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Explore the fundamental algorithms powering today's noisy intermediate-scale quantum computers and tomorrow's fault-tolerant quantum systems in this comprehensive lecture from the CEMRACS Summer School on Quantum Computing. Begin with an introduction to gate-based quantum computation before diving into essential algorithmic primitives including the quantum Fourier transform and quantum phase estimation. Discover how these building blocks enable breakthrough algorithms like Shor's factoring algorithm and energy estimation methods for quantum physics applications. Examine the critical challenges posed by decoherence in quantum systems and learn about variational algorithms designed to mitigate these effects, with particular focus on the variational quantum eigensolver (VQE). Understand the limitations of current approaches and explore counter-measures such as error mitigation techniques that make quantum computing viable in the near term. The lecture sets the foundation for practical implementation, preparing you for hands-on sessions involving phase estimation algorithms and VQE applications to quantum chemistry problems.
