July 22, 2024
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Quantum – Theory – Qubits and Bloch Sphere

This post describes the quantum theory behind qubits.

https://logosconcarne.com/2021/03/15/qm-101-bloch-sphere/

Coins in the Air

Consider qubits to be like coins that are tossed into the air. While the coins are spinning, it is possible to change their spin in a way that can change the possibility of a coin landing on heads or tails. Quantum computers are analogue computers. Consequently, they are subject to physical errors. As a result quantum programs typically run many times (For example 1024 times). The state of the qubits are recorded after each run. This helps eliminate any physical errors.

Quantum, Qubits and Bloch Sphere

Wikipedia describes a qubit as follows:

‘A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization.’

Bloch Sphere

Wikipedia describes the Bloch Sphere as: ‘the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit).’

The Bloch sphere is in 3 dimensions but the mathematics that supports qubits operate with the 2 states |0> and |1>. The qubit is in a mix of these 2 states until measurement takes place.

A qubit can only be either 0 or 1 when measured. 1 represents it being located in the northern hemisphere of the Bloch Sphere and 0 in the southern hemisphere. This gives 2 fundamental states in of   |0> and |1>. 

Use cosines and sines to identify locations in the x-z plane . The cosine of a point on this plane gives the x-axis co-ordinate and the sine gives the z-axis co-ordinate. To locate a point that is not in the x-y plane, but elsewhere in the sphere, it is necessary to be able to include a reference to the y-axis. The i value coordinates of points on the y-axis represent the fact that the points are only reachable by some rotation of i out of the x-z plane.

Normalised quantum states are pure states and therefore are on the surface of the Bloch Sphere

The Cartesian coordinates for the x, y and z-axes have been added to the Bloch Sphere diagram, such as the coordinates (0, 0, 1) which are beside the z-axis.

Next: Wave Number Bloch Sphere

Previous: Introduction

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