Farid Ablayev

Kazan (Volga Region) Federal University · Institute of Computer Mathematics and Information Technologies

In the paper, we investigate two problems on strings. The first one is the String matching problem, and the second one is the String comparing problem. We provide a quantum algorithm for the String matching problem that uses exponentially less quantum memory than existing ones. The algorithm uses the hashing technique for string matching, quantum p...

In the paper, we investigate two problems on strings. The first one is the String matching problem, and the second one is the String comparing problem. We provide a quantum algorithm for the String matching problem that uses exponentially less quantum memory than existing ones. The algorithm uses the hashing technique for string matching, quantum p...

Fingerprinting and cryptographic hashing have quite different usages in computer science, but have similar properties. Interpretation of their properties is determined by the area of their usage: fingerprinting methods are methods for constructing efficient randomized and quantum algorithms for computational problems, while hashing methods are one...

This is a review of quantum methods for machine learning problems that consists of two parts. The first part, “quantum tools”, presented some of the fundamentals and introduced several quantum tools based on known quantum search algorithms. This second part of the review presents several classification problems in machine learning that can be accel...

This is a review of quantum methods for machine learning problems that consists of two parts. The first part, “quantum tools”, presents the fundamentals of qubits, quantum registers, and quantum states, introduces important quantum tools based on known quantum search algorithms and SWAP-test, and discusses the basic quantum procedures used for quan...

Compact random number generator (RNG) is presented and demonstrated on the basis of field effect transistor connected in a such a way that avalanche electron current emerges when an input voltage exceeds the threshold value. The avalanche character of this phenomenon provides true randomness and large noise potential in wide spectral band at extrem...

We investigate the branching program complexity of quantum hashing. We consider a quantum hash function that maps elements of a finite field into quantum states. We require that this function is preimage-resistant and collision-resistant. We consider two complexity measures for Quantum Branching Programs (QBP): a number of qubits and a number of co...

In this paper we propose a model of a programmable quantum processing device realizable with existing nano-photonic technologies. It can be viewed as a basis for new high performance hardware architectures. Protocols for physical implementation of device on the controlled photon transfer and atomic transitions are presented. These protocols are des...

Bitcoin and blockchain in general is hot topic nowadays. In the paper we propose a quantum empowering of this technology and show how to speed-up the mining procedure using the modified Grover's algorithm.

We consider quantum version of known computational model Ordered Read-$k$-times Branching Programs or Ordered Binary Decision Diagrams with repeated test ($k$-QOBDD). We get lower bound for quantum $k$-OBDD for $k=o(\sqrt{n})$. This lower bound gives connection between characteristics of model and number of subfunctions for function. Additionally,...

We propose a model of a programmable quantum processing device realizable with existing nanophotonic technologies and which can be viewed as a basis for new high performance hardware architectures. We present protocols and their physical implementation on the controlled photon transfer for executing basic single-qubit and multi-qubit gates. The pos...

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 su...

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 resi...

In the letter we define the notion of a quantum resistant (-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...

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...

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 ǫ-universal hash family and a given family of functions – quantum hash generators. The relationship between ǫ-universal hash families and error-co...

In this paper we show a computational aspect of the quantum hashing technique. In particular we apply it for computing Boolean functions in the model of read-once quantum branching programs based on the properties of specific polynomial presentation of those functions.

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...

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...

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 $\epsilon$-universal hash family and a given family of functions -- quantum hash generator. The proposed construction combines the propert...

We present a version of quantum hash functions based on non-binary discrete functions. The proposed quantum procedure is ``classical-quantum'', that is, it takes a classical bit string as an input and produces a quantum state. The resulting function has the property of a one-way function (pre-image resistance), in addition it has the properties ana...

We present a version of quantum hash function based on non-binary discrete functions. The proposed quantum procedure is "classical-quantum", that is, it takes a classical bit string as an input and produces a quantum state. The resulting function has the property of a one-way function (pre-image resistance), in addition it has the properties analog...

In this paper we explore the well-known k-OBDD model of branching programs. We develop a method of representation of the k-OBDD computation process as an “automata-communication protocol” computation process. Our method allows us to extend the hierarchy proved by Bolling-Sauerhoff-Sieling-Wegener in 1996 for k-OBDDs. Moreover, using the PJM functio...

We propose an effective realization of the universal set of elementary quantum gates in solid state quantum computer based on macroscopic (or mesoscopic) resonance systems - multi-atomic coherent ensembles, squids or quantum dots in quantum electrodynamic cavity. We exploit an encoding of logical qubits by the pairs of the macroscopic two- or three...

We propose an effective realization of a complete set of elementary quantum gates in the solid-state quantum computer based on the multi-atomic coherent (MAC-) ensembles in the QED cavity. Here, we use the two-ensemble qubit encoding and swapping-based operations that together provide implementation of any encoded single-qubit operation by three el...

We propose an effective set of elementary quantum gates which provide an encoded universality and demonstrate the physical feasibility of these gates for the solid-state quantum computer based on the multi-atomic systems in the QED cavity. We use the two-qubit encoding and swapping-based operations to simplify a physical realization of universal qu...

This volume contains the proceedings of the Workshop on High Productivity Computations (HPC 2010) which took place on June 21-22 in Kazan, Russia. This workshop was held as a satellite workshop of the 5th International Computer Science Symposium in Russia (CSR 2010). HPC 2010 was intended to organize the discussions about high productivity computin...

