From bd26d401d54311b353562673bc8762d4636083cb Mon Sep 17 00:00:00 2001 From: Stefano Pigozzi Date: Wed, 6 Nov 2024 13:41:36 +0100 Subject: [PATCH] Study quantum --- .../1 - Concetti base/qbit e quaternioni.md | 9 +++ .../1 - Concetti base/★ concetti base.canvas | 6 +- .../circuito quantistico.md | 12 ---- .../★ gates semplici.canvas | 25 ------- .../Hadamard gate.md | 0 .../Identity gate.md | 0 .../Pauli X gate.md | 2 +- .../Pauli Y gate.md | 2 +- .../Pauli Z gate.md | 2 +- .../Pauli gate.md | 0 .../2 - Gates/Toffoli gate.md | 25 +++++++ .../2 - Gates/X-axis rotation gate.md | 19 ++++++ .../2 - Gates/Y-axis rotation gate.md | 19 ++++++ .../2 - Gates/Z-axis rotation gate.md | 19 ++++++ .../circuito quantistico a caso.png | Bin .../2 - Gates/circuito quantistico.md | 61 ++++++++++++++++++ .../2 - Gates/controlled Pauli X gate.md | 28 ++++++++ .../gate quantistico universale.md | 0 .../gate quantistico.md | 2 +- .../2 - Gates/phase gate.md | 46 +++++++++++++ .../2 - Gates/swap gate.md | 25 +++++++ .../universal gate.md} | 3 + .../2 - 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Introduction to quantum information processing/2 - Gates semplici/Hadamard gate.md b/7 - Introduction to quantum information processing/2 - Gates/Hadamard gate.md similarity index 100% rename from 7 - Introduction to quantum information processing/2 - Gates semplici/Hadamard gate.md rename to 7 - Introduction to quantum information processing/2 - Gates/Hadamard gate.md diff --git a/7 - Introduction to quantum information processing/2 - Gates semplici/Identity gate.md b/7 - Introduction to quantum information processing/2 - Gates/Identity gate.md similarity index 100% rename from 7 - Introduction to quantum information processing/2 - Gates semplici/Identity gate.md rename to 7 - Introduction to quantum information processing/2 - Gates/Identity gate.md diff --git a/7 - Introduction to quantum information processing/2 - Gates semplici/Pauli X gate.md b/7 - Introduction to quantum information processing/2 - Gates/Pauli X gate.md similarity index 87% rename from 7 - Introduction to quantum information processing/2 - Gates semplici/Pauli X gate.md rename to 7 - Introduction to quantum information processing/2 - Gates/Pauli X gate.md index 46b7899..c753794 100644 --- a/7 - Introduction to quantum information processing/2 - Gates semplici/Pauli X gate.md +++ b/7 - Introduction to quantum information processing/2 - Gates/Pauli X gate.md @@ -44,4 +44,4 @@ Un quadrato con una X sopra, oppure occasionalmente un cerchio con una X sopra. ### Nella [[sfera di Bloch]] -Corrisponde a una [[rotazione]] sull'[[asse X]], quello da sinistra a destra. +Corrisponde a una [[rotazione]] di $\pi$ sull'[[asse X]], quello da sinistra a destra. diff --git a/7 - Introduction to quantum information processing/2 - Gates semplici/Pauli Y gate.md b/7 - Introduction to quantum information processing/2 - Gates/Pauli Y gate.md similarity index 78% rename from 7 - Introduction to quantum information processing/2 - Gates semplici/Pauli Y gate.md rename to 7 - Introduction to quantum information processing/2 - Gates/Pauli Y gate.md index 62945ea..02feff5 100644 --- a/7 - Introduction to quantum information processing/2 - Gates semplici/Pauli Y gate.md +++ b/7 - Introduction to quantum information processing/2 - Gates/Pauli Y gate.md @@ -20,4 +20,4 @@ Un quadrato con una Y sopra. ### Nella [[sfera di Bloch]] -Corrisponde a una [[rotazione]] sull'[[asse Y]], quello da dentro a fuori. +Corrisponde a una [[rotazione]] di $\pi$ sull'[[asse Y]], quello da dentro a fuori. diff --git a/7 - Introduction to quantum information processing/2 - Gates semplici/Pauli Z gate.md b/7 - Introduction to quantum information processing/2 - Gates/Pauli Z gate.md similarity index 86% rename from 7 - Introduction to quantum information processing/2 - Gates semplici/Pauli Z gate.md rename to 7 - Introduction to quantum information processing/2 - Gates/Pauli Z gate.md index 2f92b9f..b14a06a 100644 --- a/7 - Introduction to quantum information processing/2 - Gates semplici/Pauli Z gate.md +++ b/7 - Introduction to quantum information processing/2 - Gates/Pauli Z gate.md @@ -38,4 +38,4 @@ Un quadrato con una Z sopra. ### Nella [[sfera di Bloch]] -Corrisponde a una [[rotazione]] sull'[[asse Z]], quello dal basso all'alto. +Corrisponde a una [[rotazione]] di $\pi$ sull'[[asse Z]], quello dal basso all'alto. diff --git a/7 - Introduction to quantum information processing/2 - Gates semplici/Pauli gate.md b/7 - Introduction to quantum information processing/2 - Gates/Pauli gate.md similarity index 100% rename from 7 - Introduction to quantum information processing/2 - Gates semplici/Pauli gate.md rename to 7 - Introduction to quantum information processing/2 - Gates/Pauli gate.md diff --git a/7 - Introduction to quantum information processing/2 - Gates/Toffoli gate.md b/7 - Introduction to quantum information processing/2 - Gates/Toffoli gate.md new file mode 100644 index 0000000..fbcf81d --- /dev/null +++ b/7 - Introduction to quantum information processing/2 - Gates/Toffoli gate.md @@ -0,0 +1,25 @@ +[[gate quantistico complesso]]: + +$$ +\Huge +\mathbf{X}_{0 1 \overline{2}} += +\begin{bmatrix} +1 & 0 & 0 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 & 0 & 0 \\ +0 & 0 & 1 & 0 & 0 & 0 \\ +0 & 0 & 0 & 1 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 & 1 \\ +0 & 0 & 0 & 0 & 1 & 0 +\end{bmatrix} +$$ + +## Effetto + +Applica l'effetto di un [[Pauli X gate]] all'ultimo [[qbit]] in base a quanto sono attivi i primi due. + +## Visualizzazioni + +### In un [[circuito quantistico]] + +Un [[Pauli X gate]] connesso con una linea verticale ad altri due puntini neri su altri due [[qbit]]. diff --git a/7 - Introduction to quantum information processing/2 - Gates/X-axis rotation gate.md b/7 - Introduction to quantum information processing/2 - Gates/X-axis rotation gate.md new file mode 100644 index 0000000..225b0af --- /dev/null +++ b/7 - Introduction to quantum information processing/2 - Gates/X-axis rotation gate.md @@ -0,0 +1,19 @@ +[[gate quantistico]] appartenente alla famiglia dei [[rotation gates]]: +$$ +\Huge +\def \u {{\color{magenta} \theta}} + +R_{x} (\u) = \begin{bmatrix} +\cos \par{\frac{\u}{2}} & -i \sin \par{\frac{\u}{2}} \\ +- i \sin \par{\frac{\u}{2}} & \cos \par{\frac{\u}{2}} +\end{bmatrix} +$$ + +## Effetto + +Ruota lo stato del [[qbit]] a cui è applicato di $\u$ sull'[[asse X in quantum computing|asse X]]. + +## Pauli + +- Con $\u = 0$, abbiamo un [[Identity gate]]. +- Con $\u = 1$, abbiamo un [[Pauli X gate]]. diff --git a/7 - Introduction to quantum information processing/2 - Gates/Y-axis rotation gate.md b/7 - Introduction to quantum information processing/2 - Gates/Y-axis rotation gate.md new file mode 100644 index 0000000..6aab194 --- /dev/null +++ b/7 - Introduction to quantum information processing/2 - Gates/Y-axis rotation gate.md @@ -0,0 +1,19 @@ +[[gate quantistico]] appartenente alla famiglia dei [[rotation gates]]: +$$ +\Huge +\def \u {{\color{magenta} \theta}} + +R_{y} (\u) = \begin{bmatrix} +\cos \par{\frac{\u}{2}} & - \sin \par{\frac{\u}{2}} \\ +\sin \par{\frac{\u}{2}} & \cos \par{\frac{\u}{2}} +\end{bmatrix} +$$ + +## Effetto + +Ruota lo stato del [[qbit]] a cui è applicato di $\u$ sull'[[asse