OE-4.1 Math Foundations for Quantum Computing
You can work in vector spaces and Hilbert spaces, compute inner and outer products, apply unitary operators and projections, and find eigenvalues and eigenvectors.
Quantum computing is linear algebra with a physical interpretation attached, and this unit is the algebra with the physics deliberately withheld. Get comfortable with inner product as overlap and outer product as an operator you can build - the two get confused constantly and every later chapter depends on telling them apart. Unitary matters because it means reversible and norm-preserving, which is exactly the constraint every quantum gate must obey.
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Vector spaces, subspaces, linear independence and dependence
The mathematics the rest of the subject is written in, starting from the beginning. If linear algebra is already comfortable this topic is revision, and if it is not, nothing later will make sense.
Basis and finite dimensions
How a space is described by the smallest set that spans it. Dimension is the count of that set, and every quantum state below is written against a chosen one.
Orthogonality of vectors
When two vectors share nothing, which is the geometric idea behind distinguishable states. It is used constantly and rarely stated explicitly.
Inner product and outer product; Hilbert spaces
Two products that produce very different things, and the space where all of this lives. Mixing up the two products is the single most common early error.
Unitary operators and projections
Operations that preserve length, and operations that collapse onto a subspace. These are exactly what quantum gates and quantum measurement turn out to be.
Eigenvalues and eigenvectors
The vectors an operation leaves pointing the same way. They are how a measurement's possible outcomes are described, so this is not an abstract detour.
Introduction to GCD and congruence
Two pieces of number theory that look out of place here and are needed by the factoring algorithm at the end of the course.
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