the boundary for fault-tolerant quantum computing. The utilised methodology can be applied to any gate in any architecture and can, combined with a suitable model of the noise sources, become an important guide for developments required to achieve fault tolerant quantum computing. The final experiment on the path towards linear optical quantum computing is the demonstration of a pair of basic versions of Shor's algorithm which display the essential entanglement for the algorithm. The results again highlight the need for extensive measurements to reveal the fundamental quality of the implemented algorithm, which is not accessible with limited indicative measurements. In the second part of the thesis, I describe two experiments on other forms of entanglement by extending the actions of a Fock-State filter, a filter that is capable of attenuating single photon states stronger than multi-photon states, to produce entangled states. Furthermore this device can be used in conjunction with standard wave-plates to extend the range of operations possible on the bi-photonic qutrit space, showing that this setup suffices to produce any desired qutrit state, thereby giving access to new measurement capabilities and in the process creating and proving the first entanglement between a qubit and a qutrit.
The experiments described in this thesis form an investigation into the path towards establishing the requirements of quantum computing in a linear optical system. Our qubits are polarisation encoded photons for which the basic operations of quantum computing, single qubit rotations, are a well understood problem. The difficulty lies in the interaction of photons. To achieve these we use measurement induced non-linearities. The first experiment in this thesis describes the thorough characterisation of a controlled-sign gate based on such non-linearities. The photons are provided as pairs generated through parametric down-conversion, and as such share correlations unlikely to carry over into large scale implementations of the future. En route to such larger circuits, a characterisation of the actions of the controlled-sign gate is conducted, when the input qubits have been generated independently from each other, revealing a large drop in process fidelity. To explore the cause of this degradation of the gate performance a thorough and highly accurate model of the gate is derived including the realistic description of faulty circuitry, photon loss and multi-photon emission by the source. By simulating the effects of the various noise sources individually, the heretofore largely ignored multi-photon emission is identified as the prime cause of the degraded gate performance, causing a drop in fidelity nearly three times as large as any other error source. I further draw the first comparison between the performance of an experimental gate to the error probabilities per gate derived as thresholds for fault-tolerant quantum computing. In the absence of a single vigourous threshold value, I compare the gate performance to the models that yielded the highest threshold to date as an upper bound and to the threshold of the Gremlin-model, which allows for the most general errors. Unsurprisingly this comparison reveals that the implemented gate is clearly insufficient, however just remedying the multi-photon emission error will allow this architecture to move to within striking distance of the boundary for fault-tolerant quantum computing. The utilised methodology can be applied to any gate in any architecture and can, combined with a suitable model of the noise sources, become an important guide for developments required to achieve fault tolerant quantum computing. The final experiment on the path towards linear optical quantum computing is the demonstration of a pair of basic versions of Shor's algorithm which display the essential entanglement for the algorithm. The results again highlight the need for extensive measurements to reveal the fundamental quality of the implemented algorithm, which is not accessible with limited indicative measurements. In the second part of the thesis, I describe two experiments on other forms of entanglement by extending the actions of a Fock-State filter, a filter that is capable of attenuating single photon states stronger than multi-photon states, to produce entangled states. Furthermore this device can be used in conjunction with standard wave-plates to extend the range of operations possible on the bi-photonic qutrit space, showing that this setup suffices to produce any desired qutrit state, thereby giving access to new measurement capabilities and in the process creating and proving the first entanglement between a qubit and a qutrit.
Quantum computing is an exciting field that promises great improvements to our ability to solve certain intractable classes of problem. Implementing a quantum computer will require a physical system in which quantum effects are strongly evident. Many different physical systems have been proposed for this task, including ion traps, quantum dots, neutral atoms and photons. This thesis focuses on optical implementations of a quantum computer, and specifically on single-photon designs, as opposed to those utilising coherent states. Photonic systems have shown a great deal of promise for quantum computing tasks since it was shown by Knill, Laflamme and Milburn (KLM) that scalable efficient quantum computing could be performed using only linear optical elements and measurement. This was made viable through the use of error-correcting codes. The work in this thesis covers the use of error-correcting codes in single-photon linear optics quantum computing (LOQC). The central feature of these schemes is the parity code introduced by KLM and used to overcome the gate errors that are inherent in this type of LOQC. First is a discussion of an improved scheme that incorporates the fusion gates used in Browne and Rudolph's cluster state proposal in order to simplify the encoding and gate procedures. This results in high-probability gates with a resource usage comparable to the cluster state schemes. The remainder of the thesis examines the application of further error correcting codes to handle other errors that may arise in any practical implementation of an optical quantum computer. The most significant type of error in any optical system is photon loss, and hence an extension of the parity code is presented that allows loss errors to be detected and corrected. Procedures for performing universal quantum computation whilst employing this encoding are also described. These enable the establishment of loss thresholds for parity-state based LOQC, revealing this code is capable of handling up to 10\% loss in all elements during general computation, or 17\% if it is used solely as an active memory. Although loss is the most common type of optical error, other types of error will also become an issue when attempting to produce large quantum computing systems. An analysis of the error thresholds and resource usage which can be expected when applying a general error correcting code such as the Steane code to this type of LOQC system is also presented. The error thresholds and resource requirements established are greater than those found for cat state optical quantum computing, but fall below those calculated for cluster states. However, the resource requirements of the scheme are significantly smaller than the cluster state requirements for an equivalent circuit. Finally, a new experiment is proposed, achievable with today's technology, for testing the loss-tolerant encoding described earlier. Expansions of the experiment that would demonstrate tolerance of actual experimental error are also discussed.
