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index 7da6d7e..8c799c1 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -13,12 +13,12 @@
             "state": {
               "type": "markdown",
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-                "file": "SS2026/Quantum Computing/18. Fault-torerant computation/Fault-Tolerant Computation.md",
+                "file": "SS2026/Quantum Computing/19. Quantum Complexity/Quantum Complexity.md",
                 "mode": "source",
                 "source": false
               },
               "icon": "lucide-file",
-              "title": "Fault-Tolerant Computation"
+              "title": "Quantum Complexity"
             }
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@@ -218,25 +218,26 @@
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+    "SS2026/Quantum Computing/Introduction to Quantum Computing.md",
+    "SS2026/Quantum Computing/19. Quantum Complexity/Quantum Complexity.md",
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+    "SS2026/Quantum Computing/19. Quantum Complexity",
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     "SS2026/Quantum Computing/13. Quantum Physics/Quantum Physics.md",
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     "SS2026/Quantum Computing/12. Grover's Algorithm/Grover's Algorithm.md",
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diff --git a/SS2026/Quantum Computing/18. Fault-torerant computation/Fault-Tolerant Computation.md b/SS2026/Quantum Computing/18. Fault-torerant computation/Fault-Tolerant Computation.md
index b65b294..268dca8 100644
--- a/SS2026/Quantum Computing/18. Fault-torerant computation/Fault-Tolerant Computation.md	
+++ b/SS2026/Quantum Computing/18. Fault-torerant computation/Fault-Tolerant Computation.md	
@@ -10,3 +10,20 @@ We begin by defining how to apply operations to encoded qubits:
 
 ![[Pasted image 20260806135601.png]]
 ![[Pasted image 20260806135610.png]]
+
+
+## Fault tolerant Gates
+
+Now we need to reduce our error probability.
+
+![[Pasted image 20260806140451.png]]
+
+To make this circuit fault-tolerant, we replace it with:
+
+![[Pasted image 20260806140514.png]]
+
+Let's look at a circuit for CNOT:
+
+![[Pasted image 20260806140629.png]]
+![[Pasted image 20260806140639.png]]
+![[Pasted image 20260806140650.png]]
\ No newline at end of file
diff --git a/SS2026/Quantum Computing/19. Quantum Complexity/Quantum Complexity.md b/SS2026/Quantum Computing/19. Quantum Complexity/Quantum Complexity.md
new file mode 100644
index 0000000..e69de29
diff --git a/SS2026/Quantum Computing/Introduction to Quantum Computing.md b/SS2026/Quantum Computing/Introduction to Quantum Computing.md
index 1362adc..9077dcd 100644
--- a/SS2026/Quantum Computing/Introduction to Quantum Computing.md	
+++ b/SS2026/Quantum Computing/Introduction to Quantum Computing.md	
@@ -16,3 +16,4 @@
 ## 16. [[Universal Set of Gates]]
 ## 17. [[Quantum Error Correction]]
 ## 18. [[Fault-Tolerant Computation]]
+## 19. [[Quantum Complexity]]
\ No newline at end of file