The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of quantum computers. However, several other models of quantum computation exist which provide useful alternative frameworks for both discovering new quantum algorithms and devising new physical implementations of quantum computers. In this thesis, I first present necessary background material for a general physics audience and discuss existing models of quantum computation. Then, I present three results relating to various models of quantum computation: a scheme for improving the intrinsic fault tolerance of adiabatic quantum computers using quantum error detecting codes, a proof that a certain problem of estimating Jones polynomials is complete for the one clean qubit complexity class, and a generalization of perturbative gadgets which allows k-body interactions to be directly simulated using 2-body interactions. Lastly, I discuss general principles regarding quantum computation that I learned in the course of my research, and using these principles I propose directions for future research.
The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of quantum computers. However, several other models of quantum computation exist which provide useful alternative frameworks for both discovering new quantum algorithms and devising new physical implementations of quantum computers. In this thesis, I first present necessary background material for a general physics audience and discuss existing models of quantum computation. Then, I present three results relating to various models of quantum computation: a scheme for improving the intrinsic fault tolerance of adiabatic quantum computers using quantum error detecting codes, a proof that a certain problem of estimating Jones polynomials is complete for the one clean qubit complexity class, and a generalization of perturbative gadgets which allows k-body interactions to be directly simulated using 2-body interactions. Lastly, I discuss general principles regarding quantum computation that I learned in the course of my research, and using these principles I propose directions for future research.
This work presents a comprehensive and rigorous examination of quantum computing, spanning its mathematical foundations, physical principles, algorithmic developments, hardware architectures, and philosophical implications. Beginning with linear algebra over Hilbert spaces and the postulates of quantum mechanics, the manuscript develops the formal structure of quantum information theory, including qubits, entanglement, quantum channels, and noise models. The work provides in-depth treatments of the quantum circuit model, alternative computation paradigms such as measurement-based, adiabatic, and topological quantum computation, and the theoretical framework of quantum error correction, including stabilizer and surface codes. Foundational and modern quantum algorithms are analysed in detail, including Shor’s factoring algorithm, Grover’s search, variational quantum eigensolvers, QAOA, and emerging quantum machine learning approaches. Beyond algorithms and implementations, the manuscript addresses scalability challenges, fault tolerance, computational complexity limits, and the implications of quantum computing for cryptography and the extended Church–Turing thesis. The concluding sections explore philosophical questions concerning interpretation, measurement, and the relationship between computation and physical reality. This work is intended as both a reference text and a research-level synthesis for advanced students, researchers, and practitioners in quantum computing, theoretical computer science, and quantum physics.
Quantum computing is an emerging technology that has the potential to change the perspectives and applications of computing in general. A wide range of applications are enabled: from faster algorithmic solutions of classically still difficult problems to theoretically more secure communication protocols. A quantum computer uses the quantum mechanical effects of particles or particle-like systems, and a major similarity between quantum and classical computers consists of both being abstracted as information processing machines. Whereas a classical computer operates on classical digital information, the quantum computer processes quantum information, which shares similarities with analog signals. One of the central differences between the two types of information is that classical information is more fault-tolerant when compared to its quantum counterpart. Faults are the result of the quantum systems being interfered by external noise, but during the last decades quantum error correction codes (QECC) were proposed as methods to reduce the effect of noise. Reliable quantum circuits are the result of designing circuits that operate directly on encoded quantum information, but the circuit’s reliability is also increased by supplemental redundancies, such as sub-circuit repetitions. Reliable quantum circuits have not been widely used, and one of the major obstacles is their vast associated resource overhead, but recent quantum computing architectures show promising scalabilities. Consequently the number of particles used for computing can be more easily increased, and that the classical control hardware (inherent for quantum computation) is also more reliable. Reliable quantum circuits haev been investigated for almost as long as general quantum computing, but their limited adoption (until recently) has not generated enough interest into their systematic design. The continuously increasing practical relevance of reliability motivates the present thesis to investigate some of the first answers to questions related to the background and the methods forming a reliable quantum circuit design stack. The specifics of quantum circuits are analysed from two perspectives: their probabilistic behaviour and their topological properties when a particular class of QECCs are used. The quantum phenomena, such as entanglement and superposition, are the computational resources used for designing quantum circuits. The discrete nature of classical information is missing for quantum information. An arbitrary quantum system can be in an infinite number of states, which are linear combinations of an exponential number of basis states. Any nontrivial linear combination of more than one basis states is called a state superposition. The effect of superpositions becomes evident when the state of the system is inferred (measured), as measurements are probabilistic with respect to their output: a nontrivial state superposition will collapse to one of the component basis states, and the measurement result is known exactly only after the measurement. A quantum system is, in general, composed from identical subsystems, meaning that a quantum computer (the