Suppose we feed a qubit with initial state into a system consisting of cascaded quantum logic gates. Only the quantum gates and are available to manipulate the qubit. The final state results from a sequence of operations:
The two available component gates are defined below:
The gate system can be configured in different ways. The probability of observing output in the state can take distinct values. If the arithmetic mean of these probability values is , enter .
Note: denotes the floor function .
Supplemental Information:
Inspiration
Prereq #1
Prereq #2
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The following code evaluates all 128 possibilities. The process is the following:
1) For each binary permutation from 0 0 0 0 0 0 0 to 1 1 1 1 1 1 1 , assign the H matrix to 0 and the T matrix to 1
2) Multiply all 7 matrices together sequentially to yield an aggregate system matrix
3) Multiply the aggregate system matrix by ( 1 0 ) to yield a result in the form ( α β )
4) Calculate the probability of measuring the state ∣ 0 ⟩ = ( 1 0 ) by calculating ∣ α ∣ 2
5 ) Print analyses and results for all system configurations, and produce a sorted list of probabilities for measuring state ∣ 0 ⟩ = ( 1 0 )
Results (pastebin)
The probabilities for measuring the ∣ 0 ⟩ state can be the following:
0 . 0 0 0 0 0 0 0 0 0 . 1 4 6 4 4 6 6 1 0 . 5 0 0 0 0 0 0 0 0 . 5 7 3 2 2 3 3 0 0 . 7 5 0 0 0 0 0 0 0 . 8 5 3 5 5 3 3 9 1 . 0 0 0 0 0 0 0 0
Code: