(PHD, 2022)

Abstract:

Random circuit simulation, the task of replicating the output of a randomly chosen noiseless quantum computation, has been proposed as a path toward achieving quantum advantage: it is believed to be easy for quantum devices, but hard for classical ones. This thesis scrutinizes both sides of this belief. On the one hand, we investigate whether the task is classically hard—we find that, in certain non-trivial cases, it can actually be easy, complicating a potential general proof of hardness. On the other hand, we investigate whether the task can be easily accomplished on realistic quantum devices, which are subject to substantial noise rates—we find that, indeed, a version of the circuit simulation task can be salvaged even on a noisy quantum device performing the computation with low fidelity, as long as the noise meets certain conditions. Thus, this thesis emphasizes that, to construct a strong argument of quantum advantage via random circuit simulation on noisy quantum hardware, the core theoretical challenge remains proving lower bounds on the classical complexity of the task; doing so will require new ideas to circumvent the barriers presented by our work.

A key analytical technique we utilize for each of our results is the statistical mechanics method for random quantum circuits, which maps random quantum circuits made from local Haar-random gates to partition functions of classical statistical mechanical systems. This thesis demonstrates the utility of this method by applying it in several new ways. In some cases, we use it for heuristic reasoning about the behavior of random quantum circuits. In others, we go further and perform rigorous calculations of the resulting partition function, leading to precise technical conclusions about random quantum circuits, such as sharp bounds on the number of random gates needed to achieve the anti-concentration property.

]]>

(PHD, 2021)

Abstract:

In this thesis, we mainly discuss three topics in theoretical physics: a proof of the weak gravity conjecture, a basic statement in the string theory landscape using the black hole entropy, solving the critical *O*(3) model using the conformal bootstrap method involving semidefinite programming, and numerical simulation of the false vacuum decay using tensor network methods. Those topics cover different approaches to deep understanding of quantum field theories using concepts and methods of information theory, and computer science with classical and quantum computations.

]]>

(PHD, 2021)

Abstract:

In this thesis, we take a look at how quantum information theory can be used to study physical systems at both high and low energies.

In the first part of this thesis, we examine the structure of the low-energy subspaces of quantum many-body systems. We show that the existence of error-correcting properties in low-energy subspaces is a generic feature of quantum systems. Using the formalism of matrix product states, we construct explicit quantum error-detecting codes formed from the momentum eigenstates of a quantum many-body system.

We also examine how topological order can persist past the ground state space into the low-energy subspace of excited states by studying the No Low-Energy Trivial States (NLTS) conjecture. We prove a version of the NLTS conjecture under the assumption of symmetry protection. Moreover, we show that our symmetric NLTS result has implications for the performance of quantum variational optimization algorithms by using it to prove a bound on the Quantum Approximate Optimization Algorithm (QAOA).

In the second part of this thesis, we examine problems related to bulk reconstruction in holography and the black hole firewall paradox. Using the formalism of the tensor Radon transform, we devise and implement a numerical algorithm for reconstructing (perturbatively in AdS₃/CFT₂) the bulk metric tensor from a given boundary entropy profile.

We finally examine the black hole firewall problem from the perspective of quantum error-correction and quantum computational complexity. We argue that the state of the Hawking radiation has the special property of being computationally pseudorandom, meaning that it cannot be distinguished from the maximally mixed state by any efficient quantum computation. We show that this implies that each black hole has a natural structure as a quantum error-correcting code.

]]>

(PHD, 2020)

Abstract:

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.

]]>

(PHD, 2018)

Abstract:

Quantum chaos entails an entropic and computational obstruction to describing a system and thus is intrinsically difficult to characterize. An understanding of quantum chaos is fundamentally related to the mechanism of thermalization in many-body systems and the quantum nature of black holes. In this thesis we adopt the view that quantum information theory provides a powerful framework in which to elucidate chaos in strongly-interacting quantum systems.

We first push towards a more precise understanding of chaotic dynamics by relating different diagnostics of chaos, studying the time-evolution of random matrix Hamiltonians, and quantifying random matrix behavior in physical systems. We derive relations between out-of-time ordered correlation functions, spectral quantities, and frame potentials to relate the scrambling of quantum information, decay of correlators, and Haar-randomness. We give analytic expressions for these quantities in random matrix theory to explore universal aspects of late-time dynamics. Motivated by our random matrix results, we define *k*-invariance in order to capture the onset of random matrix behavior in physical systems.

We then refine our diagnostics in order to study chaotic systems with symmetry by considering Haar-randomness with respect to quotients of the unitary group, and in doing so we generalize our quantum information machinery. We further consider extended random matrix ensembles in the context of strongly-interacting quantum systems dual to black holes. Lastly, we study operator growth in classes of random quantum circuits.

]]>

(PHD, 2018)

Abstract:

The field of tensor networks, kicked off in 1992 by Steve White’s invention of the spectacularly successful density matrix renormalization group (DMRG) algorithm, has exploded in popularity in recent years. Tensor networks are poised to play a role in helping us solve some of the greatest open physics problems of our time, such as understanding the nature of high-temperature superconductivity and illuminating a theory of quantum gravity. DMRG and extensions based on a class of variational states known as tensor network states have been indispensable tools in helping us understand both numerically and theoretically the properties of complicated classical and quantum many-body systems. However, practical challenges to these techniques still remain, and algorithmic developments are needed before tensor network algorithms can be applied to more physics problems. In this thesis we present a variety of recent advancements to tensor network algorithms.

First we describe a DMRG-like algorithm for noninteracting fermions. Noninteracting fermions, naturally being gapless and therefore having high levels of entanglement, are actually a challenging setting for standard DMRG algorithms, and we believe this new algorithm can help with tensor network calculations in that setting.

Next we explain a new algorithm called the variational uniform matrix product state (VUMPS) algorithm that is a DMRG-like algorithm that works directly in the thermodynamic limit, improving upon currently available MPS-based methods for studying infinite 1D and quasi-1D quantum many-body systems.

Finally, we describe a variety of improvements to algorithms for contracting 2D tensor networks, a common problem in tensor network algorithms, for example for studying 2D classical statistical mechanics problems and 2D quantum many-body problems with projected entangled pair states (PEPS). One is a new variant of the corner transfer matrix renormalization group (CTMRG) algorithm of Nishino and Okunishi that improves the numerical stability for contracting asymmetric two-dimensional tensor networks compared to the most commonly used method. Another is the application of the VUMPS algorithm to contracting 2D tensor networks. The last is a new alternative to CTMRG, where the tensors are solved for with eigenvalue equations instead of a power method, which we call the fixed point corner method (FPCM). We present results showing the transfer matrix VUMPS algorithm and FPCM significantly improve upon the convergence time of CTMRG. We expect these algorithms will play an important role in expanding the set of 2D classical and 2D quantum many-body problems that can be addressed with tensor networks.

]]>

(PHD, 2018)

Abstract:

The results of this thesis concern the real-world realization of quantum computers, specifically how to build their “hard drives” or quantum memories. These are many-body quantum systems, and their building blocks are qubits, the same way bits are the building blocks of classical computers.

Quantum memories need to be robust against thermal noise, noise that would otherwise destroy the encoded information, similar to how strong magnetic field corrupts data classically stored in magnetic many-body systems (e.g., in hard drives). In this work I focus on a subset of many-body models, called quantum doubles, which, in addition to storing the information, could be used to perform the steps of the quantum computation, i.e., work as a “quantum processor”.

In the first part of my thesis, I investigate how long a subset of quantum doubles (qudit surface codes) can retain the quantum information stored in them, referred to as their memory time. I prove an upper bound for this memory time, restricting the maximum possible performance of qudit surface codes.

