Qubit

But what is a quantum phase factor?

An intuitive approach from absolute scratch

Laying the Foundation

On a geometric representation

The amplitudes are the coefficients in front of each state in superposition and are proportional to the probabilities of the electron being in each of those basis states, we’ll review this soon.
cos(θ) = horizontal component ; sin(θ) = vertical component
Known as Born’s Rule

Phases in Layman’s Terms

What’s the point of the amplitudes being complex?

One complex amplitude → Black: the only possibility given a real amplitude; Red: examples of unlocked possibilities with a complex dimension
Exponential form for complex numbers. Where theta is in radians

But phases are only for waves right?

Where delta-t is the phase

What’s the point of phase?

Creating another geometric representation

A recap on interference

Complete Destructive interference (COSMOS)

Why the global phase doesn’t matter

Connecting it all back into quantum states

Let the mathematics do the talking

General equation for a qubit state
How global phase vanishes when calculating probability

An aside on the Bloch sphere

cos(θ) = horizontal component ; sin(θ) = vertical component

Voila!

A three-dimensional representation of a given quantum state where Φ is the phase factor and sin(θ/2) is the amplitude for the |0> state and cos(θ/2) is the amplitude for the |1> state

General equation for a qubit state on a Bloch sphere

All you need to know

  • An arbitrary state space is a two-dimensional complex vector space, which is then four real dimensions. The fourth dimension can be factored out as a global phase, so then what remains is a three-dimensional representation which is the Bloch sphere*.

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pavanjay.com | Invested in QC + ML | EECS @UWaterloo | Seeker of rigorous truth

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