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Chapter 1

The problems in this chapter are focused on qubit states and simple untiary operators.

  1. Background Material
  2. Problems
    1. Programming Problems

Background Material

Any two-state quantum system can be used as a quantum bit (qubit); the corresponding quantum computing version of a bit but with the addition that it can be in a superposition state.

The vector space with defined inner product (i.e. Hilbert space) that describes that state of a qubit is C2\mathbb{C}^2.

The Dirac notation to denote an orthonormal qubit basis set is given as {∣0⟩,∣1⟩} \{\ket{0},\ket{1}\} with the inner product defined as ⟨0∣0⟩=⟨1∣1⟩=1 \braket{0|0}= \braket{1|1} = 1 and ⟨0∣1⟩=⟨1∣0⟩=0 \braket{0|1} = \braket{1|0} = 0 .

A superposition state can be written as:

∣ψ⟩=α∣0⟩+β∣1⟩ \ket{\psi} = \alpha \ket{0} + \beta \ket{1}

with the conditions that the inner product ⟨ψ∣ψ⟩=α∗α⟨0∣0⟩+β∗β⟨1∣1⟩=1 \braket{\psi|\psi} = \alpha^{*} \alpha \braket{0|0} + \beta^{*} \beta \braket{1|1} = 1

Examples of basis sets are:

∣0⟩=(10),∣0⟩=(01) \ket{0} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \hskip{5mm} \ket{0} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} ∣0⟩=∣+⟩=12(11),∣0⟩=∣−⟩=12(1−1) \ket{0} = \ket{+} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \hskip{5mm} \ket{0} = \ket{-} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}

Problems

Programming Problems

© S. Bringuier (stefanbringuier@gmail.com). Last modified: August 25, 2021. Website built with Franklin.jl and the Julia programming language.