Y in quantum computing|asse Y]]. + +## Pauli + +- Con $\u = 0$, abbiamo un [[Identity gate]]. +- Con $\u = 1$, abbiamo un [[Pauli Y gate]]. diff --git a/7 - Introduction to quantum information processing/2 - Gates/Z-axis rotation gate.md b/7 - Introduction to quantum information processing/2 - Gates/Z-axis rotation gate.md new file mode 100644 index 0000000..c1be3e1 --- /dev/null +++ b/7 - Introduction to quantum information processing/2 - Gates/Z-axis rotation gate.md @@ -0,0 +1,19 @@ +[[gate quantistico]] appartenente alla famiglia dei [[rotation gates]]: +$$ +\Huge +\def \u {{\color{magenta} \theta}} + +R_{z} (\u) = \begin{bmatrix} +e^{- \frac{i \cdot \u}{2}} & 0 \\ +0 & e^{\frac{i \cdot \u}{2}} +\end{bmatrix} +$$ + +## Effetto + +Ruota lo stato del [[qbit]] a cui è applicato di $\u$ sull'[[asse Z in quantum computing|asse Z]]. + +## Pauli + +- Con $\u = 0$, abbiamo un [[Identity gate]]. +- Con $\u = 1$, abbiamo un [[Pauli Z gate]]. diff --git a/7 - Introduction to quantum information processing/2 - Gates semplici/circuito quantistico a caso.png b/7 - Introduction to quantum information processing/2 - Gates/circuito quantistico a caso.png similarity index 100% rename from 7 - Introduction to quantum information processing/2 - Gates semplici/circuito quantistico a caso.png rename to 7 - Introduction to quantum information processing/2 - Gates/circuito quantistico a caso.png diff --git a/7 - Introduction to quantum information processing/2 - Gates/circuito quantistico.md b/7 - Introduction to quantum information processing/2 - Gates/circuito quantistico.md new file mode 100644 index 0000000..0d7aaae --- /dev/null +++ b/7 - Introduction to quantum information processing/2 - Gates/circuito quantistico.md @@ -0,0 +1,61 @@ +Combinazione di più [[gate quantistico|gate quantistici]]. + +In notazione matematica, vengono eseguiti da destra verso sinistra: + +$$ +\Huge +\mathbf{4\ 3\ 2\ 1} \ket{\psi} = (\ \mathbf{4} (\ \mathbf{3} (\ \mathbf {2} (\ \mathbf{1} \ket{\psi}\ )\ )\ )\ ) +$$ + +Nei diagrammi di circuito, vengono eseguiti da sinistra verso destra: + +![[circuito quantistico a caso.png]] + +## Combinazione di gate + +Più gate si possono combinare in serie, o in parallelo.[^1] + +### Serie: [[prodotto matriciale]] + +I gate vengono applicati consecutivamente uno all'altro: + +$$ +\displaylines{ + \mathbf{Y} \times \mathbf{Z} + =\\\\ + \begin{bmatrix} + 0 & -i\\ + i & 0 + \end{bmatrix} + \times + \begin{bmatrix} + 1 & 0\\ + 0 & -1 + \end{bmatrix} + =\\\\ + \begin{bmatrix} + (0 \cdot 1) + (-i \cdot 0) & (0 \cdot 0) + (-i \cdot -1)\\ + (i \cdot 1) + (0 \cdot 0) & (i \cdot 0) + (-1 \cdot 0) + \end{bmatrix} + =\\\\ + \begin{bmatrix} + 0 & i\\ + i & 0 + \end{bmatrix} + =\\\\ + i \cdot + \begin{bmatrix} + 0 & 1\\ + 1 & 0 + \end{bmatrix} + =\\\\ + i \cdot \mathbf{X} +} +$$ + +### Parallelo: [[prodotto tensoriale]] + +==TODO== + + +[^1]: https://en.wikipedia.org/wiki/Quantum_logic_gate#Circuit_composition \ No newline at end of file diff --git a/7 - Introduction to quantum information processing/2 - Gates/controlled Pauli X gate.md b/7 - Introduction to quantum information processing/2 - Gates/controlled Pauli X gate.md new file mode 100644 index 0000000..e8921eb --- /dev/null +++ b/7 - Introduction to quantum information processing/2 - Gates/controlled Pauli X gate.md @@ -0,0 +1,28 @@ +--- +aliases: + - CNOT gate + - quantum controlled NOT gate + - Feynman gate +--- +[[gate quantistico complesso]], un [[Pauli X gate]] che opera condizionalmente su un [[qbit]] in base allo stato di un altro [[qbit]]: + +$$ +\Huge +\mathbf{X}_{0 \overline{1}} += +\begin{bmatrix} +1 