Aida Gainutdinova

On Computational Power of Quantum Branching Programs

In this paper we introduce a model of a Quantum Branching Program(QBP) and study its computational power. We define several natural restrictions of a general QBP model, such as a read-once and a read-k-times QBP, noting that obliviousness is inherent in a quantum nature of such programs. In particular we show that any Boolean function can be computed deterministically (exactly) by a read-once QBP in width O(2n), contrary to the analogous situation for quantumfinite automata. Further we display certain symmetric Boolean function which is computable by a read-once QBP with O(log n) width, which requires a width Ω(n) on any deterministic read-once BP and (classical) randomized read-once BP with permanent transitions in each levels. We present a general lower bound for the width of read-once QBPs, showing that the upper bound for the considered symmetric function is almost tight.

Very Narrow Quantum OBDDs and Width Hierarchies for Classical OBDDs

In the paper we investigate a model for computing of Boolean functions – Ordered Binary Decision Diagrams (OBDDs), which is a restricted version of Branching Programs. We present several results on the comparative complexity for several variants of OBDD models. We present some results on the comparative complexity of classical and quantum OBDDs. We consider a partial function depending on a parameter k such that for any k > 0 this function is computed by an exact quantum OBDD of width 2, but any classical OBDD (deterministic or stable bounded-error probabilistic) needs width 2 k + 1. We consider quantum and classical nondeterminism. We show that quantum nondeterminism can be more efficient than classical nondeterminism. In particular, an explicit function is presented which is computed by a quantum nondeterministic OBDD with constant width, but any classical nondeterministic OBDD for this function needs non-constant width. We also present new hierarchies on widths of deterministic and nondeterministic OBDDs. We focus both on small and large widths.

Complexity of quantum uniform and nonuniform automata

We present two different types of complexity lower bounds for quantum uniform automata (finite automata) and nonuniform automata (OBDDs). We call them “metric” and “entropic” lower bounds in according to proof technique used. We present explicit Boolean functions that show that these lower bounds are tight enough. We show that when considering “almost all Boolean functions” on n variables our entropic lower bounds gives exponential (2c(δ)(n−−logn)) lower bound for the width of quantum OBDDs depending on the error δ allowed. Next we consider “generalized measure-many” quantum automata. It is appeared that for uniform and nonuniform automata (for space restricted models) their measure-once and measure-many models have different computational power.

Very narrow quantum OBDDs and width hierarchies for classical OBDDs

In the paper we investigate Ordered Binary Decision Diagrams (OBDDs)–a model for computing Boolean functions. We present a series of results on the comparative complexity for several variants of OBDDmodels.• We present results on the comparative complexity of classical and quantum OBDDs. We consider a partial function depending on a parameter k such that for any k > 0 this function is computed by an exact quantum OBDD of width 2, but any classical OBDD (deterministic or stable bounded-error probabilistic) needs width 2k+1.• We consider quantum and classical nondeterminism. We show that quantum nondeterminismcan bemore efficient than classical nondeterminism. In particular, an explicit function is presented that is computed by a quantum nondeterministic OBDD of constant width but any classical nondeterministic OBDD for this function needs non-constant width.• We also present new hierarchies on widths of deterministic and nondeterministic OBDDs.

Unary Probabilistic and Quantum Automata on Promise Problems

We continue the systematic investigation of probabilistic and quantum finite automata (PFAs and QFAs) on promise problems by focusing on unary languages. We show that bounded-error QFAs are more powerful than PFAs. But, in contrary to the binary problems, the computational powers of Las-Vegas QFAs and bounded-error PFAs are equivalent to deterministic finite automata (DFAs). Lastly, we present a new family of unary promise problems with two parameters such that when fixing one parameter QFAs can be exponentially more succinct than PFAs and when fixing the other parameter PFAs can be exponentially more succinct than DFAs.

On the Lower Bounds for One-Way Quantum Automata

In the paper we consider measured-once (MO-QFA) one-way quantum finite automaton. We prove that for MO-QFA Q that (1=2+Ɛ)-accepts (Ɛ ∈ (0, 1=2)) regular language L it holds that dim(Q) = Ω (log dim(A)/log log dim(A)). In the case Ɛ ∈ (3/8,1/2) we have more precise lower bound dim(Q) = Ω(log dim(A)) where A is a minimal deterministic finite automaton accepting L, dim(Q), and dim(A) are complexity (number of states) of automata Q and A respectively, (1=2-Ɛ) is the error of Q. The example of language presented in [2] show that our lower bounds are tight enough.

Nondeterministic Unitary OBDDs

We investigate the width complexity of nondeterministic unitary OBDDs (NUOBDDs). Firstly, we present a generic lower bound on their widths based on the size of strong 1-fooling sets. Then, we present classically “cheap” functions that are “expensive” for NUOBDDs and vice versa by improving the previous gap. We also present a function for which neither classical nor unitary nondeterminism does help. Moreover, based on our results, we present a width hierarchy for NUOBDDs. Lastly, we provide the bounds on the widths of NUOBDDs for the basic Boolean operations negation, union, and intersection.

Classical Simulation Complexity of Quantum Machines

We present a classical probabilistic simulation technique of quantum Turing machines. As a corollary of this technique we obtain several results on relationship among classical and quantum complexity classes such as: PrQP=PP, BQP ⊆ PP and PrQSPACE(S(n))=PrPSPACE(S(n)).

On the full monomial automorphism groups of Reed–Solomon codes and their MDS-extensions

We determine the full monomial automorphism groups of Reed–Solomon codes and of theirMDS-extensions by adding one, two, and three symbols.We show that several Reed–Solomon codes, extended by two symbols, are equivalent to constacyclic codes and, under certain conditions, to cyclic codes.

On classical stimulation of quantum machines

We present a classical probabilistic simulation technique of quantum Turing machines As a corollary of this technique we obtain several results on relationship among classical and quantum complexity classes such as: PrQP PP BQP PP and PrQSPACE(S(n)) PrPSPACE(S(n)).