<aside> 📌 In an analogous manner, we define a basic unit of information to be used in quantum computation called the qubit (quantum bit).

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Miscellaneous Definitions:

<aside> ✏️ Trace of a square matrix A, denoted tr, is defined to be the sum of elements on the main diagonal of A. The trace is only defined for a square matrix. It can be proved that the trace of a matrix is the sum of its eigenvalues.

</aside>

Qubit

• An object enclosed in the notation $\ket{}$ can be called a state / vector or a ket. Such a representation of vectors (including inner products and bras, which we learn about in this chapter) follows from the Dirac Notation.

  STATES

  • Can be present in one of two states $\ket{0}$ & $\ket{1}$, similar to a bit.

  • However, it can also exist in a superposition state (linear combination) of the two.

  • This, can be represented as follows ($\ket{\psi}$ being the superposition state).

    $\ket{\psi} = \alpha\ket{0} + \beta\ket{1}$

    • $\alpha$ & $\beta$ being complex numbers. ($z^{*}$ = conjugate of $z$)
    • $|\alpha|^2 = (\alpha)(\alpha^{*})$ : Probability of finding $\ket{\psi}$ in state $\ket{0}$.
    • $|\beta|^2 = (\beta)(\beta^{*})$ : Probability of finding $\ket{\psi}$ in state $\ket{1}$.
    • If, $|\alpha|^2 + |\beta|^2 = 1$ (i.e. the probability sum equals 1), the qubit is said to be normalized.

  • On measurement, however a qubit falls back to one of the two states : $\ket{0}$ & $\ket{1}$.

  • Illustration:

Vector Spaces

• OPERATIONS & AXIOMS

• REPRESENTATION & DEFINITIONS

• MATRIX OPERATIONS

Linear Combination of Vectors

For a set of complex coefficients $\alpha_{i}$ and a set of vectors $\ket{v_{i}}$, a linear combination is given by:

$\alpha_{1}\ket{v_{1}} + \alpha_{1}\ket{v_{1}} + \dots + \alpha_{n}\ket{v_{n}} = \sum_{i=1}^{n}\alpha_{i}\ket{v_{i}}$

A set of vectors $\{\ket{v_{i}}\}$ is said to span a given vector space $V$, if $\ket{v_{1}}, \ket{v_{2}} \dots \ket{v_{n}}$ can be used to represent any vector $\ket{u} \in V$. Consider, the following example for a vector space $\mathbb{C}^3$:

On inspection, one might notice the qubit $\ket{\psi} = \alpha\ket{0} + \beta\ket{1}$, can be represented as: