ECE 802-730 Quantum

Week 7: QUANTUM STATE TRANSFORMATIONS | ECE 802-730

March 16, 2021 July 28, 2023

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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