Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs

In this paper, we present a quantum algorithm for dynamic programming approach for problems on directed acyclic graphs (DAGs). The running time of the algorithm is $O(\sqrt{\hat{n}m}\log \hat{n})$, and the running time of the best known deterministic algorithm is $O(n+m)$, where $n$ is the number of vertices, $\hat{n}$ is the number of vertices with at least one outgoing edge; $m$ is the number of edges. We show that we can solve problems that use OR, AND, NAND, MAX and MIN functions as the main transition steps. The approach is useful for a couple of problems. One of them is computing a Boolean formula that is represented by Zhegalkin polynomial, a Boolean circuit with shared input and non-constant depth evaluating. Another two are the single source longest paths search for weighted DAGs and the diameter search problem for unweighted DAGs.Comment: UCNC2019 Conference pape

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.Comment: Minor correction

Quantum hashing via ∈-universal hashing constructions and classical fingerprinting

© 2015, Pleiades Publishing, Ltd. In the paper, we define the concept of the quantum hash generator and offer design, which allows to build a large amount of different quantum hash functions. The construction is based on composition of classical ∈-universal hash family and a given family of functions-quantum hash generator. In particular, using the relationship between ∈-universal hash families and Freivalds fingerprinting schemas we present explicit quantum hash function and prove that this construction is optimal in the sense of number of qubits needed for construction

Quantum hashing via ε-universal hashing constructions and Freivalds' fingerprinting schemas

We define the concept of a quantum hash generator and offer a design, which allows one to build a large number of different quantum hash functions. The construction is based on composition of a classical ε-universal hash family and a given family of functions - quantum hash generators. In particular, using the relationship between ε-universal hash families and Freivalds' fingerprinting schemas we present explicit quantum hash function and prove that this construction is optimal with respect to the number of qubits needed for the construction. © 2014 Springer International Publishing

On the concept of cryptographic quantum hashing

© 2015 Astro Ltd. In the letter we define the notion of a quantum resistant ((ε, δ)-resistant) hash function which consists of a combination of pre-image (one-way) resistance (ε-resistance) and collision resistance (δ-resistance) properties. We present examples and discussion that supports the idea of quantum hashing. We present an explicit quantum hash function which is 'balanced', one-way resistant and collision resistant and demonstrate how to build a large family of quantum hash functions. Balanced quantum hash functions need a high degree of entanglement between the qubits. We use a phase transformation technique to express quantum hashing constructions, which is an effective way of mapping hash states to coherent states in a superposition of time-bin modes. The phase transformation technique is ready to be implemented with current optical technology

Randomization and nondeterminism are comparable for ordered read-onee branching programs

© Springer-Verlag Berlin Heidelberg 1997. In [3] we exhibited a simple boolean functions fn in n variables such that: fn can be computed by polynomial size randomized ordered read-once branching program with one sided small error; any nondeterministic ordered read-once branching program that computes fn has exponential size. In this paper we present a simple boolean function gn in n variables such that: gn can be computed by polynomial size nondeterministic ordered read-once branching program; any two-sided error randomized ordered read-once branching program that computes fn has exponential size.These mean that BPP and NP are incomparable in the context of ordered read-once branching program

On the balanced quantum hashing

In the paper we define a notion of a resistant quantum hash function which combines a notion of pre-image (one-way) resistance and the notion of collision resistance. In the quantum setting one-way resistance property and collision resistance property are correlated: the "more" a quantum function is one-way resistant the "less" it is collision resistant and vice versa. We present an explicit quantum hash function which is "balanced" one-way resistant and collision resistant and demonstrate how to build a large family of balanced quantum hash functions

On computational power of classical and quantum Branching programs

We present a classical stochastic simulation technique of quantum Branching programs. This technique allows to prove the following relations among complexity classes: PrQP-BP ⊆ PP-BP and BQP-BP ⊆ PP-BP. Here BPP-BP and PP-BP stands for the classes of functions computable with bounded error and unbounded error respectively by stochastic branching program of polynomial size. BQP-BP and PrQP-BP stands the classes of functions computable with bounded error and unbounded error respectively by quantum branching program of polynomial size. Second. We present two different types. of complexity lower bounds for quantum 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

The complexity of classical and quantum branching programs: A communication complexity approach

We present a survey of the communication point of view for a complexity lower bounds proof technique for classical (deterministic, nondeterministic and randomized) and quantum models of branching programs. © Springer-Verlag Berlin Heidelberg 2005

Comparative power of quantum and classical computation models

In the talk we present results on comparitve power of classical and quantum computational models. We focus on two well known in Computer Science models: finite automata which is known as uniform computational model and branching programs which is known as nonuniform computational model