In these two lectures, I will introduce the main algorithms used in today's noisy and tomorrow's fault-tolerant quantum computers. After a quick introduction to gate-based quantum computation, I will review basic primitives like the quantum Fourier transform and their use in algorithms such as quantum phase estimation, with applications to the factoring problem (Shor's algorithm) and energy estimation in quantum physics. Then, I will turn to the challenges of decoherence in quantum computers, to the variational algorithms that have been designed to mitigate its effects (including the variational quantum eigensolver, VQE), and to their limitations and some counter-measures like error mitigation. In the hands-on session, we will implement a phase estimation algorithm as well as a VQE algorithm applied to a quantum chemistry problem.[-]

The potential of quantum algorithms for solving optimization problems has been explored since the early days of quantum computing. This course introduces some of the key ideas and algorithms developed in this context, along with their fundamental limitations. Depending on the available time, topics covered may include: quantum optimization algorithms inspired by physics (adiabatic algorithms, variational algorithms, QAOA, quantum annealing, etc.), quantum algorithms for convex optimization (acceleration of first- and second-order methods, oracular problems, etc.), applications to combinatorial optimization (graph problems, quadratic binary optimization, etc.).[-]

One of the oldest and currently most promising application areas for quantum devices is quantum simulation. Popularised by Feynman in the early 1980s, it is important for the efficient simulation – compared to its classical counterpart – of one special partial differential equation (PDE): Schrodinger's equation. This is possible because quantum devices themselves naturally obey Schrodinger's equation. Just like with large-scale quantum systems, classical methods for other high-dimensional and large-scale PDEs often suffer from the curse-of-dimensionality, which a quantum treatment might in certain cases be able to mitigate. Aside from Schrodinger's equation, can quantum simulators also efficiently simulate other PDEs? To enable the simulation of PDEs on quantum devices that obey Schrodinger's equations, it is crucial to first develop good methods for mapping other PDEs onto Schrodinger's equations.After a brief introduction to quantum simulation, I will address the above question by introducing a simple and natural method for mapping other linear PDEs onto Schrodinger's equations. It turns out that by transforming a linear partial differential equation (PDE) into a higher-dimensional space, it can be transformed into a system of Schrodinger's equations, which is the natural dynamics of quantum devices. This new method – called /Schrodingerisation/ – thus allows one to simulate, in a simple way, any general linear partial differential equation and system of linear ordinary differential equations via quantum simulation.This simple methodology is also very versatile. It can be used directly either on discrete-variable quantum systems (qubits) or on analog/continuous quantum degrees of freedom (qumodes). The continuous representation in the latter case can be more natural for PDEs since, unlike most computational methods, one does not need to discretise the PDE first. In this way, we can directly map D-dimensional linear PDEs onto a (D + 1)-qumode quantum system where analog Hamiltonian simulation on (D + 1) qumodes can be used. It is the quantum version of analog computing and is more amenable to near-term realisation.These lectures will show how this Schrodingerisation method can be applied to linear PDEs, systems of linear ODEs and also linear PDEs with random coefficients, where the latter is important in the area of uncertainty quantification. Furthermore, these methods can be extended to solve problems in linear algebra by transforming iterative methods in linear algebra into the evolution of linear ODEs. It can also be applicable to certain nonlinear PDEs. We will also discuss many open questions and new research directions.[-]

Combining the relativistic speed limit on transmitting information with linearity and unitarity of quantum mechanics leads to a relativistic extension of the no-cloning principle called spacetime replication of quantum information. We introduce continuous-variable spacetime-replication protocols, expressed in a Gaussian-state basis, that build on novel homologically constructed continuous-variable quantum error correcting codes. Compared to qubit encoding, our continuous-variable solution requires half as many shares per encoded system. We show an explicit construction for the five-mode case and how it can be implemented experimentally. As well we analyze the ramifications of finite squeezing on the protocol.[-]

In these two lectures, I will introduce the main algorithms used in today's noisy and tomorrow's fault-tolerant quantum computers. After a quick introduction to gate-based quantum computation, I will review basic primitives like the quantum Fourier transform and their use in algorithms such as quantum phase estimation, with applications to the factoring problem (Shor's algorithm) and energy estimation in quantum physics. Then, I will turn to the challenges of decoherence in quantum computers, to the variational algorithms that have been designed to mitigate its effects (including the variational quantum eigensolver, VQE), and to their limitations and some counter-measures like error mitigation. In the hands-on session, we will implement a phase estimation algorithm as well as a VQE algorithm applied to a quantum chemistry problem.[-]