The use of photons as qubits is a promising implementation for quantum computation. The inability of photons to interact, especially with the environment, makes them an ideal physical candidate. However, this also makes them a difficult system to perform two qubit gates on. Recent breakthroughs in photonic quantum computing have shown methods around the requirement of direct photon-photon interaction. In this thesis we study three recently discovered schemes for optical quantum computation. \n \n We first investigate the so called linear optical quantum computing (LOQC) scheme, exploring a method to improve the original proposal by constructing a photon-number QND detector that succeeds with a high probability. In doing this we present a new type of LOQC teleporter, one that can detect the presence of a single photon in an arbitrary polarisation state when the input state is a sum of vacuum and multi-photon terms. This new type of teleporter is an improvement on the original scheme in that the entangled states required can be made offline with fewer entangling operations. \n \n We next investigate the so called quantum bus (qubus) scheme for photonic quantum computing. We show a scheme to measure the party of n qubit states by using a single qubus mode, controlled rotations and displacements. This allows for the syndrome measurements of any stabilizer quantum error correcting code. We extend these results to a fault tolerant scheme to measure an arbitrary Pauli operator of weight n, incorporating so called single bit teleportations. We investigate the construction of a Toffoli gate by using a single qubus mode, controlled rotations and displacements that works with a success probability of at least 25%. We also investigate the use of single bit teleportations to construct a universal set of gates on coherent state type logic and in the construction of cluster states. \n \nWe finally investigate the optical Zeno gate, a gate that uses the Zeno effect in the form of two photon absorbers to induce a CSIGN. We model realistic two photon absorption and include realistic single photon loss for this gate, examining the use of encoding to overcome the single photon loss, showing that our results are competitive with the leading equivalent LOQC scheme.
Photonics is a promising architecture for the realisation of quantum information processing, since the two-photon interaction, or non-linearity, necessary to build logical gates can efficiently be realised by the use of interference with ancillary photons and detection 1 Knill, E, Laflamme, R and Milburn, GJ. 2001. Nature, 409: 46–52. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]. Although single-photon sources and detectors are pivotal in realisations of such systems, clear guidelines for the required performance of realistic systems are yet to be defined. We present our detailed numerical simulation of several quantum optics circuits including sources and detectors all represented in multi-dimensional Fock-spaces, which allows us to obtain experimentally realistic performance bounds for these devices. In addition, the single-photon source based on switched parametric down-conversion is studied, which in principle could reach the required performance. Three approaches for implementing the switching hierarchy of the photons are simulated, and their anticipated performance is obtained. Our results define the bar for the optical devices needed to achieve the first level of linear-optics quantum computing outside the coincidence basis.
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.
In this thesis, I present experimental and theoretical work which assesses and develops a range of tools that are required for performing quantum information processing, particularly in photonic systems. I investigate the three degrees of freedom of a single photon—its polarisation, spatialmomentum and time-frequency distributions. For polarisation, I show that using wave plates to implement arbitrary, single-qubit rotations is more complicated than is commonly appreciated. In the spatial-momentum and time-frequency domains, I develop ways to perform tomographic analysis of quantum states, and I report the first demonstrations of these techniques. In the time-frequency domain, the tomography technique utilises entanglement in the photon polarisation as a resource to store and provide access to the time-frequency information. In my first two experiments, I use spontaneous parametric down-conversion to produce entanglement between pairs of single photons in all three degrees of freedom. I demonstrate the first characterisation of entanglement in spatial modes and the time-frequency domain, the first quantitative measurement of entangled qutrit states, and the highest quality entangled states yet measured in both polarisation and spatial modes. I also report the first realisation of complete hyperentanglement, and a full, black-box tomography of a 36-dimensional two-photon state—the largest system to be characterised in this way to date. In my final experiment, I model and implement a new architecture for a controlled-Z gate which is much simpler to align than previous implementations. This gate requires only one non-classical interference condition, the visibility of which is the main limitation to its performance. I show that the gate operates effectively as a means of both creating entanglement and discriminating between the four elements in a basis of maximally entangled, Bell-type states. Indeed, its observed performance as a Bell analyser would be sufficient to build a quantum state teleporter which would guarantee that the recipient would be left with a better copy of an unknown input state than any eavesdropper. In a series of numerical simulations, I investigate some of the practicalities that arise when using tomographic reconstruction techniques, including how to estimate errors, which measurements to make and what is actually the optimal reconstruction. In particular, I show that tomographies perform better when based on the results from over-complete sets of measurements. Finally, I discuss the important issue of how to compare two processes, particularly when trying to assess the quality of a measured process by comparing it to some expected ideal. Judging possible candidate measures against a set of experimentally and theoretically motivated
A major challenge in practical quantum computation is the ineludible errors caused by the interaction of quantum systems with their environment. Fault-tolerant schemes, in which logical qubits are encoded by several physical qubits, enable to the output of a higher probability of correct logical qubits under the presence of errors. However, strict requirements to encode qubits and operators render the implementation of a full fault-tolerant computation challenging even for the achievable noisy intermediate-scale quantum technology. Especially the threshold for fault-tolerant computation still lacks experimental verification. Here, based on an all-optical setup, we experimentally demonstrate the existence of the threshold for the fault-tolerant protocol. Four physical qubits are represented as the spatial modes of two entangled photons, which are used to encode two logical qubits. The experimental results clearly show that when the error rate is below the threshold, the probability of correct output in the circuit, formed with fault-tolerant gates, is higher than that in the corresponding non-encoded circuit. In contrast, when the error rate is above the threshold, no advantage is observed in the fault-tolerant implementation. The developed high-accuracy optical system may provide a reliable platform to investigate error propagation in more complex circuits with fault-tolerant gates.