37 lines
1.2 KiB
Markdown
37 lines
1.2 KiB
Markdown
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I think it is extremely important after learning linear algebra exactly what context that quantum bits become useful. Otherwise if we focus too much on the math, it can start becoming a bit complicated. Here are some essential diagrams to understand.
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---
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The TL;DR version of this section is: given that qubits are connected to complex numbers, use imaginary complex sphere for their representation, and worry about angles and lattitudes.
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Let’s see why is this the case. Consider the case of a quantum state:
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$$
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|\psi\rangle = c_{0}\cdot|0\rangle + c_{1} \cdot |1\rangle
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$$
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Reminder:
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$$
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|c_{0}|^2 + |c_{1}|^2 = 1
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$$
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where
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$$
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c_{i} \in \mathbb{C}
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$$
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Using the polar representation of a complex number, we have the following two equivalent representations:
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$$
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c_{0} = r_{0} ^{ej\varphi_{0}} + r_{1}e^{j\varphi_{1}}
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$$
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c0=r0ejφ0 and c1=r1ejφ1c0=r0ejφ0 and c1=r1ejφ1
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where $$riri$$ are the amplitudes and φiφi are the angles. Thus, each individual number cici can be represented with a unit imaginary circle.
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The unit circle (including imaginary complex numbers)
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![[Pasted image 20230105170239.png]]
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The bloch sphere representation also depicts the same information as above:
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