Then, I analyze the structure of quantum doubles, and find two interesting properties. First, that the high-level description of doubles, utilizing only their quasi-particles to describe their states, disregards key components of their microscopic properties. In short, quasi-particles (anyons) of quantum doubles are not in a one-to-one correspondence with the energy eigenstates of their Hamiltonian. Second, by investigating phase transitions of a simple quantum double, D(S_{3}), I map its phase diagram, and interpret the physical processes the theory undergoes through terms borrowed from the Landau theory of phase transitions.

]]>

(PHD, 2018)

Abstract:

Combining quantum information theory (QIT) with thermodynamics unites 21st-century technology with 19th-century principles. The union elucidates the spread of information, the flow of time, and the leveraging of energy. This thesis contributes to the theory of quantum thermodynamics, particularly to QIT thermodynamics. The thesis also contains applications of the theory, wielded as a toolkit, across physics. Fields touched on include atomic, molecular, and optical physics; nonequilibrium statistical mechanics; condensed matter; high-energy physics; and chemistry. I propose the name *quantum steampunk* for this program. The term derives from the steampunk genre of literature, art, and cinema that juxtaposes futuristic technologies with 19th-century settings.

]]>

(PHD, 2018)

Abstract:

Understanding many-body quantum systems is one of the most challenging problems in contemporary condensed-matter physics. Tensor network representation of quantum states and operators are taking central stage in this pursuit and beyond. They prove to be a powerful numerical and conceptual tool, and indeed a new language altogether. This thesis investigates various aspects of these representations by focusing on two specific problems: the first half of the thesis is devoted to examining how ‘stable’ a tensor network representation is for two-dimensional quantum states with topological order, and the second half explores the representability of various unitary loop operators with tensor networks.

In the numerical usage of the tensor networks, the tensor is varied as to find the representation of the ground states of the given Hamiltonian. In chapter two and three of this thesis we show that such a numerical program for topological phases can be ‘ill-posed’. We show that tensor network can be an unstable representation for a topological phase: even an infinitesimal variation in the representation results in the loss of topological order, completely or partially. We diagnose this problem by identifying the exact causes of this instability, and find that it is only tensor variations in certain directions that result in instability, because they result in the condensation of bosonic quasi-particles of the phase. Such unstable variations are characterized by two properties: (1) they can replace a tensor in the tensor network without making the network collapse, and (2) their presence in the network represents the presence of a non-trivial topological charge. We prove that the general tensor representation of all string-net models suffer with such instabilities. We propose an exact mathematical operator to project out all such unstable variations and show its efficacy for a few models by direct calculations. Such an operator can be useful in numerical programs involving such tensor representations. We also point out that such variations play a crucial role in simulating topological phase transitions and their presence can be vital in an accurate simulation.

In chapter four and five of this thesis we focus on the representability of unitary loop operators by tensor networks. Such operators not only provide an important tool in the study of dynamical process in one-dimensional systems, but also in understanding and classification of symmetry protected topological phases in two dimensions. To characterize all such operators, we find a necessary and sufficient condition for any loop tensor network operator of a given length to represent a unitary operator. In particular, it is shown that all unitary operators that map local operators to local operators (locality-preserving) can always be represented by a tensor network. Locality-preserving unitary loop operators are classified by a rational index called the GNVW index defined in Ref. [1] which measures how much information ‘flows’ along the loop. We define Rank-Ratio index for tensor network operators and show that it is completely equivalent to the GNVW index. Therefore, GNVW index of a unitary operator can be easily extracted from its tensor network representation. We find that, other than representing locality-preserving unitary maps, tensor networks can also represent unitary operators that map local operators to global (non-local) operators. These tensor network operators are found to have a long-ranged order similar to tensors that represent topological tensor network states in two dimensions.

]]>

(PHD, 2018)

Abstract:

This thesis is devoted to studying a class of quantum error-correcting codes — topological quantum codes. We explore the question of how one can achieve fault- tolerant quantum computation with topological codes. We treat quantum error-correcting codes not only as a compelling ingredient needed to build a quantum computer, but also as a useful theoretical tool in other areas of physics. In particular, we explore what insights topological codes can provide into challenging questions, such as the classification of quantum phases of matter.

In this thesis, we focus on a family of topological codes — color codes, which are particularly intriguing due to the rich physics they display and their computational power. We start by introducing color codes and explaining their basic properties. Then, we show how to perform fault-tolerant universal quantum computation with three-dimensional color codes by transverse gates and code switching. We later compare the resource overhead of the code-switching approach with that of a state distillation scheme. We discuss how to perform error correction with the toric and color codes, as well as introduce local decoders for those two families of codes. By exploiting a connection between error correction and statistical mechanics we estimate the storage threshold error rates for bit-flip and phase-flip noise in the three-dimensional color code. We finish by showing that the color and toric code families in d dimensions are equivalent in a sense of local unitary transformations and explore implications of this equivalence.

]]>

(PHD, 2018)

Abstract: This thesis introduces a new phase transitions in three dimensional quantum gravity. The main technical tools comes from the spectral theory of hyperbolic manifolds.

]]>

(PHD, 2016)

Abstract: The work in this thesis splits naturally into two parts: (1) experimentally oriented work consisting of experimental proposals for systems that could be used to implement quantum information tasks with current technology, and (2) theoretical work focusing on universal fault-tolerant quantum computers which we hope can be scaled as experimental capabilities continue to move forward.

]]>

(PHD, 2013)

Abstract: In this thesis, I will discuss how information-theoretic arguments can be used to produce sharp bounds in the studies of quantum many-body systems. The main advantage of this approach, as opposed to the conventional field-theoretic argument, is that it depends very little on the precise form of the Hamiltonian. The main idea behind this thesis lies on a number of results concerning the structure of quantum states that are conditionally independent. Depending on the application, some of these statements are generalized to quantum states that are approximately conditionally independent. These structures can be readily used in the studies of gapped quantum many-body systems, especially for the ones in two spatial dimensions. A number of rigorous results are derived, including (i) a universal upper bound for a maximal number of topologically protected states that is expressed in terms of the topological entanglement entropy, (ii) a first-order perturbation bound for the topological entanglement entropy that decays superpolynomially with the size of the subsystem, and (iii) a correlation bound between an arbitrary local operator and a topological operator constructed from a set of local reduced density matrices. I also introduce exactly solvable models supported on a three-dimensional lattice that can be used as a reliable quantum memory.

]]>

(PHD, 2013)

Abstract:

This thesis addresses whether it is possible to build a robust memory device for quantum information. Many schemes for fault-tolerant quantum information processing have been developed so far, one of which, called topological quantum computation, makes use of degrees of freedom that are inherently insensitive to local errors. However, this scheme is not so reliable against thermal errors. Other fault-tolerant schemes achieve better reliability through active error correction, but incur a substantial overhead cost. Thus, it is of practical importance and theoretical interest to design and assess fault-tolerant schemes that work well at finite temperature without active error correction.

In this thesis, a three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum information can be retrieved faithfully for a memory time which grows exponentially with the square of the inverse temperature. In contrast, for previously known types of topological quantum storage in three or fewer spatial dimensions the memory time scales exponentially with the inverse temperature, rather than its square.

This spin model exhibits a previously unexpected topological quantum order, in which ground states are locally indistinguishable, pointlike excitations are immobile, and the immobility is not affected by small perturbations of the Hamiltonian. The degeneracy of the ground state, though also insensitive to perturbations, is a complicated number-theoretic function of the system size, and the system bifurcates into multiple noninteracting copies of itself under real-space renormalization group transformations. The degeneracy, the excitations, and the renormalization group flow can be analyzed using a framework that exploits the spin model’s symmetry and some associated free resolutions of modules over polynomial algebras.

