New results on classical and quantum counter automata

We show that one-way quantum one-counter automaton with zero-error is more powerful than its probabilistic counterpart on promise problems. Then, we obtain a similar separation result between Las Vegas one-way probabilistic one-counter automaton and one-way deterministic one-counter automaton. We also obtain new results on classical counter automata regarding language recognition. It was conjectured that one-way probabilistic one blind-counter automata cannot recognize Kleene closure of equality language [A. Yakaryilmaz: Superiority of one-way and realtime quantum machines. RAIRO - Theor. Inf. and Applic. 46(4): 615-641 (2012)]. We show that this conjecture is false, and also show several separation results for blind/non-blind counter automata.Comment: 21 page

On calculation of monomial automorphisms of linear cyclic codes

A description of the monomial automorphisms group of an arbitrary linear cyclic code in term of polynomials is presented. This allows us to reduce a task of code's monomial automorphisms calculation to a task of solving some system of equations (in general, nonlinear) over a finite field. The results are illustrated with examples of calculating the full monomial automorphisms groups for two codes

New results on classical and quantum counter automata

We show that one-way quantum one-counter automaton with zero-error is morepowerful than its probabilistic counterpart on promise problems. Then, weobtain a similar separation result between Las Vegas one-way probabilisticone-counter automaton and one-way deterministic one-counter automaton. We also obtain new results on classical counter automata regarding languagerecognition. It was conjectured that one-way probabilistic one blind-counterautomata cannot recognize Kleene closure of equality language [A. Yakaryilmaz:Superiority of one-way and realtime quantum machines. RAIRO - Theor. Inf. andApplic. 46(4): 615-641 (2012)]. We show that this conjecture is false, and alsoshow several separation results for blind/non-blind counter automata.Comment: 21 page

New results on classical and quantum counter automata

We show that one-way quantum one-counter automaton with zero-error is more powerful than its probabilistic counterpart on promise problems. Then, we obtain a similar separation result between Las Vegas one-way probabilistic one-counter automaton and one-way deterministic one-counter automaton. We also obtain new results on classical counter automata regarding language recognition. It was conjectured that one-way probabilistic one blind-counter automata cannot recognize Kleene closure of equality language [A. Yakaryilmaz: Superiority of one-way and realtime quantum machines. RAIRO - Theor. Inf. and Applic. 46(4): 615-641 (2012)]. We show that this conjecture is false, and also show several separation results for blind/non-blind counter automata

On the Computational Power of Probabilistic and Quantum Branching Programs

In this paper we show that one-qubit polynomial time computations are as powerful as NC 1 circuits. More generally, we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power. We show that any NC 1 language can be accepted exactly by a width-2 quantum branching program of polynomial length, in contrast to the classical case where width 5 is necessary unless NC 1 = ACC. This separates width-2 quantum programs from width-2 doubly stochastic programs as we show the latter cannot compute the middle bit of multiplication. Finally, we show that bounded-width quantum and stochastic programs can be simulated by classical programs of larger but bounded width, and thus are in NC 1. For read-once quantum branching programs (QBPs), we give a symmetric Boolean function which is computable by a read-once QBP with O(log n) width, but not by a deterministic read-once BP with o(n) width, or by a classical randomized read-once BP with o(n) width which is “stable ” in the sense that its transitions depend on the value of the queried variable but do not vary from step to step. Finally, we present a general lower bound on the width of read-once QBPs, showing that our O(log n) upper bound for this symmetric function is almost tight

On the Computational Power of Probabilistic and Quantum Branching Programs (Revised Version)

In this paper we show that one-qubit polynomial time computations are as powerful as NC 1 circuits. More generally, we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power. We show that any NC 1 language can be accepted exactly by a width-2 quantum branching program of polynomial length, in contrast to the classical case where width 5 is necessary unless NC 1 = ACC. This separates width-2 quantum programs from width-2 doubly stochastic programs as we show the latter cannot compute the middle bit of multiplication. Finally, we show that bounded-width quantum and stochastic programs can be simulated by classical programs of larger but bounded width, and thus are in NC 1. For read-once quantum branching programs (QBPs), we give a symmetric Boolean function which is computable by a read-once QBP with O(log n) width, but not by a deterministic read-once BP with o(n) width, or by a classical randomized read-once BP with o(n) width which is “stable ” in the sense that its transitions depend on the value of the queried variable but do not vary from step to step. Finally, we present a general lower bound on the width of read-once QBPs, showing that our O(log n) upper bound for this symmetric function is almost tight