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Lecture 1.2 - Introduction to Quantum Circuits

Last updated Mar 27, 2022

# Notes

πŸ•’ 00:20 Circuit model is a sequence of blocks that carry out elementary computations, called gates.

πŸ•’ 01:22 One-Qubit Quantum Gates

πŸ•’ 02:56 Pauli X-gate

πŸ•’ 06:49 Pauli Z-Gate

πŸ•’ 08:31 Pauli Y-Gate

πŸ•’ 11:09 Hadamard Gate

πŸ•’ 14:44 S-gate

It is defined as: $S = \begin{pmatrix}1 & 0 \\ 0 & i \end{pmatrix}$ . It adds $90^\circ$ to the phase $\phi$ . $$S\ket{+} = \ket{+i}, \qquad S\ket{-} = \ket{-i}.$$
πŸ•’ 15:26 $SH$ is applied to change from $Z$ to $Y$ basis.

πŸ•’ 16:34 Multipartite Quantum States

πŸ•’ 21:46 Two-Qubit Quantum Gates

πŸ•’ 29:34 Entanglement

πŸ•’ 31:45 Bell states

These are four so called Bell states that are Maximally entangled and build an orthonormal basis:
$$\begin{aligned} \ket{\psi^{00}} &= \frac{1}{\sqrt 2}(\ket{00} + \ket{11}), \qquad \ket{\psi^{01}} = \frac{1}{\sqrt 2}(\ket{01} + \ket{10}), \newline \ket{\psi^{10}} &= \frac{1}{\sqrt 2}(\ket{00} - \ket{11}), \qquad \ket{\psi^{11}} = \frac{1}{\sqrt 2}(\ket{01} - \ket{10}). \end{aligned}$$
In general, we can write $\ket{\psi^{ij}} = (\mathbb{I} \otimes \sigma_x^j \cdot \sigma_z^i) \ket{\psi^{00}}$ .

πŸ•’ 35:42 Creation of Bell states

Lecture 1.2 - Introduction to Quantum Circuits_creation_of_bell_states.excalidraw.svg

πŸ•’ 42:04 Quantum Teleportation

πŸ•’ 42:16 Goal

πŸ•’ 44:41 Initial state of the total system

\ket{\psi}S \otimes \ket{\phi^{00}}{AB} &= \frac{1}{\sqrt 2}(\alpha \ket{000}{SAB} + \alpha\ket{011}{SAB} + \beta\ket{100}{SAB} + \beta\ket{111}{SAB}) \newline
&= \frac{1}{2 \sqrt 2}[(\ket{00}{SA} + \ket{11}{SA}) \otimes (\alpha\ket{0}B + \beta \ket{1}B) \newline
&\qquad + (\ket{01}
{SA} + \ket{10}
{SA}) \otimes (\alpha\ket{1}B + \beta \ket{0}B) \newline
&\qquad + (\ket{00}
{SA} - \ket{11}
{SA}) \otimes (\alpha\ket{0}B - \beta \ket{1}B) \newline
&\qquad + (\ket{01}
{SA} + \ket{10}
{SA}) \otimes (\alpha\ket{1}_B - \beta \ket{0}B)] \newline
&= \frac{1}{2}[\ket{\psi^{00}}
{SA} \otimes \ket{\phi}B + \ket{\psi^{01}}{SA} \otimes (\sigma_x \ket{\phi}B) \newline
&\qquad + \ket{\psi^{10}}
{SA} \otimes (\sigma_z \ket{\phi}B) + \ket{\psi^{11}}{SA} \otimes (\sigma_x \sigma_z \ket{\phi}_B)].

πŸ•’ 51:33 Protocol

Lecture 1.2 - Introduction to Quantum Circuits_quantum_teleportation_protocol.excalidraw.svg
πŸ•’ 57:19 Note, that Alice’s state collapsed during the measurement, so she does not have the initial state $\ket{\phi}$ anymore. This is expected due to the no-cloning theorem, as she cannot copy her state, but can just send her state to Bob while destroying her own.

πŸ•’ 59:03 An unexpected reference to one of my favorite movies, The Prestige πŸ™‚