]]>

(PHD, 2013)

Abstract:

Quantum computing offers powerful new techniques for speeding up the calculation of many classically intractable problems. Quantum algorithms can allow for the efficient simulation of physical systems, with applications to basic research, chemical modeling, and drug discovery; other algorithms have important implications for cryptography and internet security.

At the same time, building a quantum computer is a daunting task, requiring the coherent manipulation of systems with many quantum degrees of freedom while preventing environmental noise from interacting too strongly with the system. Fortunately, we know that, under reasonable assumptions, we can use the techniques of quantum error correction and fault tolerance to achieve an arbitrary reduction in the noise level.

In this thesis, we look at how additional information about the structure of noise, or “noise bias,” can improve or alter the performance of techniques in quantum error correction and fault tolerance. In Chapter 2, we explore the possibility of designing certain quantum gates to be extremely robust with respect to errors in their operation. This naturally leads to structured noise where certain gates can be implemented in a protected manner, allowing the user to focus their protection on the noisier unprotected operations.

In Chapter 3, we examine how to tailor error-correcting codes and fault-tolerant quantum circuits in the presence of dephasing biased noise, where dephasing errors are far more common than bit-flip errors. By using an appropriately asymmetric code, we demonstrate the ability to improve the amount of error reduction and decrease the physical resources required for error correction.

In Chapter 4, we analyze a variety of protocols for distilling magic states, which enable universal quantum computation, in the presence of faulty Clifford operations. Here again there is a hierarchy of noise levels, with a fixed error rate for faulty gates, and a second rate for errors in the distilled states which decreases as the states are distilled to better quality. The interplay of of these different rates sets limits on the achievable distillation and how quickly states converge to that limit.

]]>

(PHD, 2011)

Abstract:

We study two novel paradigms in quantum error correction and quantum cryptography — approximate quantum error correction and noisy-storage cryptography — which explore alternate approaches for dealing with quantum noise. Approximate quantum error correction seeks to relax the constraint of perfect error correction and construct codes that might be better adapted to correct for specific noise models. Noisy-storage cryptography relies on the power of quantum noise to execute two-party cryptographic tasks securely.

Motivated by examples of approximately correcting codes, which make use of fewer physical resources than perfect codes and still obtain comparable levels of fidelity, we study the problem of finding and characterizing such codes in general. We construct for the first time a universal, near-optimal recovery map for approximate quantum error correction (AQEC), with optimality defined in terms of worst-case fidelity. Using the analytical form of this recovery, we also obtain easily verifiable conditions for AQEC. This in turn leads to a simple algorithm for identifying good approximate codes, without having to perform a difficult optimization over all recovery maps for every possible encoding.

Noisy-storage cryptography envisions a setting where two-party cryptographic protocols can be securely implemented based solely on the assumption that the quantum storage device possessed by either party is noisy and bounded. Here, we construct two-party protocols (using higher-dimensional states) that are secure even when a dishonest player can store all but a small fraction of the information transmitted during the protocol, in his noiseless quantum memory. We also show that when his memory is noisy, security can be extended to a larger class of noisy quantum memories. Our result demonstrates that the physical limits of the quantum noisy-storage model are indeed achievable, albeit asymptotically.

We also describe our investigations on obtaining strong entropic uncertainty relations using symmetric complementary bases. Uncertainty relations are an important and useful resource in analyzing the security of quantum cryptographic protocols, in addition to being of interest from a foundational standpoint. We present a novel construction of sets of symmetric, complementary bases in dimension d = 2^{n}, which are cyclically permuted under the action of a unitary transformation. We also obtain new lower bounds for uncertainty relations in terms of the min-entropy, which are tight for specific instances of our construction.

]]>

(PHD, 2011)

Abstract: In this thesis, we address the problem of solving for the properties of interacting quantum many-body systems in thermal equilibrium. The complexity of this problem increases exponentially with system size, limiting exact numerical simulations to very small systems. To tackle more complex systems, one needs to use heuristic algorithms that approximate solutions to these systems. Belief propagation is one such algorithm that we discuss in chapters 2 and 3. Using belief propagation, we demonstrate that it is possible to solve for static properties of highly correlated quantum many-body systems for certain geometries at all temperatures. In chapter 4, we generalize the multiscale renormalization ansatz to the anyonic setting to solve for the ground state properties of anyonic quantum many-body systems. The algorithms we present in chapters 2, 3, and 4 are very successful in certain settings, but they are not applicable to the most general quantum mechanical systems. For this, we propose using quantum computers as we discuss in chapter 5. The dimension reduction algorithm we consider in chapter 5 enables us to prepare thermal states of any quantum many-body system on a quantum computer faster than any previously known algorithm. Using these thermal states as the initialization of a quantum computer, one can study both static and dynamic properties of quantum systems without any memory overhead.

]]>

(PHD, 2010)

Abstract:

Quantum coherence is the key ingredient for characteristically quantum effects. It allows for radically different technologies than those using classical systems, including quantum communication, quantum computation, and other devices with quantum control. Quantum coherences are, however, extremely fragile and susceptible to damage from environmental noise. The success of any experiment or technology based on quantum phenomena demands careful preservation of quantum coherences within the system. The study of the effects of noise on a quantum system, and how to prevent loss of coherence is the central theme of this thesis.

Starting from basic principles behind how information is stored in a system and what it means for it to be preserved, we build up a framework that allows one to understand what kind of information can survive through a noise process. The resulting elegant matrix-algebraic description of information-preserving structures within a quantum system characterizes codes that can perfectly preserve information in the presence of noise. Our framework encompasses examples like pointer states, noiseless subsystems and error-correcting codes. Furthermore, it leads to a simple, analytical approach to approximate quantum error correction. While perfect quantum error correction is a standard method used to protect information from noise, approximate error correction allows for the use of a smaller quantum system to store the same information, without sacrificing much in resilience against noise.

Asking what happens to information stored in a quantum system when the encoding and recovery procedures in error correction are also noisy leads to the concept of fault tolerance. Fault tolerance provides schemes, built upon quantum error correction, that enable accurate simulation of a quantum computation even when the elementary gates are imperfect. Realistic gates used to build a fault-tolerant circuit, however, often require additional noise-suppression techniques in order for any quantum effects to be observed at all. A common technique is dynamical decoupling. We demonstrate how dynamical decoupling in elementary gates can be rigorously accounted for in the fault-tolerance analysis, and show how, under the right conditions, it can lead to fault-tolerant circuits with less stringent noise and resource requirements.

]]>

(PHD, 2009)

Abstract:

Nonlocality refers to correlations between spatially separated parties that are stronger than those explained by the existence of local hidden variables. Quantum mechanics is known to allow some nonlocal correlations between particles in a phenomena known as entanglement. We explore several aspects of nonlocality in general and how they relate to quantum mechanics.

First, we construct a hierarchy of theories with nonlocal correlations stronger than those allowed in quantum mechanics and derive several results about these theories. We show that these theories include codes that can store an amount of information exponential in the number of physical bits used. We use this result to demonstrate an unphysical consequence of theories with stronger-than-quantum correlations: learning even an approximate description of states in such theories would be practically impossible.

Next, we consider the difficult problem of determining whether specific correlations are nonlocal. We present a novel learning algorithm and show that it provides an outer bound on the set of local states, and can therefore be used to identify some nonlocal states.

Finally, we put nonlocal correlations to work by showing that the entanglement present in the vacuum of a quantum field can be used to detect spacetime curvature. We quantify how the entangling power of the quantum field varies as a function of spacetime curvature.

]]>

(PHD, 2009)

Abstract:

In this work, we describe a method to achieve fault tolerant measurement based quantum computation in two and three dimensions. The proposed scheme has an threshold of 7.8*10^-3 and poly-logarithmic overhead scaling. The overhead scaling below the threshold is also studied. The scheme uses a combination of topological error correction and magic state distillation to construct a universal quantum computer on a qubit lattice. The chapters on measurement based quantum computation are written in review form with extensive discussion and illustrative examples.

In addition, we describe and analyze a family of entanglement purification protocols that provide a flexible trade-off between overhead, threshold and output quality. The protocols are studied analytically, with closed form expressions for their threshold.

]]>

(PHD, 2008)

Abstract:

All systems are open to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. In some cases the environment’s only effect is to add a weak relaxation mechanism and thus can be ignored for short timescales. In others, however, the presence of the environment can fundamentally alter the behavior of the system. Such is the case in mesoscopic superconductors where the environment can stabilize superconductivity and in spin-boson systems where the environment induces a localization transition. Likewise, in technological applications we are often interested in systems operating far from equilibrium. Here the environment might act as a particle reservoir or strong driving force.

In all these examples, we need accurate methods to describe the influence of the environment on the system and to solve for the resulting dynamics or equilibrium states. In this thesis, we develop computational and conceptual approaches to efficiently simulate quantum systems in contact with an environment. Our starting point is the use of numerical renormalization techniques. Thus, we restrict our attention to one-dimensional lattices or small quantum systems coupled to an environment. We have developed several complementary algorithms: a superoperator renormalization algorithm for simulating real-time Markovian dynamics and for calculating states in thermal equilibrium; a blocking algorithm for simulating integro-differential equations with long-time memory; and a tensor network algorithm for branched lattices, which can be used to simulate strongly dissipative systems. Further, we provide support for an idea that to generically and accurately simulate the real-time dynamics of strongly dissipative systems, one has to include all or part of the environment within the simulation. In addition, we discuss applications and open questions.

]]>

(PHD, 2007)

Abstract:

Computers have led society to the information age revolutionizing central aspects of our lives from production and communication to education and entertainment. There exist, however, important problems which are intractable with the computers available today and, experience teaches us, will remain so even with the more advanced computers we can envision for tomorrow.

Quantum computers promise speedups to some of these important but classically intractable problems. Simulating physical systems, a problem of interest in a diverse range of areas from testing physical theories to understanding chemical reactions, and solving number factoring, a problem at the basis of cryptographic protocols that are used widely today on the internet, are examples of applications for which quantum computers, when built, will offer a great advantage over what is possible with classical computer technology.

The construction of a quantum computer of sufficient scale to solve interesting problems is, however, especially challenging. The reason for this is that, by its very nature, operating a quantum computer will require the coherent control of the quantum state of a very large number of particles. Fortunately, the theory of quantum error correction and fault-tolerant quantum computation gives us confidence that such quantum states can be created, can be stored in memory and can also be manipulated provided the quantum computer can be isolated to a sufficient degree from sources of noise.

One of the central results in the theory of fault-tolerant quantum computation, the quantum threshold theorem shows that a noisy quantum computer can accurately and efficiently simulate any ideal quantum computation provided that noise is weakly correlated and its strength is below a critical value known as the quantum accuracy threshold. This thesis provides a simpler and more transparent non-inductive proof of this theorem based on the concept of level reduction. This concept is also used in proving the quantum threshold theorem for coherent and leakage noise and for quantum computation by measurements. In addition, the proof provides a methodology which allows us to establish improved rigorous lower bounds on the value of the quantum accuracy threshold.

]]>

(PHD, 2007)

Abstract: This thesis is primarily a study of the measurement theory of non-Abelian anyons through interference experiments. We give an introduction to the theory of anyon models, providing all the formalism necessary to apply standard quantum measurement theory to such systems. This formalism is then applied to give a detailed analysis of a Mach-Zehnder interferometer for arbitrary anyon models. In this treatment, we find that the collapse behavior exhibited by a target anyon in a superposition of states is determined by the monodromy of the probe anyons with the target. Such measurements may also be used to gain knowledge that would help to properly identify the anyon model describing an unknown system. The techniques used and results obtained from this model interferometer have general applicability, and we use them to also describe the interferometry measurements in a two point-contact interferometer proposed for non-Abelian fractional quantum Hall states. Additionally, we give the complete description of a number of important examples of anyon models, as well as their corresponding quantities that are relevant for interferometry. Finally, we give a partial classification of anyon models with small numbers of particle types.

]]>

(PHD, 2007)

Abstract:

Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality.

Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck’s inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck’s inequality.

In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB’s and AC’s violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson’s bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.

]]>

(PHD, 2006)

Abstract: We present the unification of many previously disparate results in noisy quantum Shannon theory and the unification of all of noiseless quantum Shannon theory. More specifically we deal here with bipartite, unidirectional, and memoryless quantum Shannon theory. We find all the optimal protocols and quantify the relationship between the resources used, both for the one-shot and for the ensemble case, for what is arguably the most fundamental task in quantum information theory: sharing entangled states between a sender and a receiver. We find that all of these protocols are derived from our one-shot superdense coding protocol and relate nicely to each other. We then move on to noisy quantum information theory and give a simple, direct proof of the “mother” protocol, or rather her generalization to the Fully Quantum Slepian-Wolf protocol(FQSW). FQSW simultaneously accomplishes two goals: quantum communication-assisted entanglement distillation, and state transfer from the sender to the receiver. As a result, in addition to her other “children,” the mother protocol generates the state merging primitive of Horodecki, Oppenheim, and Winter as well as a new class of distributed compression protocols for correlated quantum sources, which are optimal for sources described by separable density operators. Moreover, the mother protocol described here is easily transformed into the so-called “father” protocol, demonstrating that the division of single-sender/single-receiver protocols into two families was unnecessary: all protocols in the family are children of the mother.

]]>

(PHD, 2006)

Abstract:

This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution. The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively. The next four chapters present achievable rates for quantum codes in various scenarios. The final chapter is dedicated to an upper bound on the quantum channel capacity.

Chapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists. These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is impossible.

Chapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes. By studying the scaling of the optimal block length as a function of the channel’s parameters, we develop a heuristic for designing even better codes.

Chapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called “twisted states.”

In Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible.

Finally, Chapter 7 presents a new upper bound on the quantum channel capacity that is both additive and convex, and which can be interpreted as the capacity of the channel for communication given access to side channels from a class of zero capacity “cloning” channels. This “clone assisted capacity” is equal to the unassisted capacity for channels that are degradable, which we use to find new upper bounds on the capacity of a depolarizing channel.

]]>

(PHD, 2005)

Abstract:

Following their divorce, Alice and Bob would like to split some of their possessions by flipping a coin. Unwilling to meet in person, and without a trusted third party, they must figure out a scheme to flip the coin over a telephone that guarantees that neither party can cheat.

The preceding scenario is the traditional definition of two-party coin-flipping. In a classical setting, without limits on the available computational power, one player can always guarantee a coin-flipping victory by cheating. However, by employing quantum communication it is possible to guarantee, with only information-theoretic assumptions, that neither party can win by cheating, with a probability greater than two thirds. Along with the description of such a protocol, this thesis derives a tight lower bound on the bias for a large family of quantum weak coin-flipping protocols, proving such a protocol optimal within the family. The protocol described herein is an improvement and generalization of one examined by Spekkens and Rudolph. The key steps of the analysis involve Kitaev’s description of quantum coin-flipping as a semidefinite program whose dual problem provides a certificate that upper bounds the amount of cheating for each party.

In order for such quantum protocols to be viable, though, a number of practical obstacles involving the communication and processing of quantum information must be resolved. In the second half of this thesis, a scheme for processing quantum information is presented, which uses non-abelian anyons that are the magnetic and electric excitations of a discrete-group quantum gauge theory. In particular, the connections between group structure and computational power are examined, generalizing previous work by Kitaev, Ogburn and Preskill. Anyon based computation has the advantage of being topological, which exponentially suppresses the rate of decoherence and the errors associated with the elementary quantum gates. Though no physical systems with such excitations are currently known to exist, it remains an exciting open possibility that such particles could be either engineered or discovered in exotic two-dimensional systems.

