Abstract:
Starting with this lecture, we begin discussing computations on quantum systems. The simplest quantum system is a quantum bit (qubit). For the mathematical description of a qubit, we use the Hilbert space $\mathbb{C}^2$ with the chosen computational basis $\{\vert 0\rangle,\vert 1\rangle\}$. The set of pure states on a quantum bit is encoded by two nonzero complex coordinates modulo multiplication by a scalar, mixed states are encoded by density matrices, and reversible transformations are represented by unitary matrices. When describing a single qubit, it is convenient to use Pauli matrices and visualize the states of the qubit on a Bloch ball; unitary operations correspond to rotations of the ball. We discussed the simplest operations on a quantum bit: states $\vert 0 \rangle, \vert 1 \rangle, \vert + \rangle, \vert - \rangle, \vert +i \rangle, \vert -i \rangle$ and gates $I,X,Y,Z,H,S,T$. We showed that using the vocabulary $\{H,T\}$, it is possible to approximate any rotation of the Bloch sphere.
