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(S3), I map its phase diagram, and interpret the physical processes the theory undergoes through terms borrowed from the Landau theory of phase transitions.
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(S3), I map its phase diagram, and interpret the physical processes the theory undergoes through terms borrowed from the Landau theory of phase transitions.
This thesis is devoted to the study of coherent storage of quantum information as well as its potential applications.Quantum memories are crucial to harnessing the potential of quantum physics for information processing tasks.They are required for almost all quantum computation proposals.However, despite the large arsenal of theoretical techniques and proposals dedicated to their implementation, the realization of long-lived quantum memories remains an elusive task.Encoding information in quantum states associated to many-body topological phases of matter and protecting them by means of a static Hamiltonian is one of the leading proposals to achieve quantum memories.While many genuine and well publicized virtues have been demonstrated for this approach, equally real limitations were widely disregarded.In the first two projects of this thesis, we study limitations of passive Hamiltonian protection of quantum information under two different noise models.Chapter 2 deals with arbitrary passive Hamiltonian protection for a many body system under the effect of local depolarizing noise.It is shown that for both constant and time dependent Hamiltonians, the optimal enhancement over the natural single-particle memory time is logarithmic in the number of particles composing the system.The main argument involves a monotonic increase of entropy against which a Hamiltonian can provide little protection.Chapter 3 considers the recoverability of quantum information when it is encoded in a many-body state and evolved under a Hamiltonian composed of known geometrically local interactions and a weak yet unknown Hamiltonian perturbation.We obtain some generic criteria which must be fulfilled by the encoding of information.For specific proposals of protecting Hamiltonian and encodings such as Kitaev's toric code and a subsystem code proposed by Bacon, we additionally provide example perturbations capable of destroying the memory which imply upper bounds for the provable memory times.vi
Since the seminal work by Shor who proposed a quantum algorithm factorizing integers into prime factors, it has become manifest that the laws of quantum mechanics provide resources for computation that overpower classical physics. The computational advantages that quantum physics offers have stimulated a tremendous amount of theoretical and experimental research. In this context, spin systems have played a major role, given that the spin degree of freedom -- with the paradigmatic case of the spin-1/2 of electrons -- represents an obvious candidate for the encoding of an elementary bit of quantum information (qubit). \n \nOn the other hand, however, quantum objects are very fragile entities, being very susceptible to the environment they reside in. This fragility of qubits is one of the main obstacles in the realization of a quantum computer. \n \nIn this thesis, we mainly address the two following questions relevant to quantum computation. \n \ni) How is it possible to realize quantum gates both in a reliable and scalable way? \n \nii) How can we store quantum information in a way that is resilient to the errors caused by the thermal environment? \n \nWe focus on spin systems and demonstrate that long-range spin-spin interactions in the models considered can have beneficial effects. \n \nIn their pioneering work, Loss and DiVincenzo proposed a way to perform quantum computation in a semiconductor-based architecture where the spin state of an electron trapped in a quantum dot is chosen to encode the elementary qubit. In this proposal, the spins are required to lie spatially close to each other, and this might complicate the realization of a scalable architecture. \n \nIn the first part of the thesis we thus propose a scheme that allows the constraint on the positioning of the qubits to be relaxed. This is achieved by introducing a ferromagnetic coupler between the distant qubits, to which it is coupled via a dipolar interaction. Most importantly, our proposal is applicable to any type of spin qubits and in particular to the technologically very relevant silicon-based qubits and NV-centers in diamond to which previous coupling schemes do not apply. \n \nAs additional key element, a quantum computer needs a memory capable of reliably storing quantum information in the presence of thermal fluctuations. \n \nThis brings us to the second part of this thesis, where we consider self-correcting memories, for which the protection against thermal noise is built-in at the hardware level. We propose physical models that exhibit these self-correcting properties, using as a starting point the well known topologically ordered toric code. In particular, we investigate how to induce long-range interactions between the spins of the toric code, since such interactions help increase the memory lifetime. \n \nAs a first step, we study a honeycomb quantum spin model coupled to delocalized cavity modes. We investigate the properties of the low-energy toric code Hamiltonian and show that the coupling to cavity modes prolongs the lifetime of the memory and offers a method to detect the presence of excitations. \nWhile the introduction of extended bosonic modes makes the model non-local, we also propose a purely local model consisting of a toric code embedded in a three-dimensional cubic lattice of hopping bosons; the low-energy sector of a toric code coupled to a three-dimensional Heisenberg ferromagnet in a broken-symmetry state realizes this model. Our analysis leads to an energy penalty for the creation of defects that grows linearly with the size L of the memory and thus to a lifetime increasing exponentially with L. \n \nIn the third part of this thesis, we study spin systems that support anyons, i.e., particles with fractional statistics. Similar to the toric code, such systems are topologically ordered: they are immune to local perturbations and quantum gates are implemented by non-local operations, namely the exchange of anyons, whose outcomes depend only on the topology of the exchange. Here again the fault-tolerance is achieved at the level of the hardware and physical systems supporting non-abelian anyons are thus promising platforms for quantum computation. \n \nWe focus on spin systems that exhibit some of these properties and specifically on variations of the honeycomb quantum spin model. We first investigate the exact solution of the honeycomb model in detail and derive an explicit formula for the projector onto the physical subspace. We use this result to study inhomogeneous open spin ladders, related to the honeycomb model, which can be tuned between topological and non-topological phases. We test the robustness of Majorana end states (MES) which emerge at the boundary between sections in different topological phases. Furthermore, we present a trijunction setup where MES can be braided. This is of interest since MES in these spin ladders potentially follow non-abelian braiding statistics. Finally, we study the ground states of the aforementioned ladders and show that they are free of vortices when the signs of the spin couplings are all positive or negative. To prove this, we use exact reflection-positivity-based methods as well as approximate methods. \n \nIn the last part of the thesis, we provide an extension of the Mermin-Wagner theorem to a system of lattice spins that are spin-coupled to itinerant and interacting charge carriers. We prove that neither (anti-) ferromagnetic nor helical long-range order is possible in one and two dimensions at any finite temperature (in the absence of spin-orbit). The fundamental question whether spontaneous ordering of the lattice spins occurs in these systems is of interest in the context of quantum computation; the polarization of nuclear spins coupled to a two-dimensional electron gas is a possible route towards the reduction of decoherence induced by the fluctuating Overhauser field in gate-defined quantum dots. \n \n