]]>

(PHD, 2004)

Abstract: Quantum information theory is concerned with identifying how quantum mechanical resources, such as entangled quantum states, can be utilized for a number of information processing tasks, including data storage, computation, communication, and cryptography. Efficient quantum algorithms and protocols have been developed for performing some tasks (e.g., factoring large numbers, securely communicating over a public channel, and simulating quantum mechanical systems) that appear to be very difficult with just classical resources. In addition to identifying the separation between classical and quantum computational power, much of the theoretical focus in this field over the last decade has been concerned with finding novel ways of encoding quantum information that are robust against errors, which is an important step toward building practical quantum information processing devices. In this thesis I present some results on the quantum error-correcting properties of oscillator codes (also described as symplectic lattice codes) and toric codes. Any harmonic oscillator system, such as a mode of light, can be encoded with quantum information via symplectic lattice codes that are robust against shifts in the system’s continuous quantum variables. I show the existence of lattice codes whose achievable rates match the one-shot coherent information over the Gaussian quantum channel. Also, I construct a family of symplectic self-dual lattices and search for optimal encodings of quantum information distributed between several oscillators. Toric codes provide encodings of quantum information into two-dimensional spin lattices that are robust against local clusters of errors and which require only local quantum operations for error correction. Numerical simulations of this system under realistic error models provide a calculation of the accuracy threshold for quantum memory using toric codes, which can be related to phase transitions in particular condensed matter models. I also present a local classical processing scheme for correcting errors on toric codes, which demonstrates that quantum information can be maintained in two dimensions by purely local quantum and classical resources.

]]>

(PHD, 2004)

Abstract:

This thesis develops restrictions governing how a quantum system, jointly held by two parties, can be altered by the local actions of those parties, under assumptions about how they may communicate. These restrictions are expressed as constraints involving the eigenvalues of the density matrix of one of the parties. The thesis is divided into two parts.

Part I (Chapters 1-4) explores what is possible if the two parties may use only classical communication. A well-known result by M. Nielsen says that this is intimately connected to the mathematical notion of majorization. If entanglement catalysis is permitted, then the relevant notion is an extension of majorization known as the trumping relation. In Part I, we study the structure of the trumping relation.

Part II (Chapters 5-9) considers the question of how a state can change as a result of quantum communication between the parties; i.e., one party sends the other a portion of the jointly held quantum system. Given the spectrum of the initial state, it turns out that the possible spectra of the final state are given by the solutions to linear inequalities. We develop a method for deriving these inequalities, using a variational principle. In order to apply this principle, we need to know when certain subvarieties of a Grassmannian variety intersect, which can be regarded as a problem in Grassmannian cohomology. We discuss this cohomology and derive the conditions for nontrivial intersections. Finally, we illustrate how these intersections give rise to the desired inequalities.

]]>

(PHD, 2004)

Abstract:

Quantum mechanical applications range from quantum computers to quantum key distribution to teleportation. In these applications, quantum error correction is extremely important for protecting quantum states against decoherence. Here I present two main results regarding quantum error correction protocols.

The first main topic I address is the development of continuous-time quantum error correction protocols via combination with techniques from quantum control. These protocols rely on weak measurement and Hamiltonian feedback instead of the projective measurements and unitary gates usually assumed by canonical quantum error correction. I show that a subclass of these protocols can be understood as a quantum feedback protocol, and analytically analyze the general case using the stabilizer formalism; I show that in this case perfect feedback can perfectly protect a stabilizer subspace. I also show through numerical simulations that another subclass of these protocols does better than canonical quantum error correction when the time between corrections is limited.

The second main topic is development of improved overhead results for fault-tolerant computation. In particular, through analysis of topological quantum error correcting codes, it will be shown that the required blowup in depth of a noisy circuit performing a fault-tolerant computation can be reduced to a factor of O(log log L), an improvement over previous results. Showing this requires investigation into a local method of performing fault-tolerant correction on a topological code of arbitrary dimension.

]]>

(PHD, 2004)

Abstract: In this thesis I explore two different types of limits on the time complexity of quantum computation—that is, limits on how much time is required to perform a given class of quantum operations on a quantum system. Upper limits can be found by explicit construction; I explore this approach for the problem of determining whether two graphs are isomorphic. Finding lower limits, on the other hand, usually requires appeal to some fundamental principle of the operation under consideration; I use this approach to derive lower limits placed by the requirements of relativistic causality on the time required for implementation of some nonlocal quantum operations. In some situations these limits are attainable, but for other physical spacetime geometries we exhibit classes of operations which do not violate relativistic causality but which are nevertheless not implementable.

]]>

(PHD, 2004)

Abstract: This thesis studies classical communication over quantum channels. Chapter 1 describes an algebraic technique which extends several previously known qubit channel capacity results to the qudit quantum channel case. Chapter 2 derives a formula for the relative entropy function of two qubit density matrices in terms of their Bloch vectors. The application of the Bloch vector relative entropy formula to the determination of Holevo-Schumacher-Westmoreland (HSW) capacities for qubit quantum channels is discussed. Chapter 3 outlines several numerical simulation results which support theoretical conclusions and conjectures discussed in Chapters 1 and 2. Chapter 4 closes the thesis with comments, examples and discussion on the additivity of Holevo Chi and the HSW channel capacity.

]]>

(PHD, 2003)

Abstract:

Entanglement is a key resource in the emerging field of Quantum Information. The strong correlations between systems described by an entangled state allow us to perform certain tasks more efficiently than it would be possible by using only classical resources. This is why the characterization of entanglement is one of the most important problems in Quantum Information.

In this thesis, we analyze several aspects of entanglement. First, we introduce a new family of criteria to determine if a bipartite mixed state is entangled or not. This family consists of a sequence of tests that can be implemented efficiently, and has the property that all entangled states can be detected by some test in the sequence.

Each test in the family can be stated as a semidefinite program, which is a class of convex optimization problems. The duality structure of these programs allows us to explicitly construct an entanglement witness that proves entanglement of a state, whenever the state fails one of the tests in the sequence. The entanglement witnesses constructed in this manner have well-defined algebraic properties that can be used to give a characterization of the interior of the set of all possible entanglement witnesses, as well as the set of strictly positive bihermitian forms and the set of strictly positive maps.

We also study deterministic transformations of three-qubit pure state when only local operations and classical communication (LOCC) are allowed. We derive strong constraints that the operations and states involved must satisfy, and we apply these results to characterize the set of real states that can be obtained from the GHZ state by LOCC.

]]>

(PHD, 2002)

Abstract: We consider several examples of metallic systems that exhibit non-Fermi-liquid behavior. In these examples the system is not a Fermi liquid due to the presence of a “hidden” order. The primary models are density waves with an odd-frequency-dependent order parameter and density waves with d-wave symmetry. In the first model, the same-time correlation functions vanish and there is a conventional Fermi surface. In the second model, the gap vanishes at the nodes. We derive the phase diagrams and study the thermodynamic and kinetic properties. We also consider the effects of competing orders on the phase diagram when the underlying microscopic interaction has a high symmetry.

]]>

(PHD, 2002)

Abstract:

Quantum information science explores ways in which quantum physical laws can be harnessed to control the acquisition, transmission, protection, and processing of information. This field has seen explosive growth in the past several years from progress on both theoretical and experimental fronts. Essential to this endeavor are methods for controlling quantum information.

