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Explore quantum computing algorithms for both current noisy and future fault-tolerant quantum computers in this comprehensive lecture from the CEMRACS Summer School. Delve into gate-based quantum computation fundamentals before examining essential primitives like the quantum Fourier transform and their applications in quantum phase estimation algorithms. Discover how these techniques apply to significant problems including Shor's factoring algorithm and energy estimation in quantum physics systems. Learn about the critical challenges posed by decoherence in quantum computers and understand variational algorithms designed to mitigate these effects, with particular focus on the variational quantum eigensolver (VQE). Examine the limitations of current approaches and explore counter-measures such as error mitigation techniques. The session includes practical implementation of both phase estimation algorithms and VQE algorithms applied to quantum chemistry problems, providing hands-on experience with these fundamental quantum computing concepts.
