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Week 7: QUANTUM STATE TRANSFORMATIONS | ECE 802-730
Week 7: QUANTUM STATE TRANSFORMATIONS | ECE 802-730

Week 7: QUANTUM STATE TRANSFORMATIONS | ECE 802-730

Coursework focused on Chapter 5 of Quantum Computing: A Gentle Introduction.

Exercise 5.12

Design a circuit that constructs the Hardy State: \frac{1}{\sqrt{12}}(3|00\rangle + |01\rangle + |10\rangle + |11\rangle)

(\alpha|0\rangle + \beta|1\rangle) \otimes |0 .....0\rangle

Basis States:

0: \frac{1}{\sqrt{12}}(3|00\rangle + |01\rangle), 1: \frac{1}{\sqrt{12}}(|01\rangle + |11\rangle)


\left(\frac{3}{\sqrt{12}}\right)^{2} + \left(\frac{1}{\sqrt{12}}\right)^{2}, \left(\frac{1}{\sqrt{12}}\right)^{2} + \left(\frac{1}{\sqrt{12}}\right)^{2} = \frac{2}{\sqrt{12}}

1st Qubit \rightarrow \left(\sqrt{\frac{10}{12}}|0\rangle + \sqrt{\frac{2}{12}}|1\rangle\right) \rightarrow R_y Gate \rightarrow REPEAT

Or simply,

Exercise 5.13

Show that the swap circuit of section 5.2.4 does indeed swap into two single-qubit values in that it sends |\psi\rangle|\phi\rangle to |\phi\rangle|\psi\rangle for all single-qubit states |\psi\rangle and |\phi\rangle

C_{NOT} = |\psi\rangle\langle\psi| \otimes I + |\phi\rangle\langle\phi| \otimes X

= |\psi\rangle\langle\psi| \otimes \left(|\psi\rangle\langle\psi| + |\phi\rangle\langle\phi|\right) + |\phi\rangle\langle\phi| \otimes \left(|\phi\rangle\langle\psi| + |\psi\rangle\langle\phi|\right)

=|\psi\psi\rangle\langle\psi\psi| + |\psi\phi\rangle\langle\psi\phi| + |\phi\phi\rangle\langle\phi\psi| + |\phi\psi\rangle\langle\phi\phi|

Therefore,

|\psi\psi\rangle \rightarrow |\psi\psi\rangle
|\psi\phi\rangle \rightarrow |\psi\phi\rangle
|\phi\psi\rangle \rightarrow |\phi\phi\rangle
|\phi\phi\rangle \rightarrow |\phi\psi\rangle

Exercise 5.13

Show how to implement the Toffoli Gate in terms of a single-qubit and C_{NOT} Gates.

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