Quantum Hamiltonian Complexity (QHC) is an emergent field which uses tools and notions from complexity theory and quantum information theory to study the physics of local Hamiltonians.

In this tutorial I will introduce basic tools and notions from this theory by concentrating on a central and largely open problem in this field: what is the complexity of approximating the ground energy of gapped local Hamiltonians? The study of this question will take us on a journey on which we will encounter fundamental results such as exponential decay of correlations, area laws, tensor networks and variational algorithms.

Since the first observation, almost 15 years ago, of first coherent quantum behaviour in a superconducting qubit there have been significant developments in the field of superconducting quantum circuits. With improvements of coherence times by over 5 order of magnitude, it is now possible to implement simple quantum algorithms with these circuits. In parallel to these developments, much effort has been invested in using superconducting qubits as artificial atoms to explore quantum optics in unconventional parameter regimes. Assuming no knowledge of superconductivity, in this tutorial we will explain how an electrical circuit can be made to behave as a qubit when operated in the right conditions. We will focus on the transmon qubit, the simplest but also the most widely used superconducting qubit. We will see how a transmon can be strongly coupled to a microwave resonator, and how the latter can be used to readout the logical state of the transmon and to realize two-qubit gates. These concepts will be illustrated with results from state of the art experiments. Finally, the current challenges and limitations of this architecture will be discussed.

Typically when executing a quantum protocol, we assume we know how particular devices operate, and the working of the protocol relies on this. However, it can be difficult to ensure that devices really work as intended. This can be a particular problem in cryptography where any unexpected behaviour of a device may be exploitable by an eavesdropper.

I will introduce the topic of device-independence, which aims to make statements based only on the input-output behaviour of any devices used, and without needing to model their internal behaviour. I will explain the model in detail and describe the tasks of quantum key distribution and randomness expansion, outlining the key ideas that make up security proofs.

Three decades have passed since Richard Feynman first proposed devising a "quantum computer" founded on the laws of quantum physics to achieve computational speed-ups over classical methods. In that time, quantum algorithms have been developed that offer fast solutions to problems in a variety of fields including number theory, chemistry, and materials science. To execute such algorithms on a quantum device will require a well-developed quantum hardware and software architecture. One of the grand challenges for the computer science community is the design and implementation of a software architecture to control and program quantum hardware. This tutorial will describe the components of a scalable software architecture for quantum computing, from the high-level programming language, to algorithms for quantum circuit synthesis and optimization, to methods of quantum error correction. It will highlight recent advances in quantum circuit decomposition, quantum error correction, and software design tools, and pose crucial open questions in quantum computer science.