HW5 - Problem 2

Problem 2: Hadamard Waves

People say quantum mechanics describes tiny objects as both a wave and a particle. Besides the use of the word ``amplitude’’, it doesn’t feel like we have seen much wavy-ness in this class. The layer of Hadamard gates gives us one look at states that look somewhat wave like.

If we apply \(H^{\otimes 4}\) to \(\ket{0000}\), we can draw the amplitudes as the following, where + means the amplitude of the term is positive, and - means it is negative.

1. Draw the amplitudes of the states \(H^{\otimes 4}\left|0001\right>\), \(H^{\otimes 4}\left|0010\right>\), \(H^{\otimes 4}\left|0100\right>\), \(H^{\otimes 4}\left|1000\right>\).
2. Draw the amplitudes of the state \(H^{\otimes 4}\left|1010\right>\). How does this relate to amplitudes you got in the previous problem?
3. Based on what you got in the previous part, draw the amplitudes for \(H^{\otimes 4}\left|1101\right>\). Verify that you have the correct signs for each term.