complete system) operates on multiple similar particles (subsystems). Entanglement expresses the impossibility of separating the state of the subsystems from the state of the complete system: the nontrivial interactions between the subsystems result into a single indivisible state. Entanglement is an additional source of probabilistic behaviour: by measuring the state of a subsystem, the states of the unmeasured subsystems will probabilistically collapse to states from a well defined set of possible states. Superposition and entanglement are the building blocks of quantum information teleportation protocols, which in turn are used in state-of-the-art fault-tolerant quantum computing architectures. Information teleportation implies that the state of a subsystem is moved to a second subsystem without copying any information during the process. The probabilistic approach towards the design of quantum circuits is initiated by the extension of classical test and diagnosis methods. Quantum circuits are modelled similarly to classical circuits by defining gate-lists, and missing quantum gates are modelled by the single missing gate fault. The probabilistic approaches towards quantum circuits are facilitated by comparing these to stochastic circuits, which are a particular type of classical digital circuits. Stochastic circuits can be considered an emulation of analogue computing using digital components. A first proposed design method, based on the direct comparison, is the simulation of quantum circuits using stochastic circuits by mapping each quantum gate to a stochastic computing sub-circuit. The resulting stochastic circuit is compiled and simulated on FPGAs. The obtained results are encouraging and illustrate the capabilities of the proposed simulation technique. However, the exponential number of possible quantum basis states was translated into an exponential number of stochastic computing elements. A second contribution of the thesis is the proposal of test and diagnosis methods for both stochastic and quantum circuits. Existing verification (tomographic) methods of quantum circuits were targeting the reconstruction of the gate-lists. The repeated execution of the quantum circuit was followed by different but specific measurement at the circuit outputs. The similarities between stochastic and quantum circuits motivated the proposal of test and diagnosis methods that use a restricted set of measurement types, which minimise the number of circuit executions. The obtained simulation results show that the proposed validation methods improve the feasibility of quantum circuit tomography for small and medium size circuits. A third contribution of the thesis is the algorithmic formalisation of a problem encountered in teleportation-based quantum computing architectures. The teleportation results are probabilistic and require corrections represented as quantum gates from a particular set. However, there are known commutation properties of these gates with the gates used in the circuit. The corrections are not applied as dynamic gate insertions (during the circuit’s execution) into the gate-lists, but their effect is tracked through the circuit, and the corrections are applied only at circuit outputs. The simulation results show that the algorithmic solution is applicable for very large quantum circuits. Topological quantum computing (TQC) is based on a class of fault-tolerant quantum circuits that use the surface code as the underlying QECC. Quantum information is encoded in lattice-like structures and error protection is enabled by the topological properties of the lattice. The 3D structure of the lattice allows TQC computations to be visualised similarly to knot diagrams. Logical information is abstracted as strands and strand interactions (braids) represent logical quantum gates. Therefore, TQC circuits are abstracted using a geometrical description, which allows circuit input-output transformations (correlations) to be represented as geometric sub-structures. TQC design methods were not investigated prior to this work, and the thesis introduces the topological computational model by first analysing the necessary concepts. The proposed TQC design stack follows a top-down approach: an arbitrary quantum circuit is decomposed into the TQC supported gate set; the resulting circuit is mapped to a lattice of appropriate dimensions; relevant resulting topological properties are extracted and expressed using graphs and Boolean formulas. Both circuit representations are novel and applicable to TQC circuit synthesis and validation. Moreover, the Boolean formalism is broadened into a formal mechanism for proving circuit correctness. The thesis introduces TQC circuit synthesis, which is based on a novel logical gate geometric description, whose formal correctness is demonstrated. Two synthesis methods are designed, and both use a general planar representation of the circuit. Initial simulation results demonstrate the practicality and performance of the methods. An additional group of proposed design methods solves the problem of automatic correlation construction. The methods use validity criteria which were introduced and analysed beforehand in the thesis. Input-output correlations existing in the circuit are inferred using both the graph and the Boolean representation. The thesis extends the TQC state-of-the-art by recognising the importance of correlations in the validation process: correlation construction is used as a sub-routine for TQC circuit validation. The presented cross-layer validation procedure is useful when investigating both the QECC and the circuit, while a second proposed method is QECC-independent. Both methods are scalable and applicable even to very large circuits. The thesis completes with the analysis of TQC circuit identities, where the developed Boolean formalism is used. The proofs of former known circuit identities were either missing or complex, and the presented approach reduces the length of the proofs and represents a first step towards standardising them. A new identity is developed and detailed during the process of illustrating the known circuit identities. Reliable quantum circuits are a necessity for quantum computing to become reality, and specialised design methods are required to support the quest for scalable quantum computers. This thesis used a twofold approach towards this target: firstly by focusing on the probabilistic behaviour of quantum circuits, and secondly by considering the requirements of a promising quantum computing architecture, namely TQC. Both approaches resulted in a set of design methods enabling the investigation of reliable quantum circuits. The thesis contributes with the proposal of a new quantum simulation technique, no
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.
