Introduction to Quantum Computation

Akira Imada Brest State Technical University, Belarus

(last modified on)

May 2, 2007

1 Basic Concept for Quantum Computation

In general, a Hilbert space of wave functions. For quantum-C, in particular, finite dimensional complex vector space with an inner product that are spanned by abstract wave function such as | →⟩ will be sufficent. (Rieffel, 2000)

Rieffel (2000) introduced an example of two basis vector | ↑⟩ and | →⟩ which are both state vectors. Then wrote “When we measure state the state ψ = a| ↑⟩+b| →⟩ is measured as | ↑⟩ with the probability ∥a∥^{2} and | →⟩ with the probability ∥b∥^{2}.

Then what means by “spanned?” Ambainis (2003) uses this word as

Let H be the subspace spanned by states |ψ_{i}⟩.

or

Consider Hilbert space spanned by |x⟩|i⟩ where x ∈ {0, 1}^{n} corresponds to a vertex of hypercube and i ∈ {1, ⋯, n} corresponds to an edge.

As for Hamiltonian

One can define walk on the line on the set of states |n⟩, n ∈ Z. To do that, we take Hamiltonian H defined by H|n⟩ = −|n − 1⟩ + 2|n⟩ − |n + 1⟩ Running quantum walk for time t is just applying the transformation U^{iHt}

Ket Notation Quantum state spaces and transformations acting on them can be descriged in terms of vectors and matices or bra/ket notation. Kets like |x⟩ denote column vectors and typically quantum states (Rieffel, 2000).

Qubit Qubit is a column vector in a two dimensional complex vector space where a paticular basis, e.g., |0⟩ and |1⟩ where |0⟩ and |1⟩ correspond to 0 and 1 in classical comuter.

However qubits is a superposition of |0⟩ and |1⟩ — a|0⟩ + b|1⟩. That is, the probility of measurment being |0⟩ is ∥a∥^{2} and |1⟩ being ∥b∥^{2} (Rieffel, 2000).

Example of state The position on the line |x⟩, and joint state of a particle at posiion x with a coin in state |L⟩, |L, x⟩ (Rieffel, 2000).

J. Kempe (2003) wrote ”Imagine a particle on a line whose position is described by a wave-packet |ψ_{x_{0}}⟩ localized around a position x_{0}, i.e. the function ⟨x|ψ_{x_{0}}⟩ corresponds to a wave-packet centered around x_{0}.

This make us a clear image. A particle with a state of its polarity, an electoron with a state of its spin ... In the case of randome walk a particle in a certain position in an n-dimensional lattice whose state is 0 or 1.

1.1 Random Walk

Now let P be the momentum operator. The translation of the particle, corresponding to a step of length l can be represented by the unitary operator U_{l} = exp(−iP l/ℏ) so that U_{l}|ψ_{x_{0}}⟩U_{l}|ψ_{x_{0}−l}⟩ (Kempe, 2003).

Ambainis (2003) further spcialized the situation. We consider the state space consisting of states |n, 0⟩ and |n, 1⟩. While classical computer uses a single binary bit. The counter part of bit of quantum computer is qubit which is a unit vector in a complex Hilbert space C^{2}.

Now that we can guess what Ambainis (2003) denote |n, 0⟩ and |n, 1⟩ in the context of random walk in n-dimensional hypercube, as

|a vertex in the hypercube, wave function of a state⟩.