inner product

computational-diffie-hellman.md

@@ -1,4 +1,4 @@
 # Computational Diffie–Hellman
 
-- computational version of [[decisional-diffie-hellman|Decisional Diffie–Hellman]]
+- computational version of [[decisional-diffie-hellman]]
 - given a generator $`g`$ and uniformly selected $`\left(g^a, g^b\right)`$, compute $`g^{ab}`$

controlled-not.md

@@ -1,6 +1,6 @@
 # Controlled Not
 
-- [[quantum-computing|Quantum Computing]]
+- [[quantum-computing]]
 - [[push]] [[CNOT]]
 - quantum equivalent of XOR
 - written $`\operatorname{CNOT}_{1 \rarr 2}`$ for when $`\left(x, y\right)`$ becomes $`\left(x, x \oplus y\right)`$

decisional-diffie-hellman.md

@@ -1,4 +1,4 @@
 # Decisional Diffie–Hellman
 
-- decisional version of [[computational-diffie-hellman|Computational Diffie–Hellman]]
+- decisional version of [[computational-diffie-hellman]]
 - given a generator $`g`$ and uniformly selected $`\left(g^a, g^b\right)`$, discriminate $`g^{ab}`$ from a different group value $`c`$

inner-product.md

@@ -0,0 +1,13 @@
+# Inner Product
+
+- operates on two vectors and produces a scalar
+- notation $`\braket{x|y}`$
+  - $`\bra{x}`$ is a [[bra]]
+  - $`\ket{y}`$ is a [[ket]]
+- follows laws
+  1. $`\braket{0|y} = 0`$ and $`\braket{x|0} = 0`$
+  2. $`\braket{x + y|z} = \braket{x|z} + \braket{y|z}`$ and $`\braket{x|y + z} = \braket{x|y} + \braket{x|z}`$
+  3. $`\braket{cx|y} = \overline{c}\braket{x|y}`$ and $`\braket{x|cy} = \braket{x|y}c`$
+  4. $`\braket{x|y} = \overline{\braket{y|x}}`$
+- the inner product $`\braket{x|y}`$ is antilinear in $`x`$ and linear in $`y`$
+  - if starting with two [[ket]]s, take the conjugate transpose of the former: $`\bra{x} = \overline{\ket{x}}^\intercal = \overline{\ket{x}^\intercal}`$

ket.md

@@ -1,4 +1,4 @@
 # Ket
 
 - $`\ket{u}`$ where $`u`$ is the _name_ of a _state_
-- [[push]] [[column-vector|Column Vector]]
+- [[push]] [[column-vector]]

np-complete.md

@@ -1,4 +1,4 @@
 # NP Complete
 
-- a problem that all other [[np|NP]] problems can be reduced to
-- if you can solve the decision version in P-time, you can solve all decision problems in [[np|NP]] in P-time
+- a problem that all other [[np]] problems can be reduced to
+- if you can solve the decision version in P-time, you can solve all decision problems in [[np]] in P-time

np-hard.md

@@ -1,4 +1,4 @@
 # NP Hard
 
-- may not necessarily be in [[np|NP]]
-- computational version of [[np-complete|NP Complete]]
+- may not necessarily be in [[np]]
+- computational version of [[np-complete]]

np.md

@@ -1,5 +1,5 @@
 # NP
 
 - class of languages whose membership proofs can be verified in polynomial time
-- the [[discrete-log|Discrete Log]] problem is an example of an NP problem
+- the [[discrete-log]] problem is an example of an NP problem
   - from $`g^x`$ in a finite field it is hard to compute $`\log_g\left(g^x\right)`$, but easy to check $`g^x = g^y`$ to validate a knowledge proof of $`x`$

quantum-state.md

@@ -0,0 +1,4 @@
+# Quantum State
+
+- [[quantum-computing]]
+- unit-norm (length 1) [[ket]]

toffoli-gate.md

@@ -1,6 +1,6 @@
 # Toffoli Gate
 
-- [[quantum-computing|Quantum Computing]]
+- [[quantum-computing]]
 - also called [[push]] [[CCNOT]]
 - quantum equivalent of AND/NAND
-- [[universal-gate|Universal Gate]] for reversible computing
+- [[universal-gate]] for reversible computing