Characteristics of a Quantum System

The 2 Characteristics of a Quantum System, which are helpful in designing better algorithms for Quantum Systems are:

1. Superposition:

   Superposition or Linear Combination of states helps perform parallel computations in a Single Circuit (In short, it speeds up the Execution).

2. Interference:

   Helps us to gain infromation about a function $f(x)$ wrt various input values of $x$.

   $\therefore$ it deduces certain global properties of the given function.

Importance of $H-Gate$

• Helps create a superposition of states in the form of COMPUTATIONAL BASIS.

• Further, $H(H\ket{\psi}) = \ket{\psi}$, for any state $\ket{\psi} = a\ket{0} + b\ket{1}$.

• Hadmard Transform : $n$ Hadamard gates acting in parallel on $n$ Qubits creating a Superposition Phenomenon.

  here, $x$ is represented in binary (one can clearly notice that the superposition has generated all possible n-qubit states with equal probability):

  $(H\ket{0})^{\otimes n} = \left (\large \frac{\ket{0} + \ket{1}}{\sqrt{2}}\normalsize \right )^{\otimes n} = \large \frac{1}{\sqrt{2^{n}}} \normalsize \sum_{x=0}^{2^n - 1}\ket{x}$

• Let’s now see , the affect of applying the Hadamard gate to each qubit of a general register of n-qubits:

Discrete Phase Gate

• Given by

  $R_k = \left[\begin{matrix}1 & 0\\0 & e^{2\pi i/2^k}\end{matrix}\right]$ where, $k = \{0, 1, 2, \dots\}$

• $R_3 = T$ (or $\pi/8$ gate)

Serial & Parallel Operations

• Serial operations are given by product of operators:

  

• Parallel operations are given by tensor product of operators to be applied in parallel:

  

Quantum Interference

Addition of Probability Amplitudes.

• For example, consider the initial state:

  $\ket{\psi} = \large \frac{\ket{0} + \ket{1}}{\sqrt{2}}$

  it’s evident that the probability of measuring either $\ket{0}$ or $\ket{1}$ is (individually) = $\frac{1}{2}$.

• However, if we apply the Hadamard Gate to this state, the resulting state will become:

  $H(\ket{\psi}) = \large\frac{1}{\sqrt{2}}\left[ \frac{\ket{0} + \ket{1}}{\sqrt{2}} + \large \frac{\ket{0} - \ket{1}}{\sqrt{2}}\right]\normalsize = \ket{0}$

  We notice a:

  • Positive interference effect wrt $\ket{0}$, & a
  • Negative interference effect wrt $\ket{1}$.

Deutsch-Jozsa Algorithm

Problem Statement:

To determine whether a given boolean function $f(x)$ is constant or balanced. A function is said to be:

• Constant, if $f(x) = 0$ or $f(x) = 1$ for all inputs. Example:

  $f(x_0) = 0$

• Balanced, if $f(x) = 0$ for half of the inputs and $f(x) = 1$ for the other half. Example:

  $f(x_0, x_1, x_2) = x_0 \oplus x_1 \oplus x_2$

  

THE CLASSICAL COMPUTING APPROACH