After decades of unhindered progress, classical computing has started facing significant hurdles both in terms of physical scalability and theoretical bounds of efficiency. Among the alternative models of computing, Quantum Computing, though proposed six decades ago, has recently started seeing potentials to progress beyond the limits of classical computing. This book discusses the theoretical bounds on the efficiency of low depth quantum circuits, one of the structurally simplest models of quantum computing. Three different properties are explored; universality in which one circuit can be used to simulate different circuits, fault detection in which certain kinds of faults in simple quantum gates can be detected and third, a fundamental theoretical limitation in the power of a popular quantum gate. These properties give us a better idea about simple quantum circuits, which are essentially building blocks for more complicated gadgets. The properties are analysed using novel techniques which will be useful to analyse other similar quantum circuits. Overall, this book will be useful to researchers of quantum circuit complexity and graduate students of theoretical computer science.
Quantum algorithms promise up to a superpolynomial factor speedup in computational tasks compared to their classical counterparts. Quantum computers are now approaching sizes where quantum algorithms can be demonstrated. As a result, the implementation of algorithms as quantum circuits has become an important research topic. In this work, we provide an overview of quantum computation and the design of quantum circuits. Properties unique to quantum computation are outlined. We introduce how a quantum algorithm is translated into a circuit and show measures used to assess the performance of quantum circuits.
Quantum computers promise to solve computational problems that are inaccessible to classical machines. The reliable execution of large-scale quantum algorithms requires robust operations in the presence of noise. Quantum error correction addresses this challenge by encoding quantum information redundantly across several physical qubits, such that errors can be detected and corrected during a computation. However, no single error-correcting code intrinsically supports an inherently robust, fault-tolerant implementation of all gates needed for universal quantum computing. Moreover, most modern approaches to both quantum error correction and the implementation of universal logical operations on encoded qubits rely on mid-circuit measurements and feed-forward control. These types of operations are technologically demanding, often slow, and prone to high error rates, and they limit the overall success probability of quantum algorithms. In this thesis, we develop fault-tolerant protocols and circuits for quantum error correction and universal logical computation, with an emphasis on implementations compatible with current hardware systems. We first focus on fault-tolerant quantum error correction on small codes. By comparing different protocols, we identify trade-offs between compact circuit constructions and their robustness to noise. We then assess their practical performance by comparing numerical simulations of noisy circuits with experimental data. Beyond the preservation of encoded information, we study how universal logic can be implemented in a fault-tolerant manner on encoded qubits. A central approach considered in this thesis is code switching, in which logical information is transferred between two distinct error-correcting codes that support complementary sets of inherently fault-tolerant logical gates. In combination, these codes give access to a complete universal fault-tolerant logical gate set. In this thesis, we determine conditions that allow us to construct fault-tolerant protocols for code switching. We adapt these protocols to the capabilities of current hardware, thereby enabling the first experimental realization of universal fault-tolerant logic by means of code switching. However, the overall performance of these schemes is limited by slow and noisy mid-circuit measurements. To address this bottleneck, we construct a suite of measurement-free fault-tolerant universal operations that do not require any measurements during the execution of an algorithm. In this framework, error correction and logical operations are implemented coherently as part of the quantum circuit, without mid-circuit measurements or in-sequence classical control. We introduce measurement-free code-switching schemes that can be scaled to arbitrary code distances through concatenation, and which thereby provide, in principle, arbitrarily strong error suppression. We further tailor our approach to the capabilities of current trapped-ion quantum processors. New circuit constructions support the first experimental implementation of a logical quantum algorithm without mid-circuit measurements, demonstrated through Grover's search on three encoded qubits. Together, the results of this thesis expand the toolbox for early fault-tolerant quantum computation. They show how protocols for quantum error correction and universal logical gates can be designed without mid-circuit measurements with realistic hardware constraints in mind, opening new pathways towards universal quantum logic.