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From Classical To Quantum Computing

These are some key concepts that make quantum computing unique from classical computing.

Quantum States

  1. Quantum computers represent information as qubits which can represent binary states (0’s and 1s) like classical computers but because qubits are actually wave functions (Quantum Wave Function) they can be conceptualized as spherical objects with directional coordinates called the Bloch vector.

More information can be found on my writings on Pauli Spin Matrices

\[\displaylines{ \\ \ket{\psi} = cos(\theta/2)\ket{0} + e^{i\phi}\ sin(\theta/2)\ket{1} = cos(\theta/2)\ket{0} + (cos\ \phi + i\ sin \phi) \ sin(\theta/2) \ \ket{1}\\ where \ 0 \le \theta \le \pi\ \\ and\\\ \ 0 \le \phi < 2\pi }\]
  1. As such a qubit is said to be in the 0 state when its orientation is pointing up and it is said to be in the 1 state when its orientation is pointing down.

cbits-qubits

  1. Mathematically we can say that each qubit as a linear combination of ket 0 and ket 1

    \[\displaylines{ \ket{\psi} = \alpha \ket{0} + \beta \ket{1}\\ \ \ket{0} = \begin{bmatrix} 1\\ 0\\ \end{bmatrix} \\ \ket{1} = \begin{bmatrix} 0\\ 1\\ \end{bmatrix} }\]

Superposition

  1. A qubit has the ability to be in a superposition state where the arrow is neither pointing up or down. Which makes it a combination of 0 and 1 until the time it is measured which then makes it either a 0 or a 1 depending on the probability as set by the direction of the arrow such that:

    • If the arrow is slightly pointing up it has a higher probability that it will be a 0.
    • If it is slightly pointing down it has a higher probability that it will be measured as a 1.
    • If it’s exactly on the equator then either state has a 50% probability.

    From the second axiom of probability theory we can represent this as:

    \[P(\Omega) = |\alpha|^{2} + |\beta|^{2} = 1\]

    superposition

Quantum Entanglement

  1. In classical computing each bit is independent of each other meaning one bit is not influenced by other bits and vice versa. In quantum computing qubits can be entangled with each other, meaning one qubit can influence the state of another qubit. This means that a set of qubits can be part of a much larger quantum state.

    We can see this in the following example. Here we have 2 qubits that are not yet entangled so they both have individual probabilities that are independent of each other.

    untangled

  2. When we entangle the bits then the probability distribution becomes the state.

    \[\displaylines{ \begin{array}{|c|c|} \hline \ket{00} & 22.5\%\\ \hline \ket{01} & 32.5\%\\ \hline \ket{10} & 15.5\%\\ \hline \ket{11} & 27.5\%\\ \hline \end{array} }\]

    Once the qbits are entangled then the direction of one qubit will alter the probability distribution of all the possible states. Thus, increasing the number of qubits in the calculation will allow for a larger number of possible states at \(S = 2^n\)

    \[\displaylines{ \begin{array}{|c|c|} \hline qubits & states\\ \hline 1 & 2\\ 2 & 4\\ 3 & 8\\ \vdots & \vdots\\ n & 2^{n}\\ \hline \end{array} }\]

  3. Entanglement is one of the most signiicant difference between quantum computing and Classical computing. While a classical computer can be in one single state at a given time, a quantum computer can have multiple states at a single given time. This gives way to some very powerful computing algorithms that can never be realized in a single state classical computer.

Quantum Interference

  1. In a quantum computer with many number of qubits that are entangled together. Their individual wavefunctions are added together into an overall wavefunction which becomes the state of the computer.
  2. The entanglement is basically the interference of all the individual wavefunctions of every qubit affecting each other. This is akin to dropping a pebble in a pond and then dropping a second pebble so that the ripples produced from each pebble would interfere with each other creating a different pattern of ripples on the water. Each ripple can either amplify (constructive interference) or stop the ripples from propagating (destructive interference).
  3. In a quantum computation scenario, we would use constructive interference increase the probability computing the correct answer while using destructive interference to decrease the probability of computing the incorrect answer.