1. Languages Recognizable by Quantum Finite Automata with cut-point 0
Lelde Lace
Oksana Scegulnaja-Dubrovska Supervisor: Rusins Freivalds University of Latvia

2. Definition MM-QFA States – Q , q0Q, Qacc  Q, Qrej Q, Qacc  Qrej=0
Halting states Qacc Qrej , Non halting states Qnon= Q–( Qacc Qrej)
Input alphabet – , Working alphabet - =   {#,$}
State of M – superposition of states in Q
= qiQ ai|qi = qiQacc i|qi+ qjQrej j|qj+ qkQnon k|qk
Transition function V=  l2(Q) l2(Q)
a  , Va(x) – unitary transformation
Word acceptance with cut-point 0
w L, probacc>0, w L, probacc=0

3. PALINDROMES PALINDROMES = {x | x {0,1}* and x = xrev }
Notation x1x2x3...xn  0.0{n}x1x2x3...xn
Input word coding – x = x1x2x3...xn
k1 (n)= 0.0{n}x1x2x3...xn k2 (n)= 0.0{n}xnxn-1xn-2...x1
ki (n+1)= ki (n)*c1+c2(n)
w L, k1= k2 , w  L, k1 k2
Hadamard operation
Co-PALINDROMES can be recognized by finite automata

4. Postselection MM-QFA + postselection set of states and state q+
= qiQpost i|qi+ qjQ-Qpost j|qj
= qiQpost k*i|qi = |q+
 |i|2=0
 |i|2  0
Co-PALINDROMES can be recognized with probability 1
Complement of language can be recognized with probability 1
PALINDROMES can be recognized with probability 1