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

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

PALINDROMES PALINDROMES = {x | x {0,1}* and x = xrev } Notation 0.000...000 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

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