In this thesis, I present three new approaches for controlling quantum information. First, I present a new protocol for continuously protecting unknown quantum states from noise. This protocol combines and expands ideas from the theories of quantum error correction and quantum feedback control. The result can outperform either approach by itself. I generalize this protocol to all known quantum stabilizer codes, and study its application to the three-qubit repetition code in detail via Monte Carlo simulations.

Next, I present several new protocols for controlling quantum information that are fault-tolerant. These protocols require only local quantum processing due to the topological properties of the quantum error correcting codes upon which they are built. I show that each protocol’s fault-dependence behavior exhibits an order-disorder phase transition when mapped onto an associated statistical-mechanical model. I review the critical error rates of these protocols found by numerical study of the associated models, and I present new analytic bounds for them using a self-avoiding random walk argument. Moreover, I discuss fault-tolerant procedures for encoding, error-correction, computing, and decoding quantum information using these protocols, and calculate the accuracy threshold of fault-tolerant quantum memory for protocols using them.

I end by presenting a new class of quantum algorithms that solve combinatorial optimization problems solely by measurement. I compute the running times of these algorithms by establishing an explicit dynamical model for the measurement process. This model, the digitized version of von Neumann’s measurement model, is recognized as Kitaev’s phase estimation algorithm. I show that the running times of these algorithms are closely related to the running times of adiabatic quantum algorithms. Finally, I present a two-measurement algorithm that achieves a quadratic speedup for Grover’s unstructured search problem.

]]>

(PHD, 2002)

Abstract:

In chapter 2 various parameterizations for the orbits under local unitary transformations of three-qubit pure states are analyzed. It is shown that the entanglement monotones of any multipartite pure state uniquely determine the orbit of that state. It follows that there must be an entanglement monotone for three-qubit pure states which depends on the Kempe invariant defined in [1]. A form for such an entanglement monotone is proposed. A theorem is proved that significantly reduces the number of entanglement monotones that must be looked at to find the maximal probability of transforming one multipartite state to another.

In chapter 3 Grover’s unstructured quantum search algorithm is generalized to use an arbitrary starting superposition and an arbitrary unitary matrix. A formula for the probability of the generalized Grover’s algorithm succeeding after n iterations is derived. This formula is used to determine the optimal strategy for using the unstructured quantum search algorithm. The speedup obtained illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. The analysis is extended to the case of a society of k quantum searches acting in parallel.

In chapter 4 the positive map Г : p → (Trρ) - ρ is introduced as a separability criterion. Any separable state is mapped by the tensor product of Г and the identity in to a non-negative operator, which provides a necessary condition for separability. If Г acts on a two-dimensional subsystem, then it is equivalent to partial transposition and therefore also sufficient for 2 x 2 and 2 x 3 systems. Finally, a connection between this map for two qubits and complex conjugation in the “magic” basis [2] is displayed.

]]>

(BS, 2000)

Abstract: The physics of information and computation has been a recognized discipline for several decades. This is not surprising. Information is, after all, encoded in the state of a physical system. Our abilities to compute and process information depend directly on the physics of the system. A computation is something that can be carried out on an actual physically realizable device. Hence the study of information and computation is linked to the study of the underlying physical process. From the perspective of developing state-of-the-art computing technology, study of the principles of physics and material science is essential. From a more abstract and theoretical point of view, there have been noteworthy milestones in our understanding of how physics constrains our ability to use and manipulate information e.g. Landauer’s Principle, Reversible Computation, Explanation of Maxwell’s Daemon, etc.

]]>

(PHD, 1998)

Abstract: The study of baryogenesis introduces a new datum in the study of physics beyond the Standard Model. Some physics relevant to baryogenesis is reviewed. Electroweak baryogenesis due to scalar particles (such as squarks) possessing baryon number is estimated and found to be considerable within a broad and plausible range of parameter values. Transport properties of squarks in the electroweak plasma are found to support this conclusion. If baryonic scalars are discovered in physics beyond the Standard Model, these results will be useful in determining whether they are consistent with cosmological observations.

]]>

(PHD, 1997)

Abstract:

The general formulation of quantum statistical mechanics hints at interesting generalizations of the usual Bose/Fermi framework in two spatial dimensions. Anyon statistics, which is essentially a continuous interpolation between Bose and Fermi statistics, is relevant to the Fractional Quantum Hall Effect in two-dimensional (i.e., thin layer) condensed matter systems. In addition, the possibility of non-abelian statistics, in which the multi-particle wavefunction transforms as a representation of a non-abelian group under the exchange of indistinguishable particles, has been explored. Spontaneously broken non-abelian gauge theories in (2 + 1) dimensions often have stable topological defects, called non-abelian vortices, that experience non-abelian statistics. In addition, it has been suggested that degenerate quasihole multiplets in Quantum Hall systems also transform as non-abelian representations of the braid group under particle exchange. In this thesis, I explore the braiding properties of systems of two-cycle flux vortices in a residual S₃ discrete gauge group. The individual vortices are uncharged, but multi-vortex states can have Cheshire charge. The uncharged sectors all have non-vanishing bosonic subspaces, as do the non-abelian charged trivial flux sectors. A kinetic Hamiltonian for vortices on a periodic lattice is constructed. There is a modification to the translational symmetry in the periodically identified direction for non-trivial Z₂ charged sectors. The ground state energies for various three and four vortex sectors is numerically determined. Typically, the ground state is bosonic, with a gap separating it from a non-abelian subspace.

]]>

(PHD, 1997)

Abstract:

We probe string dualities by using the orientifold and F-theory, and by investigating world volume actions of super D-branes and super M-branes.

We first study orientifolds in various dimensions. We construct orientifolds dual to M-theory compactified on the Klein bottle and on the Möbius band, respectively. Six-dimensional orientifolds with N=1 supersymmetry are constructed. They have multiple tensor multiplets, which cannot be obtained by the conventional Calabi-Yau compactifications. We find F-theory duals for some of these models, thereby making manifest the phase transitions involving the tensionless strings these models can have.

We construct orientifold and F-theory duals of the heterotic string models constructed by Chaudhuri, Hockney and Lykken (CHL) and study N=2 supersymmetric F-theory vacua in six dimensions.

Next, we construct the supersymmetric world volume action of the M-theory 5-brane in a flat eleven-dimensional background. Finally, dual D-brane actions are obtained by carrying out a duality transformation of the world volume gauge field of the D-brane and their properties are studied.

]]>

(PHD, 1997)

Abstract: Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated subclass of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specific codes and classes of codes. I will give an overview of the field of quantum error correction and the formalism of stabilizer codes. In the context of stabilizer codes, I will discuss a number of known codes, the capacity of a quantum channel, bounds on quantum codes, and fault-tolerant quantum computation

]]>

(PHD, 1997)

Abstract: Heavy Quark Effective Theory (HQET) is reviewed and applied to extracting the
fundamental parameters of the Standard Model from experimental data. The main
focus is on precision measurements of the Cabibbo-Kobayashi-Maskawa matrix
element |V_{cb}|, and the charm and bottom quark masses m_{c} and m_{b}. We discuss the
model-independent extraction of |V_{cb}| from the B → D*lv decay rate and show that
the corresponding theoretical uncertainties, although small, cannot be further
reduced. The theory of the inclusive B → X _{c}lv decay is described and then used to
extract |V_{cb}|, m_{c}, and m_{b} from the available data. We also determine the HQET
parameters Λ and λ_{1} which appear in the expressions for the heavy meson decay rates
and the relations between the meson and quark masses. At present, the accuracy of
the inclusive measurement of |V_{cb}| is comparable to that from the exclusive B → D*lv
decay, but could be improved if the bottom quark mass m

]]>

(PHD, 1996)

Abstract: This thesis examines several examples of systems in which non-Abelian magnetic
flux and non-Abelian forms of the Aharonov-Bohm effect play a role. We consider
the dynamical consequences in these systems of some of the exotic phenomena associated
with non-Abelian flux, such as Cheshire charge holonomy interactions and
non-Abelian braid statistics. First, we use a mean-field approximation to study a
model of U(2) non-Abelian anyons near its free-fermion limit. Some self-consistent
states are constructed which show a small SU(2)-breaking charge density that vanishes
in the fermionic limit. This is contrasted with the bosonic limit where the SU(2)
asymmetry of the ground state can be maximal. Second, a global analogue of Chesire
charge is described, raising the possibility of observing Cheshire charge in condensedmatter
systems. A potential realization in superfluid He-3 is discussed. Finally, we
describe in some detail a method for numerically simulating the evolution of a network
of non-Abelian (S_{3}) cosmic strings, keeping careful track of all magnetic fluxes
and taking full account of their non-commutative nature. I present some preliminary
results from this simulation, which is still in progress. The early results are suggestive
of a qualitatively new, non-scaling behavior.

