IQC

• Every Qubit is in superposition between 0 or 1
• A variable x can be in superposition between all values
• Any probalistic process can be written as nxn-matrix A Pr[output j | input i] = A_ji
• Probabilistic -> Quantum :

  • distributions -> quantum state
  • probabilities -> amplitudes
  • deterministic possibilities -> classical possibilities/states
  • processes (stoch. matrices) -> unitary matrices
• Probabilistically -> Quantumly

  distribution is a vector because it "mixes" deterministic possabilities ->

  quantum state is a vector because superposition "mixes" classical possabilities
• Quantum systems are described by a list of n classical possibilities
• A quantum state is a vector $\psi \in \mathbb{C}^n$with $\sum |\psi_i|^2 =$$1$or $||\psi|| =$$1$
• Valid quantum processes are unitary matrices
• A (square) matrix U is unitary if $||U_\psi||=$$1$when$||\psi||=$$1$
• U is unitary if $U^t U =$$I -> (U^t = konjugiert + transponiert)$
• Description of Systems

  • List of deterministic possibilities 1,...,N -> Probabilistic
  • List of classical possibilities 1,...,N -> Quantum
• State of Systems


  • N-vector $\psi \in \mathbb{R}^n =$$$probabilistic distribution $\sum \psi_i =$$1, \psi_i >=$$0$
  • N-vector $\psi \in \mathbb{C}^n =$quantum state $\sum |\psi_i|^2 =$$1$
• Development of system

  Probabilistic:

  • Stochastic matrix $\in R^{NxN}$(Every column is probabilistic distribution)

  Quantum

  • Unitary matrix U $U\in \mathbb{C}^{NxN}$ maps quantum state to quantum state
  • $U^t U =$$I$
• Given distribution $\mu$, if we observe it , get outcome i with prob $\mu_i$and in that : new distribution is : $e_i$:= (0,...,0,1,0,...0) at the i-th element
• Measurments do change the system
• Observation : Probabilitic -> Quantum

  Update of knowledge ->

  State becomes e_i -> physical effect -> no superposition -> no interference
• ∣∣ϕ∣∣= sqrt($\sum$|$\phi_i$|^2)
• Hadamard H = $\frac 1 {\sqrt2}$ ($\begin{matrix} 1&1\\1&-1 \end{matrix}$)
• You are given a n-dimensional probability distribution vector x.What is the average (a.k.a. mean value) of the entries x_i? :


  1/n
• A quantum measurement changes the outcome probabilities of future observations.
• Consider a vector x∈R^n and a vector y∈R^m. Then the tensor product x⊗y is in R^k.

  The value of k is n*m
• |+> :=$\frac 1 {\sqrt2}$|0> + $\frac 1 {\sqrt2}$|1> = H(|0>)
• |-> := $\frac 1 {\sqrt2}$|0> - ​$\frac 1 {\sqrt2}$|1> = H(|1>)
• H(|+>) = H($\frac 1 {\sqrt2}|0>$+ $\frac 1 {\sqrt2}|1>$) = 1/2(|0> + |1>) + 1/2(|0> - |1>) = |0>
• • H(|->) = H($\frac 1 {\sqrt2}|0>$-$\frac 1 {\sqrt2}|1>$) = 1/2(|0> + |1>) - 1/2(|0> - |1>) = |1>