A Taste of Quantum Computing

"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:

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

"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.

"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 M_m^† M_m \lvert Ψ ⟩}}

"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.