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 \\ +0 & 0 & 0 & 1 \\ +0 & 0 & 1 & 0 +\end{bmatrix} +$$ +## Effetto + +Applica l'effetto di un [[Pauli X gate]] all'ultimo [[qbit]] in base a quanto è attivo il primo. + +## Visualizzazioni + +### In un [[circuito quantistico]] + +Un [[Pauli X gate]] connesso con una linea verticale a un puntino nero su un altro [[qbit]]. diff --git a/7 - Introduction to quantum information processing/2 - Gates semplici/gate quantistico universale.md b/7 - Introduction to quantum information processing/2 - Gates/gate quantistico universale.md similarity index 100% rename from 7 - Introduction to quantum information processing/2 - Gates semplici/gate quantistico universale.md rename to 7 - Introduction to quantum information processing/2 - Gates/gate quantistico universale.md diff --git a/7 - Introduction to quantum information processing/2 - Gates semplici/gate quantistico.md b/7 - Introduction to quantum information processing/2 - Gates/gate quantistico.md similarity index 96% rename from 7 - Introduction to quantum information processing/2 - Gates semplici/gate quantistico.md rename to 7 - Introduction to quantum information processing/2 - Gates/gate quantistico.md index b1057ae..dda9fa5 100644 --- a/7 - Introduction to quantum information processing/2 - Gates semplici/gate quantistico.md +++ b/7 - Introduction to quantum information processing/2 - Gates/gate quantistico.md @@ -9,7 +9,7 @@ aliases: ## Visualizzazioni -### [[Circuito quantistico]] +### [[circuito quantistico]] Un gate quantistico è rappresentato in un circuito quantistico come un quadrato con scritto dentro il nome del gate, da cui entra l'input a sinistra ed esce l'output a destra. diff --git a/7 - Introduction to quantum information processing/2 - Gates/phase gate.md b/7 - Introduction to quantum information processing/2 - Gates/phase gate.md new file mode 100644 index 0000000..ec55a94 --- /dev/null +++ b/7 - Introduction to quantum information processing/2 - Gates/phase gate.md @@ -0,0 +1,46 @@ +--- +aliases: + - S gate + - square root of Z +--- +[[gate quantistico]] corrispondente alla radice quadrata del [[Pauli Z gate]]: +$$ +\Huge +\sqrt{\mathbf{Z}} = \begin{bmatrix} +1 & 0 \\ +0 & i +\end{bmatrix} +$$ + +## Effetto + +Si comporta come mezzo [[Pauli Z gate]]: +$$ +\sqrt{\mathbf{Z}} \ket{0} = \ket{0} +$$ +$$ +\sqrt{\mathbf{Z}} \ket{1} = i \cdot \ket{1} +$$$$ +\sqrt{\mathbf{Z}}\ \sqrt{\mathbf{Z}} \ket{1} = \mathbf{Z} \ket{1} +$$ +$$ +\sqrt{\mathbf{Z}} +\begin{bmatrix} + \alpha \\ + \beta +\end{bmatrix} += +\begin{bmatrix} + \alpha \\ + i \cdot \beta +\end{bmatrix} +$$ + +## Visualizzazioni +### In un [[circuito quantistico]] + +Un quadrato con una S sopra, o un quadrato con una radice quadrata di Z. + +### Nella [[sfera di Bloch]] + +Corrisponde a una [[rotazione]] di $\frac{\pi}{2}$ sull'[[asse Z]], quello dal basso all'alto. diff --git a/7 - Introduction to quantum information processing/2 - Gates/swap gate.md b/7 - Introduction to quantum information processing/2 - Gates/swap gate.md new file mode 100644 index 0000000..f061825 --- /dev/null +++ b/7 - Introduction to quantum information processing/2 - Gates/swap gate.md @@ -0,0 +1,25 @@ + +[[gate quantistico complesso]]: + +$$ +\Huge +\mathbf{S}_{0 \leftrightarrow 1} += +\begin{bmatrix} +1 & 0 & 0 & 0 \\ +0 & 0 & 1 & 0 \\ +0 & 1 & 0 & 0 \\ +0 & 0 & 0 & 1 +\end{bmatrix} +$$ + + +## Effetto + +Scambia gli stati di due [[qbit]]. + +## Visualizzazioni + +### In un [[circuito quantistico]] + +Due croci sui [[qbit]] affetti, connesse da una linea verticale. diff --git a/7 - Introduction to quantum information processing/3 - Gates complessi/gate quantistico controllato universale.md b/7 - Introduction to quantum information processing/2 - Gates/universal gate.md similarity index 97% rename from 7 - Introduction to quantum information processing/3 - Gates complessi/gate quantistico controllato universale.md rename to 7 - Introduction to quantum information processing/2 - Gates/universal gate.md index dc1cabf..43cd763 100644 --- a/7 - Introduction to quantum information processing/3 - Gates complessi/gate quantistico controllato universale.md +++ b/7 - Introduction to quantum information processing/2 - Gates/universal gate.md @@ -1,4 +1,5 @@ [[gate quantistico universale]] che opera condizionalmente su un [[qbit]] in base allo stato di un altro [[qbit]], in modo simile a un [[controlled Pauli X gate]]: + $$ \def \varX {{\color{coral} a}} \def \varY {{\color{cornflowerblue} b}} @@ -15,3 +16,5 @@ $$ - e^{\varI \varZ} \sin \left( \frac{\varX}{2} \right) \end{bmatrix} $$ + +==Deutsch gate?== diff --git a/7 - Introduction to quantum information processing/2 - Gates/★ gates.canvas b/7 - Introduction to quantum information processing/2 - Gates/★ gates.canvas new file mode 100644 index 0000000..49ffbd9 --- /dev/null +++ b/7 - Introduction to quantum information processing/2 - Gates/★ gates.canvas @@ -0,0 +1,41 @@ +{ + "nodes":[ + {"id":"d1d88e7c08769a0c","type":"file","file":"7 - 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Introduction to quantum information processing/3 - Gates complessi/Swap gate.md b/7 - Introduction to quantum information processing/3 - Gates complessi/Swap gate.md deleted file mode 100644 index 09c5076..0000000 --- a/7 - Introduction to quantum information processing/3 - Gates complessi/Swap gate.md +++ /dev/null @@ -1,13 +0,0 @@ -[[gate quantistico complesso]] che scambia gli [[qbit|stati di due qbit]]. - -$$ -\Huge -\mathbf{S}_{0 \leftrightarrow 1} -= -\begin{bmatrix} -1 & 0 & 0 & 0 \\ -0 & 0 & 1 & 0 \\ -0 & 1 & 0 & 0 \\ -0 & 0 & 0 & 1 -\end{bmatrix} -$$ diff --git a/7 - Introduction to quantum information processing/3 - Gates complessi/controlled Pauli X gate.md b/7 - Introduction to quantum information processing/3 - Gates complessi/controlled Pauli X gate.md deleted file mode 100644 index 2a7ffb1..0000000 --- a/7 - Introduction to quantum information processing/3 - Gates complessi/controlled Pauli X gate.md +++ /dev/null @@ -1,18 +0,0 @@ ---- -aliases: - - CNOT gate - - quantum controlled NOT gate - 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Ci troviamo nello stato configurato dal gate $\usecond$ per $\ket{0}_\noteb$: $$ diff --git a/7 - Introduction to quantum information processing/★ introduction to quantum information processing.md b/7 - Introduction to quantum information processing/★ introduction to quantum information processing.md index 4ddd93b..5530a91 100644 --- a/7 - Introduction to quantum information processing/★ introduction to quantum information processing.md +++ b/7 - Introduction to quantum information processing/★ introduction to quantum information processing.md @@ -6,7 +6,7 @@ Appunti del corso di ***Algoritmi distribuiti***, tenuto da [[Andrea Bertoni]] e > Gli appunti sono nell'ordine in cui mi sembra più naturale capire le cose, non nell'ordine utilizzato dai docenti per la didattica. 1. [[7 - Introduction to quantum information processing/1 - Concetti base/★ concetti base.canvas|★ concetti base]] -2. [[★ gates semplici.canvas|★ gates semplici]] +2. [[★ gates.canvas|★ gates semplici]] 3. [[★ gates complessi.canvas|★ gates complessi]] 4. [[★ modello computazionale.canvas|★ modello computazionale]] 5. [[★ cose strane.canvas|★ cose strane]]