Advanced Quantum Algorithms

This is not an introductory course. MIT’s advanced quantum algorithms offering picks up where most other courses leave off and covers the algorithmic techniques that underlie the most significant quantum speedups known today. Students who complete it will have the background needed to read current research papers and to evaluate whether a proposed quantum algorithm represents a genuine computational advantage.

The course reflects MIT’s research-first culture: lectures are dense, proofs are given in full, and the problem sets are genuinely hard. The reward is a level of understanding that most short courses do not come close to providing.

What you’ll learn

The course begins with a rigorous treatment of quantum phase estimation (QPE), covering the precision-complexity tradeoffs and the role QPE plays as a subroutine in nearly every major quantum speedup. From there it covers the HHL algorithm for solving linear systems, including a careful analysis of the conditions under which the quantum speedup is real and the significant caveats around state preparation and readout.

Quantum walks are treated both in their continuous and discrete forms, with applications to element distinctness and graph problems. Amplitude estimation generalizes Grover’s algorithm and is analyzed in the context of Monte Carlo speedups. The final weeks introduce quantum singular value transformation (QSVT), the unifying framework that subsumes QPE, HHL, quantum walks, and many other algorithms as special cases. Students leave with a modern, unified picture of the algorithmic landscape.

Course structure

Twelve weeks of graduate-level lectures, each approximately two hours. Lecture notes are provided in addition to video content. Weekly problem sets require written proofs and circuit constructions; solutions are provided after submission deadlines. The course includes a final project in which students analyze a research paper on a quantum algorithm of their choosing and present a technical critique.

The verified certificate track includes instructor feedback on problem sets and access to the course community forums.

Who is this for?

• Graduate students in physics, computer science, or mathematics who need a thorough grounding in quantum algorithms
• Quantum computing researchers who want to understand the QSVT framework and its implications
• Industry practitioners in quantum software who want to move beyond Grover and Shor
• Anyone who has completed a solid introductory course and is ready for graduate-level material

Prerequisites

A strong introductory quantum computing course is required: comfort with quantum circuits, basic algorithms, and Dirac notation is assumed from the first lecture. Graduate-level linear algebra (spectral theorem, singular value decomposition) and familiarity with computational complexity theory are also expected.