This dissertation investigates the theory of quantum stochastic processes and its applications in quantum many-body physics. The main goal is to analyse complexity-theoretic aspects of both static and dynamic properties of physical systems modelled by quantum stochastic processes. The thesis consists of two parts: the first one addresses the computational complexity of certain quantum and classical divisibility questions, whereas the second one addresses the topic of Hamiltonian complexity theory. In the divisibility part, we discuss the question whether one can efficiently sub-divide a map describing the evolution of a system in a noisy environment, i.e. a CPTP- or stochastic map for quantum and classical processes, respectively, and we prove that taking the nth root of a CPTP or stochastic map is an NP-complete problem. Furthermore, we show that answering the question whether one can divide up a random variable X into a sum of n iid random variables Yi, i.e. X=∑i=1nYi, is poly-time computable; relaxing the iid condition renders the problem NP-hard. In the local Hamiltonian part, we study computation embedded into the ground state of a many-body quantum system, going beyond "history state" constructions with a linear clock. We first develop a series of mathematical techniques which allow us to study the energy spectrum of the resulting Hamiltonian, and extend classical string rewriting to the quantum setting. This allows us to construct the most physically-realistic QMAEXP-complete instances for the LOCAL HAMILTONIAN problem (i.e. the question of estimating the ground state energy of a quantum many-body system) known to date, both in one- and three dimensions. Furthermore, we study weighted versions of linear history state constructions, allowing us to obtain tight lower and upper bounds on the promise gap of the LOCAL HAMILTONIAN problem in various cases. We finally study a classical embedding of a Busy Beaver Turing Machine into a low-dimensional lattice spin model, which allows us to dictate a transition from a purely classical phase to a Toric Code phase at arbitrarily large and potentially even uncomputable system sizes.
Building a working quantum computer that is able to perform useful calculations remains a challenge. With this thesis, we are trying to contribute a small piece to this puzzle by addressing three of the many fundamental questions one encounters along the way of reaching that goal. These questions are: (i) What is an easy way to create highly entangled states as a resource for quantum computation? (ii) What can we do to efficiently quantify states of noisy entanglement in systems coupled to the outside world? (iii) How can we protect and store fragile quantum states for arbitrary long times? \n \nThe first two questions are the subject of part one of this thesis, `Entanglement Measures & Highly Entangled States'. We devise a particular proposal for generating entanglement within a solid-state setup, starting first with the tripartite case and continuing with a generalization to four and more qubits. The main idea there is to realize systems with highly entangled ground states in order for entanglement to be created by merely cooling to low enough temperatures. We have addressed the issue of quantifying entanglement in these systems by numerically calculating mixed-state entanglement measures and maximizing the latter as a function of the external magnetic field strength. The research along these lines has led to the development of the numerical library 'libCreme'. \n \nThe second part of the thesis, 'Self-Correcting Quantum Memories', addresses the question how to reliably store quantum states long enough to perform useful calculations. Every computer, be it classical or quantum, needs the information it processes to be protected from corruption caused by faulty gates and perturbations from interactions with its environment. However, quantum states are much more susceptible to these adverse effects than classical states, making the manipulation and storage of quantum information a challenging task. Promising candidates for such 'quantum memories' are systems exhibiting topological order, because they are robust against local perturbations, and information encoded in their ground state can only be manipulated in a non-local fashion. We extend the so-called toric code by repulsive long-range interactions between anyons and show that this makes the code stable against thermal fluctuations. Furthermore, we investigate incoherent effects of quenched disorder in the toric code and similar systems.
All the data stored and processed by our computers is encoded as sequences of zeros and ones, called bits. Quantum computers offer an alternative to this traditional way of encoding and manipulating information. The information in a collection of qubits can be a combination, also called a superposition, over many possible sequences of zeros and ones. Where classical computation consists of flipping zeros and ones, quantum computation consists of manipulating high-dimensional vectors called quantum states. Researchers have found ways to use this to engineer quantum algorithms that solve some tasks much faster than any classical computer would. These are called quantum advantages, and are proven for specific tasks, such as breaking some encryption schemes and solving certain large linear systems. The goal of this thesis was to find new tasks for which quantum computers provably have an advantage over classical computers. In this thesis, I present two such advantages and associated results. The first one revolves around a physically relevant learning task. In other words, I explore how quantum computers can be used to help solve specialized machine learning tasks. The second one addresses the simulation of classical systems. One of the first high-potential and natural uses of quantum computers was the simulation of quantum systems. A lot of real-world problems, however, are ruled by classical physics. It turns out that quantum computers are well suited for the simulation of some exponentially large classical systems. In this thesis, I add a new problem to this list.
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.