associative means that operations can be grouped in any way as long as order is preserved.

That is:

\begin{equation} (AB)C = A(BC) \end{equation}

Intuition:

• \(O\): \(\leq\)
• \(\theta\): \(=\)
• \(\Omega\): \(\geq\)

Definitions:

• \(f(n) = O(g(n)) \implies \exists n_{0}: \forall n > n_0, f(n) \leq c (g(n))\)
• \(f(n) = \Omega(g(n)) \implies \exists n_{0}: \forall n > n_0, f(n) \geq c (g(n))\)
• \(f(n) = \theta(g(n)) \implies \exists n_{0}: \forall n > n_0, f(n) \geq 1 (g(n)), f(n) \leq c (g(n))\)

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Given functions \(f(n)\) and \(g(n)\), if:

\begin{equation} \lim_{n\to \infty} \left(\frac{f(n)}{g(n)}\right) = 1 \end{equation}

we say that \(f \sim g\).

You can use atoms as many different types of qubits.

manipulating physical qubits

To make physical qubits go to different states, we will again use something in the ancillary states. Rotating it to \(z\) — leverage one lazer to make it fall; \(rx\), \(ry\), we leverage combinations of two light.

various qubit implementations

Implementations of physical qubits

possible uses for qubits

Here are some possible uses for physical qubits

AFIB is a heart conditinos, which is augmented during heart surgery.

“Endogeneous extrolluculor proteases damage to Kv1.5 in the atria contributes to AFIB”

• WBC’s secretion of proteases, such as calpain when it is inflamed