An introduction to this research project

Description

This has been an ongoing project for many months now. I have worked on it actively and inactively over this time, and have made significant progress in linear algebra and working in Hilbert Space. While I know that many topics here will require significantly more reading and problem-solving, it is healthy to create goals along the way, so that is just what I have done. The book that I am reading to learn and take note of this information is Quantum Computation and Quantum Information by Michael A. Nielsen and Isaac I. Chuang. This is not an easy project, and I, with complete knowledge, expect this to take a significant amount of time. Though this is a serious research topic, I am treating it as a casual activity, and will add research whenever I have time or whenever I feel like learning some new material. The material that I am working on is incredibly rigour-heavy, so it is rare that I will update with anything purely theory, yet it in and of itself is incredibly fascinating and I highly recommend the dear reader investigate on their own.

Some words about notation/nomenclature and prerequisites

Though these notes are intended to be as accessible to a general audience as possible, I will not be developing anything in elementary linear algebra. I am still in the process of learning quite a bit of linear algebra myself, so the more complex ideas will certainly be introduced, but everything else will be assumed (a first course in linear algebra will more than suffice). Unlike in standard linear algebra, the linear algebra that we will do here involves a completely new system of notation, especially for vectors. We will introduce some notational continuities throughout these notes now.

As I cannot construct a table in \(\LaTeX\) (I am very sad) in this website due to unsupported packages, a paragraph form will have to suffice. Some specifics may be ommitted here. We will always use “\(^{*}\)” to denote conjugation. We denote \(|v\rangle\) to be a ket , and \(\langle v|\) to be a bra , otherwise known as the vector dual to \(|v\rangle\). We denote \(\langle v|w \rangle\) to be the inner product between two vectors \(|v \rangle\) and \(|w \rangle\). We denote \(|v \rangle \bigotimes |w\rangle = |v \rangle|w \rangle\) to be the tensor product between two vectors \(|v \rangle\) and \(|w \rangle\). We denote \(A^{*}\) to be the conjugate, \(A^{T}\) to be the transpose, and \(A^{\dagger}\) to be the Hermitian conjugate or adjoint for some matrix \(A\). The matrix \(A\) is assumed to be a square matrix unless otherwise specified. We denote \(M_{n\times n}(\mathbb{R})\) to be the set of \(n\times n\) matrices with real entries. The symbol \(\delta_{ij}\) denotes the Kronecker delta: if \(i=j\), then \(\delta_{ij}=1\), and \(\delta_{ij}=0\) otherwise (it has uses beyond this elementary definition). We will also be making use of a nesting matrix notation known as submatrices or matrix block form. This means that we are placing a matrix inside another matrix as a continuation of its entries. For example,

\[\begin{bmatrix} a_{11} & \cdots & a_{n1} \\ \vdots & & \\ a_{1n} & & B \end{bmatrix} = \begin{bmatrix} a_{11} & \cdots & a_{n1} \\ \vdots & & \\ a_{1n} & & \begin{bmatrix} b_{11} & \cdots & b_{n1} \\ \vdots & \ddots & \vdots \\ b_{1n} & \cdots & b_{nn} \end{bmatrix} \end{bmatrix}.\]

Finally, when we refer to a “number” in these notes, we are more often than not referring to some complex number from \(\mathbb{C}\). When we speak of a finite-dimensional vector space, we are largely talking about a finite-dimensional vector space over the field \(\mathbb{C}\).