In these three lectures on quantum information and complexity, we will (1) review the basic concepts of quantum information processing units (QPUs), (2) prove a version of the claim that almost all quantum circuits are very complex in the sense that they are exponentially expensive to realize in the quantum circuit model of computation and (3) that the quantum complexity of a random quantum circuit grows linearly with the size of the circuit up to exponentially large circuits.

The underlying proof technique uses a versatile proof strategy from high-dimensional probability theory that can (and has been) readily extended to other problems within quantum computing theory and beyond.[-]

In these three lectures on quantum information and complexity, we will (1) review the basic concepts of quantum information processing units (QPUs), (2) prove a version of the claim that almost all quantum circuits are very complex in the sense that they are exponentially expensive to realize in the quantum circuit model of computation and (3) that the quantum complexity of a random quantum circuit grows linearly with the size of the circuit up to exponentially large circuits.

The underlying proof technique uses a versatile proof strategy from high-dimensional probability theory that can (and has been) readily extended to other problems within quantum computing theory and beyond.[-]

In these three lectures on quantum information and complexity, we will (1) review the basic concepts of quantum information processing units (QPUs), (2) prove a version of the claim that almost all quantum circuits are very complex in the sense that they are exponentially expensive to realize in the quantum circuit model of computation and (3) that the quantum complexity of a random quantum circuit grows linearly with the size of the circuit up to exponentially large circuits.

The underlying proof technique uses a versatile proof strategy from high-dimensional probability theory that can (and has been) readily extended to other problems within quantum computing theory and beyond.[-]

The remarkable progress in control and readout of atomic and solid-state qubits has led to an accelerated race towards building a useful quantum computer. A portion of the recent developments deal with noisy quantum bits and aim at proving an advantage with respect to classical processors. However, in order to fully exploit the power of quantum physics in computation, developing fault-tolerant processors is unavoidable. In such a processor, quantum bits and logical gates are dynamically and continuously protected against noise by means of quantum error correction. While a theory of quantum error correction has existed and developed since mid 1990s, the first experiments are being currently investigated in the physics labs around the world. I will review the main approach pursued in this direction and state of progress towards error corrected qubits. I will also present some shortcut approaches that are pursued to reduce the significant hardware overhead of error correction.[-]

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.[-]

Quantum computing promises to transform computational capabilities across diverse fields. The rapid advancement of quantum algorithms has expanded the potential of quantum computing for tackling a broad spectrum of scientific computing challenges. In this lecture, we will present fundamental concepts of quantum algorithms, focusing on solving large-scale numerical linear algebra problems as well as addressing high-dimensional linear and nonlinear differential equations. We will start with basic notions of quantum states, unitary operators, no-cloning theorem and measurements. After introducing block-encoding and linear combination of unitaries (LCU), we will discuss various quantum algorithms for scientific computing, i.e., Quantum Linear System Problem (QLSP), Quantum Singular Value (Eigenvalue) Transformation (QSVT), Hamiltonian Simulation and Trotterization, Adiabatic Quantum Computation (AQC), Variational Quantum Eigensolver (VQE), Quantum Krylov Algorithms and Quantum (linear) Differential Equation Solvers.[-]

Quantum computing promises to transform computational capabilities across diverse fields. The rapid advancement of quantum algorithms has expanded the potential of quantum computing for tackling a broad spectrum of scientific computing challenges. In this lecture, we will present fundamental concepts of quantum algorithms, focusing on solving large-scale numerical linear algebra problems as well as addressing high-dimensional linear and nonlinear differential equations. We will start with basic notions of quantum states, unitary operators, no-cloning theorem and measurements. After introducing block-encoding and linear combination of unitaries (LCU), we will discuss various quantum algorithms for scientific computing, i.e., Quantum Linear System Problem (QLSP), Quantum Singular Value (Eigenvalue) Transformation (QSVT), Hamiltonian Simulation and Trotterization, Adiabatic Quantum Computation (AQC), Variational Quantum Eigensolver (VQE), Quantum Krylov Algorithms and Quantum (linear) Differential Equation Solvers.[-]

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.[-]