Quantum Computing

Introduction

Run on superconducting circuits that require very low temperature to work.

Hardware

Technologies

• Quantum-dot electron-spin qubits
• Nitrogen-vacancy centers in diamons
• Transmon qubits (superconducting circuits used by IBM)
• Topological qubits (marjorana fermions)

These can all compute the same thing, they are all ‘quantum complete’. But have different efficiency.

Limitations

• All early in implementation
• Must be called to near absolute zero
• Scalability (number of qubits)
• Quantum coherence is very short, so programs must run fast.

Quantum mechanics concepts

Superposition

A quantum sytem is in a superposition of many states

Measurement

measruement causes the wave-function to collapse

Entanglement

If I have more than one qubits, then they can be entangled making their wave-function intertwined

Interference

This is fundamental. Several superimposed states of a system can be made to reach the same final state via two different paths.

Spin

Fundamental quantum physical properties. Each particle with either be spin-up or spin-down.

Splitter

Randomly flips the spin. Eg. partial mirror.

Quantum operations

Must be:

• linear,
• reversible
• preserve total probability
• “unitary”

NOT gate

Also called Pauli X gate.
${\sigma }_X|0⟩=⟨1|$
${\sigma }_x|1⟩=⟨0|$
${\sigma }_X|\psi ⟩=\sigma (\alpha |0⟩+\beta |1⟩)=\alpha |1⟩+\beta |0⟩$

Represented as

$${\sigma }_X=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$

Z gate

$${\sigma }_Z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$

Hadamar gate

$$\begin{array}{r}H|0⟩=\frac{|0⟩+|1⟩}{\sqrt{2}}\\ H|1⟩=\frac{|0⟩-|1⟩}{\sqrt{2}}\\ H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\end{array}$$

Applications

Breaks certain classical encryption such as RSA using Shor’s Algorithm
Grover’s Search Algorithm
Quantum Fourier Transform
Quantum Fast Factorization