Quantum Country exercise 11

Posted on October 19, 2021 by hecker

This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.

Exercise 11. Find single-qubit states $\vert a \rangle$ and $\vert b \rangle$ such that applying the CNOT gate to the combined state $\vert ab \rangle$ changes the first qubit, i.e., the control qubit.

Answer: This is a placeholder for when I get around to looking at this problem in depth.

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1 Response to Quantum Country exercise 11

C says:

August 18, 2022 at 3:23 am

H∣+−⟩ = ∣−−⟩

Proof (coefficients ignored)

LHS:
H∣+−⟩
= H∣+, −⟩
= H∣∣0⟩ + ∣1⟩, ∣0⟩ – ∣1⟩⟩
= H(∣0⟩∣0⟩ + ∣0⟩∣-1⟩ + ∣1⟩∣0⟩ + ∣1⟩∣-1⟩)
= H(∣00⟩ – ∣01⟩ + ∣10⟩ – ∣11⟩)
= H∣00⟩ – H∣01⟩ + H∣10⟩ – H∣11⟩
= ∣00⟩ – ∣01⟩ + ∣11⟩ – ∣10⟩

RHS:
∣−−⟩
= ∣−,−⟩
= ∣∣0⟩ – ∣1⟩, ∣0⟩ – ∣1⟩⟩
= ∣0⟩∣0⟩ + ∣0⟩∣-1⟩ + ∣-1⟩∣0⟩ + ∣-1⟩∣-1⟩
= ∣00⟩ – ∣01⟩ – ∣10⟩ + ∣11⟩
= ∣00⟩ – ∣01⟩ + ∣11⟩ – ∣10⟩

LHS = RHS

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