outer products

controlled-not.md

@@ -3,4 +3,4 @@
 - [[quantum-computing]]
 - [[push]] [[CNOT]]
 - quantum equivalent of XOR
-- written $`\operatorname{CNOT}_{1 \rarr 2}`$ for when $`\left(x, y\right)`$ becomes $`\left(x, x \oplus y\right)`$
+- written $`\operatorname{CNOT}_{1 \rightarrow 2}`$ for when $`\left(x, y\right)`$ becomes $`\left(x, x \oplus y\right)`$

fourier-basis.md

@@ -0,0 +1,6 @@
+# Fourier Basis
+
+- [[quantum-computing]]
+- the basis with base vectors $`\ket{+}`$ and $`\ket{-}`$
+- $`\ket{+} = \frac{1}{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix}`$
+- $`\ket{-} = \frac{1}{\sqrt{2}}\begin{bmatrix}1\\-1\end{bmatrix}`$

hadamard-gate.md

@@ -0,0 +1,6 @@
+# Hadamard Gate
+
+- [[quantum-computing]]
+- [[unitary-operator]]
+- change-of-basis into the [[fourier-basis]]
+- written $`\mathbf{H}`$

inner-product.md

@@ -4,6 +4,7 @@
 - notation $`\braket{x|y}`$
   - $`\bra{x}`$ is a [[bra]]
   - $`\ket{y}`$ is a [[ket]]
+- [[matrix-multiplication]], associates with [[outer-product]]
 - follows laws
   1. $`\braket{0|y} = 0`$ and $`\braket{x|0} = 0`$
   2. $`\braket{x + y|z} = \braket{x|z} + \braket{y|z}`$ and $`\braket{x|y + z} = \braket{x|y} + \braket{x|z}`$

norm.md

@@ -0,0 +1,5 @@
+# Norm
+
+- "length" of a vector
+- norm of a state $`u`$ is square root of [[inner-product]] $`\sqrt{\braket{u|u}}`$
+- written $`\left\lVert \ket{u} \right\rVert`$

orthonormal.md

@@ -0,0 +1,5 @@
+# Orthonormal
+
+- property of a set of vectors
+- [[norm]] of all vectors must be 1
+- [[inner-product]] of all vectors must be 0

outer-product.md

@@ -0,0 +1,8 @@
+# Outer Product
+
+- operates on two vectors and produces a matrix
+- notation is $`\ket{x}\bra{y}`$
+  - $`\ket{x}`$ is a [[ket]]
+  - $`\bra{y}`$ is a [[bra]]
+- [[matrix-multiplication]], associates with [[inner-product]]
+- a sum $`\ket{x}\bra{0} + \ket{y}\bra{1}`$ where $`\ket{x}`$ and $`\ket{y}`$ are [[orthonormal]] produces a [[unitary-operator]] sending $`\ket{0}`$ to $`\ket{x}`$ and $`\ket{1}`$ to $`\ket{y}`$

unitary-operator.md

@@ -0,0 +1,12 @@
+# Unitary Operator
+
+- [[quantum-computing]]
+- a matrix $`\mathcal{U}`$ operating on a [[quantum-state]] to produce another [[quantum-state]]
+  - maps unit vectors to unit vectors
+  - [[norm]]
+- other equivalent definitions:
+  - columns of $`\mathcal{U}`$ are [[orthonormal]]
+  - rows of $`\mathcal{U}`$ are [[orthonormal]]
+  - $`\mathcal{U}^{-1} = \mathcal{U}^\dagger = \overline{\mathcal{U}^\intercal} = \overline{\mathcal{U}}^\intercal`$
+  - preserves [[norm]]
+  - preserves [[inner-product]]