]]>

(PHD, 1994)

Abstract:

This thesis deals with some exotic phenomena in non-Abelian gauge theories. More specifically, we study aspects of non-Abelian vortices, non-Abelian Chern-Simons particles, wormhole physics and electroweak strings. Non-Abelian vortices are capable of carrying charges without apparent sources (Cheshire charge). They obey exotic statistics—they generally form irreducible representations of the braid groups of dimensions larger than one. Owing to topological interactions, two vortices scatter non-trivially with each other even in the absence of any classical forces. As a function of the scattering angle, the exclusive cross-section for the vortex-vortex scattering process in the “group eigenstates” is generally multi-valued. Moreover, there can be an exchange contribution even if the two vortices have distinct initial quantum numbers. Thus, two vortices can be indistinguishable without being the same! We also construct exact wave functions for systems of non-Abelian Chern-Simons particles. In wormhole physics, we analyze the measurements of charge and magnetic flux in a wormhole background and show that they are complementary observables. For one thing, this investigation illustrates clearly how charge is conserved in the presence of a wormhole. Finally, we discuss the scattering of fermions from an electroweak string.

]]>

(PHD, 1994)

Abstract:

Gauge theory with a finite gauge group (or with a gauge group that has disconnected components) is systematically studied, with emphasis on the case of a non-Abelian gauge group. An operator formalism is developed, and an order parameter is constructed that can distinguish the various phases of a gauge theory. The non-Abelian Aharonov-Bohm interactions and holonomy interactions among cosmic string loops, vortices, and charged particles are analyzed; the detection of Cheshire charge and the transfer of charge between particles and string loops (or vortex pairs) are described. Non-Abelian gauge theory on a surface with non-trivial topology is also discussed. Interactions of vortices with “handles” on the surface are discussed in detail. The electric charge of the mouth of a “wormhole” and the magnetic flux “linked” by the wormhole are shown to be non-commuting observables. This observation is used to analyze the color electric field that results when a colored object traverses a wormhole.

]]>

(PHD, 1994)

Abstract:

We explore some exotic phenomena in charged black holes, arising from the second quantization of matter fields or from the first quantization of fundamental strings. Spherically symmetric magnetic black holes admit special modes of electrically charged fermions, known as Callan-Rubakov modes, which can be quantized efficiently. We find that the chargeless sector generates non-thermal quantum radiations from extremal Reissner-NordstrOm black holes, thereby reducing the black hole masses below the familiar classical bound in spite of the vanishing Hawking temperature. On the other hand, the charged sector induces a vacuum energy distribution and the gravitational backreaction thereof, which are particularly pronounced for extremal dilatonic black holes. Implications of these quantum effects are studied in great detail. Also considered is a string-inspired two-dimensional gravity with non-singular charged black holes among its solutions. After a lengthy discussion on the stability of these novel space-times, we speculate on the possibility and implications of nonsingular black holes in full-blown string theories.

]]>

(PHD, 1993)

Abstract: This thesis deals with a global analogue of the Aharonov-Bohm effect previously pointed out by other authors. The effect was not well understood because the pure Aharonov-Bohm cross section was thought to be merely an approximate low energy limit. This thesis provides a detailed analysis and reveals that in the particular model considered, there is an exact Aharonov-Bohm cross section over the energy range that a mass splitting occurs. At energies slightly above the mass splitting, the effect has completely disappeared and there is effectively no scattering at large distances. This is a curious observation as it was previously thought that a global theory would not act exactly like a local one over an extended range of energies. It begs the heretical speculation that experimentally observed forces modelled with Lagrangians possessing local symmetries may have an underlying global theory.

]]>

(PHD, 1992)

Abstract:

I present a summary of the developments in wormhole physics. I then investigate the (Euclidean time) decay of axion charge that occurs in a 3-sphere of constant volume when there is a small charge violating operator perturbing the Hamiltonian. I demonstrate that in the limit of large Euclidean time T, axion charge decays like CT[superscript -1], where C depends only logarithmically on the coefficient of the charge-violating operator. I apply this result to axionic wormholes, and argue that small wormholes will destabilize large wormholes because of this charge decay. In another model, I demonstrate the existence of wormhole solutions with topology S[superscript 1]xS[superscript 2]xR. I interpret these wormholes in terms of topological charge violation on flat R[superscript 4].

]]>

(PHD, 1991)

Abstract: The thesis deals with the theory of non-Abelian vortices in two spatial dimensions and cosmic strings in three spatial dimensions that arise when a non-Abelian gauge symmetry G is broken to a non-Abelian unbroken symmetry group H by the condensation of a Higgs field. The first part of the thesis discusses the case in which H is discrete. In this case all of the gauge bosons acquire a large mass; however, at low energies discretely charged particles experience non-Abelian Aharonov-Bohm scattering off vortices, which can be used to measure the flux of the vortices. Vortices also experience non-Abelian Aharonov-Bohm scattering with each other. When there are more than three vortices in a system, the Aharonov-Bohm interaction, which is described by a path integral involving sums over elements of the braid group, becomes extremely complicated. The vortices are subject to a new kind of exotic statistics. The second part of the thesis discusses the physics that arises when the requirement that H be discrete is relaxed to allow H to have one continuous generator. The vortices or strings that result change the sign of the charge for charged particles. Loops of Alice string or pairs of vortices can carry charge without any apparent source. A quantization condition for the charge carried by such pairs and loops is derived. It is also found that loops of Alice string can carry magnetic charge and that topologically stable monopoles exist in any theory with Alice symmetry breaking.

]]>

(PHD, 1990)

Abstract:

We calculate the coupling constant and energy dependence of the scattering amplitudes for baryon- and lepton-number violating processes in the context of the standard model, in the semiclassical approximation. It is found that, to leading order in this expansion, the spin-averaged total cross sections for these processes grow as a power of the CM-energy and thus violate the bound imposed by unitarity. This result has a twofold implication: first, perturbation theory in the instanton sector of the electroweak theory must break down at high energies and, second, it strongly suggests that baryon and lepton number non-conservation might be observed experimentally at energies accesible in the near future.

]]>

(PHD, 1990)

Abstract: We study here some models of quantum gravity. In Euclidean quantum gravity, some of the possible consequences of including topology changes in the path integral are studied in the semiclassical approximation. The effects of wormhole interactions on the semiclassical sum are considered. The effects of wormholes in the Yang-Mills-Einstein system on the phase structure of these theories is discussed. Also, we perform the computation of some partition and correlation functions in conformal gauge, in a two dimensional model of quantum gravity, i.e., the sub-critical Polyakov string.

]]>

(PHD, 1990)

Abstract:

The construction of covariant string field theories is an important step toward a deeper understanding of string theories. This thesis discusses the general formulation of bosonic and supersymmetric covariant open string field theories and their second quantization. A particular emphasis is given to the perturbative calculation in the framework of string field theory. The use of string wave functional and the technique of conformal field theory are illustrated by explicit calculations of on- and off-shell string amplitudes. The background independent cubic actions for open strings are described briefly.

