# A Taste of Quantum Computing

## Basics

This section is my notes for Mark Oskin's Quantum Computing - Lecture Notes.

### Postulates

#### Qubit

"Associated to any isolated physical system is a complex vector space with inner product (i.e. a Hilbert space) known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the system’s state space."

Qubit is a 2-dimensional state space:

$\lvert \Psi \rangle = a \lvert 0 \rangle + b \lvert 1 \rangle, \textrm{ where } a, b \in \mathbb{C}, and |a|^2 + |b|^2 = 1$

#### Evolution of Quantum Systems

"The evolution of a closed quantum system is described by a unitary transformation. That is, the state $$\lvert \Psi \rangle$$ of the system at time $$t_1$$ is related to the state of $$\lvert \Psi' \rangle$$ of the system at time $$t_2$$ by a unitary operator $$U$$ which depends only on times $$t_1$$ and $$t_2$$."

$\lvert \Psi' \rangle = U \lvert \Psi \rangle$

where $$U$$ must be unitary, $$U^{\dag} U = I$$. The fact that $$U$$ is not related to $$\Psi$$ is disappointing because otherwise NP-complete problem is solvable by quantum computers.

#### Measurement

"Quantum measurements are described by a collection $$M_m$$ of measurement operators. These are operators acting on the state space of the system being measured. The index $$m$$ refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is $$\lvert \Psi \rangle$$ immediately before the measurement then the probability that result $$m$$ occurs is given by:

$p(m) = \langle \Psi \rvert M_m^{\dag} M_m \lvert \Psi \rangle$

and the state of the system after measurement is:

\frac{M_m \lvert \Psi \rangle}{\sqrt{⟨ Ψ \rvert Mm Mm \lvert Ψ ⟩}}

#### Multi-Qubit Systems

"The state space of a composite physical system is the tensor product of the state spaces of the component physical systems. [sic] e.g. suppose systems $$1$$ through $$n$$ and system $$i$$ is in state $$\lvert \Psi_i\rangle$$, then the joint state of the total system is $$\lvert \Psi_1\rangle \otimes \lvert \Psi_2\rangle \otimes \cdots \lvert \Psi_n\rangle$$"

This is not a proof but a postulate for quantum computing.

## Compilers

Email: expye@outlook.com

Date: 2022-11-14 Mon 00:00