The potential of quantum algorithms for solving optimization problems has been explored since the early days of quantum computing. This course introduces some of the key ideas and algorithms developed in this context, along with their fundamental limitations. Depending on the available time, topics covered may include: quantum optimization algorithms inspired by physics (adiabatic algorithms, variational algorithms, QAOA, quantum annealing, etc.), quantum algorithms for convex optimization (acceleration of first- and second-order methods, oracular problems, etc.), applications to combinatorial optimization (graph problems, quadratic binary optimization, etc.).[-]

One of the oldest and currently most promising application areas for quantum devices is quantum simulation. Popularised by Feynman in the early 1980s, it is important for the efficient simulation – compared to its classical counterpart – of one special partial differential equation (PDE): Schrodinger's equation. This is possible because quantum devices themselves naturally obey Schrodinger's equation. Just like with large-scale quantum systems, classical methods for other high-dimensional and large-scale PDEs often suffer from the curse-of-dimensionality, which a quantum treatment might in certain cases be able to mitigate. Aside from Schrodinger's equation, can quantum simulators also efficiently simulate other PDEs? To enable the simulation of PDEs on quantum devices that obey Schrodinger's equations, it is crucial to first develop good methods for mapping other PDEs onto Schrodinger's equations.After a brief introduction to quantum simulation, I will address the above question by introducing a simple and natural method for mapping other linear PDEs onto Schrodinger's equations. It turns out that by transforming a linear partial differential equation (PDE) into a higher-dimensional space, it can be transformed into a system of Schrodinger's equations, which is the natural dynamics of quantum devices. This new method – called /Schrodingerisation/ – thus allows one to simulate, in a simple way, any general linear partial differential equation and system of linear ordinary differential equations via quantum simulation.This simple methodology is also very versatile. It can be used directly either on discrete-variable quantum systems (qubits) or on analog/continuous quantum degrees of freedom (qumodes). The continuous representation in the latter case can be more natural for PDEs since, unlike most computational methods, one does not need to discretise the PDE first. In this way, we can directly map D-dimensional linear PDEs onto a (D + 1)-qumode quantum system where analog Hamiltonian simulation on (D + 1) qumodes can be used. It is the quantum version of analog computing and is more amenable to near-term realisation.These lectures will show how this Schrodingerisation method can be applied to linear PDEs, systems of linear ODEs and also linear PDEs with random coefficients, where the latter is important in the area of uncertainty quantification. Furthermore, these methods can be extended to solve problems in linear algebra by transforming iterative methods in linear algebra into the evolution of linear ODEs. It can also be applicable to certain nonlinear PDEs. We will also discuss many open questions and new research directions.[-]

Quantum optical systems provides one of the best physical settings to engineer quantum many-body systems of atoms and photons, which can be controlled and measured on the level of single quanta. In this course we will provide an introduction to quantum optics from the perspective of control and measurement, and in light of possible applications including quantum computing and quantum communication.
The first part of the course will introduce the basic quantum optical systems and concepts as 'closed' (i.e. isolated) quantum systems. We start with laser driven two-level atoms, the Jaynes-Cummings model of Cavity Quantum Electro-dynamics, and illustrate with the example of trapped ions control of the quantum motion of atoms via laser light. This will lead us to the model system of an ion trap quantum computer where we employ control ideas to design quantum gates.
In the second part of the course we will consider open quantum optical systems. Here the system of interest is coupled to a bosonic bath or environment (e.g. vacuum modes of the radiation field), providing damping and decoherence. We will develop the theory for the example of a radiatively damped two-level atom, and derive the corresponding master equation, and discuss its solution and physical interpretation. On a more advanced level, and as link to the mathematical literature, we establish briefly the connection to topics like continuous measurement theory (of photon counting), the Quantum Stochastic Schrödinger Equation, and quantum trajectories (here as as time evolution of a radiatively damped atom conditional to observing a given photon count trajectory). As an example of the application of the formalism we discuss quantum state transfer in a quantum optical network.
Parts of this video related to ongoing unpublished research have been cut off.[-]