]]>

(PHD, 1990)

Abstract: We study some aspects of conformal field theory, wormhole physics and two-dimensional
random surfaces. Inspite of being rather different, these topics serve
as examples of the issues that are involved, both at high and low energy scales,
in formulating a quantum theory of gravity. In conformal field theory we show
that fusion and braiding properties can be used to determine the operator product
coefficients of the non-diagonal Wess-Zumino-Witten models. In wormhole physics
we show how Coleman’s proposed probability distribution would result in wormholes
determining the value of ^{θ}QCD. We attempt such a calculation and find the most
probable value of ^{θ}QCD to be π. This hints at a potential conflict with nature.
In random surfaces we explore the behaviour of conformal field theories coupled to
gravity and calculate some partition functions and correlation functions. Our results
throw some light on the transition that is believed to occur when the central charge
of the matter theory gets larger than one.

]]>

(PHD, 1988)

Abstract:

The covariant path integral formalism for theories of open and closed strings is used to study the first order of string perturbation theory beyond tree level for the closed-string states, in which the string world sheet has the topology of the disk or the real projective plane. We find that scattering amplitudes (in flat spacetime) confirm these surfaces’ contribution to the low-energy effective action for the bosonic string theory, as derived by another method, demanding consistency of string propagation in background gravitational and dilaton fields (the “sigma model approach”). However, we are not able to obtain results consistent with this effective action by demanding that amplitudes in a curved background be finite; this is an unresolved puzzle. Decoupling of spurious tachyon states from the superstring S-matrix is discussed, and finiteness of amplitudes for the disk plus projective plane is demonstrated for a large class of external states, when the gauge group is SO(32).

]]>

(PHD, 1988)

Abstract:

In this work two major topics in Conformal Field Theory are discussed. First a detailed investigation of N=2 Superconformal theories is presented. The structure of the representations of the N=2 superconformal algebras is investigated and the character formulae are calculated. The general structure of N=2 superconformal theories is elucidated and the operator algebra of the minimal models is derived. The first minimal system is discussed in more detail. Second, applications of the conformal techniques are studied in the Ashkin-Teller model. The c = 1 as well as the c = ½ critical lines are discussed in detail.

]]>

(PHD, 1987)

Abstract:

We examine the connection between anomalous quantum numbers, symmetry breaking patterns and topological properties of some field theories. The main results are the following: In three dimensions the vacuum in the presence of abelian magnetic field configurations behaves like a superconductor. Its quantum numbers are exactly calculable and are connected with the Atiyah-Patodi-Singer index theorem. Boundary conditions, however, play a nontrivial role in this case. Local conditions were found to be physically preferable than the usual global ones. Due to topological reasons, only theories for which the gauge invariant photon mass in three dimensions obeys a quantization condition can support states of nonzero magnetic flux. For similar reasons, this mass induces anomalous angular momentum quantum numbers to the states of the theory. Parity invariance and global flavor symmetry were shown to be incompatible in such theories. In the presence of massless flavored fermions, parity will always break for an odd number of fermion flavors, while for even fermion flavors it may not break but only at the expense of maximally breaking the flavor symmetry. Finally, a connection between these theories and the quantum Hall effectwas indicated.

]]>

(PHD, 1987)

Abstract:

The subject of this thesis is the description of the Very Early Universe, from the Big Bang to the beginning of the radiation-dominated Friedman-Roberston-Walker era. We examine a pure gravity inflationary model for the Universe which is based on adding ƐR^{2} term to the usual gravitational Lagrangian (“improved Starobinsky model”). We find the classical inflationary solution essentially independent of initial conditions. The model has only one free parameter, which is bounded from above by observational constraints on scalar and tensorial perturbations and from below by both the need for standard baryogenesis and the need for galaxy formation. This requires 10^{11}GeV < Ɛ^{-1/2} < 10^{13}GeV.

The model is interpreted as a Chaotic Inflationary model, with initial conditions for classical evolution being generated by the quantum fluctuations in metric and curvature in Very Early Universe. We discuss those fluctuations using a particular solution of the Wheeler-De Witt equation and find that the inflationary phase is a highly typical event.

]]>

(PHD, 1987)

Abstract:

We discuss some problems that arise when one tries to quantize a theory that possesses gauge degrees of freedom. First, we identify the Gribov problem that is encountered when the Faddeev-Popov procedure of fixing the gauge is employed to define a perturbation expansion. We propose a modification of the procedure that takes this problem into account. We then apply this method to two-dimensional gauge theories where the exact answer is known. Second, we try to build chiral theories that are consistent in the presence of anomalies, without making use of additional degrees of freedom. We are able to solve the model exactly in two dimensions, arriving at a gauge-invariant theory. We discuss the four-dimensional case and also the application of this method to string theory. In the latter, we obtain a model that lives in arbitrary dimensions. However, we do not compute the spectrum of the model. Third, we investigate the possibility of compactifying the unwanted dimensions of superstrings on a group manifold. We give a complete list of conformally invariant models. We also discuss one-loop modular invariance. We consider both type-II and heterotic superstring theories. Fourth, we discuss quantization of string field theory. We start by presenting the lagrangian approach, to demonstrate the non-uniqueness of the measure in the path-integral. It is fixed by demanding unitarity, which manifests itself in the hamiltonian formulation, studied next.

]]>

(PHD, 1987)

Abstract:

The method of effective Lagrangian flow provides the most physically illuminating discussion of renormalisation theory. At distance scales much larger than some physical cutoff, the physics is described by a small number of parameters, which can be identified purely by dimensional analysis. For scalar theories a rigorous yet simple proof of renormalisability, based on this concept, was given by Polchinski, and this work forms the bedrock of this thesis.

For gauge theories there is the extra issue of the unitarity of the renormalised S-matrix, which can only be guaranteed by proving renormalised Ward identities, and this is what we carry out for all cases of interest in d = 4. In particular we cover the case of N = 1 super Yang-Mills.

We prove that the cancellation of anomalies at the one-loop level is a sufficient as well as necessary condition for a theory to be perturbatively quantisable, and hence that there are no higher-loop anomalies.

]]>

(PHD, 1986)

Abstract:

We explore gauge fields in the functional Schrödinger representation. We first consider perturbatively solving quantum electrodynamics using known free field wave functionals. Failures of ordinary perturbative techniques force us to develop techniques to solve nontrivial functional differential equations. These techniques can also be used for Yang-Mills as we also demonstrate. We regularize QED in a new fashion using functional directional derivatives. This may also be generalized to Yang-Mills. We carry out mass renormalization in QED using wave functionals since no one has explicitly done it before. We briefly look at magnetic flux tubes in the Abelian Higgs model to illustrate renormalization in a variational calculation. We also perform a variational calculation using wave functionals in Yang-Mills to see if quantum fluctuations can produce electric flux tubes.

]]>

(PHD, 1986)

Abstract:

We will attempt to understand the *Delta I Equals One Half* pattern of the nonleptonic weak decays of the Kaons. The calculation scheme employed is the Strong Coupling Expansion of lattice QCD. Kogut-Susskind fermions are used in the Hamiltonian formalism. We will describe in detail the methods used to expedite this calculation, almost all of which was done by computer algebra.

The final result is very encouraging. Even though an exact interpretation is clouded by the presence of irrelevant operators, a distinct signal of the *Delta I Equals One Half* is observed. With an appropriate choice of the one free parameter, enhancements as great as those observed experimentally can be obtained along with a qualitative prediction for the relative magnitudes of the CP violating phases.

We also point out a number of surprising results which we turn up in the course of the calculation. The computer methods employed are briefly described.

]]>