Lecture 003 - Quantum vs Probabilistic Computing

H Gate

\begin{align*} Hat(0) =& \begin{cases} 0 & \text{w.p } \sqrt{1/2}\\ 1 & \text{w.p } \sqrt{1/2}\\ \end{cases}\\ Hat(1) =& \begin{cases} 0 & \text{w.p } \sqrt{1/2}\\ 1 & \text{w.p } -\sqrt{1/2}\\ \end{cases} \end{align*}

The above is a quantum automata with the following probability matrix:

\begin{bmatrix} \sqrt{1/2} & \sqrt{1/2}\\ \sqrt{1/2} & -\sqrt{1/2}\\ \end{bmatrix}

Quantum vs Probabilistic

Quantum Automata:

When you calculate, treat amplitude like probability. In the middle of amplitudes, you might have negative amplitudes. But after you combine, squared, and sum all the leaves, you get probability.

In probabilistic code, you have the following property:

In quantum code, you have the following property:

The fundamental difference between classical and quantum computation is that combining the same outcome probability/amplitude could cancel probability/amplitude out. You can imagine that nature is keeping track of forking tree since it knows which state to go back to (un-mix probability).

To fully capture the joint state, you need to keep track of the probability of every possible states. To simulate a random algorithm, you need only the transition matrix (all column sum to one) and track down one path. To simulate a quantum algorithm, you need the transition matrix (that is unitary), since you need to do addition and subtraction between branches.

You cannot track each bits individually because they might not be independent.

Also, although you can write probabilistic quantum instructions (do quantum instructions according to the result of fliping coins), but the calculation will be different. (You can't use the same matrix multiplication. Since probability square does not necessarily sum to one.)

Some Observations

(Add_1_to_Coin A)
Hat A

The above code prints exactly the same whether we run the instruction in "()". Because when we calculate the probability, we squared it therefore dropping the negative sign that come from 1 \to 1.

(Add_1_to_Coin A)
Hat A
Hat A

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