Simplifying quantum double Hamiltonians using perturbative gadgets

Perturbative gadgets were originally introduced to generate effective k-local interactions in the low-energy sector of a 2-local Hamiltonian. Extending this idea, we present gadgets which are specifically suited for realizing Hamiltonians exhibiting non-abelian anyonic excitations. At the core of our construction is a perturbative analysis of a widely used hopping-term Hamiltonian. We show that in the low-energy limit, this Hamiltonian can be approximated by a certain ordered product of operators. In particular, this provides a simplified realization of Kitaev's quantum double Hamiltonians.Comment: 34 pages, v2: updated to match published versio

Toric codes and quantum doubles from two-body Hamiltonians

We present here a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the low-energy limits of entirely two-body Hamiltonians. Our construction makes use of a new type of perturbation gadget based on error-detecting subsystem codes. The procedure is motivated by a projected entangled pair states (PEPS) description of the target models, and reproduces the target models' behavior using only couplings that are natural in terms of the original Hamiltonians. This allows our construction to capture the symmetries of the target models

Schrieffer-Wolff transformation for quantum many-body systems

The Schrieffer-Wolff (SW) method is a version of degenerate perturbation theory in which the low-energy effective Hamiltonian H_{eff} is obtained from the exact Hamiltonian by a unitary transformation decoupling the low-energy and high-energy subspaces. We give a self-contained summary of the SW method with a focus on rigorous results. We begin with an exact definition of the SW transformation in terms of the so-called direct rotation between linear subspaces. From this we obtain elementary proofs of several important properties of H_{eff} such as the linked cluster theorem. We then study the perturbative version of the SW transformation obtained from a Taylor series representation of the direct rotation. Our perturbative approach provides a systematic diagram technique for computing high-order corrections to H_{eff}. We then specialize the SW method to quantum spin lattices with short-range interactions. We establish unitary equivalence between effective low-energy Hamiltonians obtained using two different versions of the SW method studied in the literature. Finally, we derive an upper bound on the precision up to which the ground state energy of the n-th order effective Hamiltonian approximates the exact ground state energy.Comment: 47 pages, 3 figure

Many-body models for topological quantum information

We develop and investigate several quantum many-body spin models of use for topological quantum information processing and storage. These models fall into two categories: those that are designed to be more realistic than alternative models with similar phenomenology, and those that are designed to have richer phenomenology than related models. In the first category, we present a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the perturbative low-energy limits of entirely two-body Hamiltonians. This construction reproduces the target models' behavior using only couplings which are natural in terms of the original Hamiltonians. As an extension of this work, we construct parent Hamiltonians involving only local 2-body interactions for a broad class of Projected Entangled Pair States (PEPS). We define a perturbative Hamiltonian with a finite order low energy effective Hamiltonian that is a gapped, frustration-free parent Hamiltonian for an encoded version of a desired PEPS. For topologically ordered PEPS, the ground space of the low energy effective Hamiltonian is shown to be in the same phase as the desired state to all orders of perturbation theory. We then move on to define models that generalize the phenomenology of several well-known systems. We first define generalized cluster states based on finite group algebras, and investigate properties of these states including their PEPS representations, global symmetries, relationship to the Kitaev quantum double models, and possible applications. Finally, we propose a generalization of the color codes based on finite groups. For non-Abelian groups, the resulting model supports non-Abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure

Combinatorial algorithms for perturbation theory and application on quantum computing

Quantum computing is an emerging area between computer science and physics. Numerous problems in quantum computing involve quantum many-body interactions. This dissertation concerns the problem of simulating arbitrary quantum many-body interactions using realistic two-body interactions. To address this issue, a general class of techniques called perturbative reductions (or perturbative gadgets) is adopted from quantum complexity theory and in this dissertation these techniques are improved for experimental considerations. The idea of perturbative reduction is based on the mathematical machinery of perturbation theory in quantum physics. A central theme of this dissertation is then to analyze the combinatorial structure of the perturbation theory as it is used for perturbative reductions

Universal Hamiltonians for quantum simulation and their applications to holography

Recent work has demonstrated the existence of universal Hamiltonians – simple spin lattice models that can simulate any other quantum many body system. These universal Hamiltonians have applications for developing quantum simulators, as well as for Hamiltonian complexity, quantum computation, and fundamental physics. In this thesis we extend the theory of universal Hamiltonians. We begin by developing a new method for proving that a given family of Hamiltonians is indeed universal. We then use this method to construct two new universal models – both of which consist of translationally invariant interactions acting on a 1D spin chain. But the benefit of our method doesn’t just lie in the simple universal models it allows us to construct. It also gives deeper insight into the origins of universality – and demonstrates a link between the universality and complexity. We make this insight rigorous, and derive a complexity theoretic classification of universal Hamiltonians which encompasses all known universal models. This classification provides a new, simplified route to checking whether a particular family of Hamiltonians meets the conditions to be a universal simulator. We also consider the practical use of analogue Hamiltonian simulation. Under- standing the effect of noise on Hamiltonian simulation is a key issue in practical implementations. The first step to tackling this issue is characterising the noise processes affecting near term quantum devices. Motivated by this, we develop and numerically benchmark an algorithm which fits noise models to tomographic data from quantum devices to enable this process. This algorithm has applicability beyond analogue simulators, and could be used to investigate the physical noise processes in any quantum computing device. Finally, we apply the theory of universal Hamiltonians to high energy physics by using them to construct toy models of holographic duality which capture more of the expected features of the AdS/CFT correspondence

Superconducting qubits for quantum annealing applications

Over the last two decades, Quantum Annealing (QA) has grown to be a commercial technology with machines reaching the scale of 5000 interconnected qubits. Two reasons for this progress are the relative ease of implementing adiabatic Hamiltonian control and QA’s partial robustness against errors caused by decoherence. Despite the success of this approach to quantum computation, proving a scaling advantage over classical computation remains an elusive goal to this date. Different strategies are therefore being considered to boost the performance of quantum annealing. These include using more coherent qubit architectures and error-suppression to limit the effect of environmental noise, implementing non-stoquastic driver terms and tailored annealing schedules to enhance the success probability of the algorithm, and using many-body couplers to embed higher-order binary optimisation problems with less resource overhead. This thesis contributes to these efforts in two different ways. The first part provides a detailed numerical analysis and a physical layout for a threebody coupler for flux qubits based on ancillary spins. The application of the coupler in a coherence-signature QA Hamiltonian is also considered and the results of the simulated quantum evolution are compared to the outcomes of classical optimisation on the problem Hamiltonian showing that the classical algorithms cannot correctly reproduce the state distribution at the end of QA. In the second part of the thesis, we develop a numerical method for mapping the Hamiltonian of a composite superconducting circuit to an effective many-qubit Hamiltonian. By overcoming drawbacks of standard reduction methods, this protocol can be used to guide the design of non-stoquastic and many-body Hamiltonian terms, as well as to get a more precise evaluation of the QA schedule parameters, which can greatly improve the outcomes of the optimisation. This numerical work is accompanied by a proposal for an experimental verification of the predictions of the reduction protocol and by some preliminary experimental results

Hamiltonian Complexity in Many-Body Quantum Physics

The development of quantum computers has promised to greatly improve our understanding of quantum many-body physics. However, many physical systems display complex and unpredictable behaviour which is not amenable to analytic or even computational solutions. This thesis aims to further our understanding of what properties of physical systems a quantum computer is capable of determining, and simultaneously explore the behaviour of exotic quantum many-body systems. First, we analyse the task of determining the phase diagram of a quantum material, and thereby charting its properties as a function of some externally controlled parameter. In the general case we find that determining the phase diagram to be uncomputable, and in special cases show it is P^{QMA_{EXP}}-complete. Beyond this, we examine how a common method for determining quantum phase transitions --- the Renormalisation Group (RG) --- fails when applied to a set of Hamiltonians with uncomputable properties. We show that for such Hamiltonians (a) there is a well-defined RG procedure, but this procedure must fail to predict the uncomputable properties (b) this failure of the RG procedure demonstrates previously unseen and novel behaviour. We also formalise in terms of a promise problem, the question of computing the ground state energy per particle of a model in the limit of an infinitely large system, and show that approximating this quantity is likely intractable. In doing this we develop a new kind of complexity question concerned with determining the precision to which a single number can be determined. Finally we consider the problem of measuring local observables in the low energy subspace of systems --- an important problem for experimentalists and theorists alike. We prove that if a certain kind of construction exists for a class of Hamiltonians, , the results about hardness of determining the ground state energy directly implies hardness results for measuring observables at low energies