In this paper we focus on how the classical and quantum parallelism are combined in the quantum fingerprinting technique we proposed earlier. We also show that our method can be used not only to efficiently compute Boolean functions with linear polynomial presentations but also can be adapted for the functions with nonlinear presentations of bounde...

In function theory the superposition problem is known as the problem of representing a continuous function f(x1, … ,xk) in k variables as the composition of “simpler” functions. This problem stems from the Hilbert's thirteenth problem. In computer science good formalization for the notion of composition of functions is formula. In the paper we cons...

In this paper we review our current results concerning the computational power of quantum read-once branching programs. First of all, based on the circuit presentation of quantum branching programs and our variant of quantum fingerprinting technique, we show that any Boolean function with linear polynomial presentation can be computed by a quantum...

In this paper, we develop the fingerprinting technique of calculation of Boolean functions in quantum calculation models.The use of the fingerprinting technique is demonstrated on the example of calculation of the function MODm in the class of quantum OBDD (oblivious read-once branching programs). Next, the potentialities of the fingerprinting tech...

In the paper we develop a method for constructing quantum algorithms for computing Boolean functions by quantum ordered read-once branching programs (quantum OBDDs). Our method is based on fingerprinting technique and representation of Boolean functions by their characteristic polynomials. We use circuit notation for branching programs for desired...

We consider the problems of computing certain types of boolean functions which we call Equal-ity, Semi-Simon and Periodicity functions. For all these problems, we prove linear lower complexity bounds on oblivious Ordered Read-Once Quantum Branching Programs (quantum Ordered Binary Decision Diagrams). We present also two different approaches to prov...

We develop quantum fingerprinting technique for constructing quantum branching programs (QBPs), which are considered as circuits with an ability to use classical bits as control variables. We demonstrate our approach constructing optimal quantum ordered binary decision diagram (QOBDD) for M OD m and DM U LT n Boolean functions. The construction of...

We consider the Hidden Subgroup, and Equality-related problems in the context of quantum Ordered Binary Decision Diagrams. For the decision versions of considered problems we show polynomial upper bounds in terms of quantum OBDD width. We apply a new modification of the fingerprinting technique and present the algorithms in circuit notation. Our al...

We present classical simulation techniques for measure once quantum branching programs. For bounded error syntactic quantum branching program of width $w$ that computes a function with error $delta$ we present a classical deterministic branching program of the same length and width at most $(1+2/(1-2delta))^{2w}$ that computes the same function. Se...

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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 progr...

In [3] we exhibited a simple boolean functions f n in n variables such that: 1) f n can be computed by polynomial size randomized ordered read-once branching program with one sided small error; 2) any nondeterministic ordered read-once branching program that computes f n has exponential size. In this paper we present a simple boolean functi...

We define the notion of a randomized branching program in the natural way similar to the definition of a randomized circuit. We exhibit an explicit function f n for which we prove that:1) fn can be computed by polynomial size randomized read-once ordered branching program with a small one-sided error; 2) fn cannot be computed in polynomial size by...

We prove three different types of complexity lower bounds for the one-way unbounded-error and bounded-error error probabilistic communication protocols for boolean functions. The lower bounds are proved for arbitrary boolean functions in the common way in terms of the deterministic communication complexity of functions and in terms of the notion pr...

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 p...

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.

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 “al...

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.

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)).

We prove an exponential lower bound on the size of any randomized ordered read-once branching program computing integer multiplication. Our proof depends on proving a new lower bound on Yao’s randomized one-way communication complexity of certain Boolean functions. It generalizes to some other models of randomized branching programs. In contrast, w...

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)).

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 compu...

We prove upper and lower bounds on the power of quantum and stochastic branching programs of bounded width. We show any NC1 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 NC1 = ACC. This separates width-2 quantum programs from width-2...

In this paper we show that one qubit polynomial time computations are at least as powerful as $\NC^1$ circuits. More precisely, we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power. We show any $\NC^1$ language can be accepted exactly by a width-2 quantum branching...

The superposition (or composition) problem is a problem of representation of a function f by a superposition of “simpler” (in a different meanings) set Ω of functions. In terms of circuits theory this means a possibility of computing f by a finite circuit with 1 fan-out gates Ω of functions. Using a discrete approximation and communication approac...

We investigate the relationship between probabilistic and nondeterministic complexity classes PP, BPP, NP and coNP with respect to ordered read-once branching programs (OBDDs). We exhibit two explicit Boolean functions qn; Rn such that: (1) qn : {0,1}n → { 0,1} belongs to BPP (NP (semi-circle up) coNP) in the context of OBDDs; (2) Rn : {0,1}n → {0,...

In the paper we consider measured-once (MO-QFA) oneway quantum finite automaton. We prove that for MO-QFA Q that (1/2+ε)-accepts (ε ∈ (0,1/2)) regular language L it holds that dim(Q) = Ω (log dim (A)/log log dim (A)). In the case ε ∈ (3/8, 1/2) we have more precise lower bound dim(Q) = Ω (log dim (A)) where A is a minimal deterministic finite autom...

S On the Power of Randomized Branching Programs Farid Ablayev Kazan University (joint work with Marek Karpinski, Universitat Bonn) We define a notion of randomized branching programs in a natural way similar to the notion of randomized circuits. We present two explicit boolean functions f n : f0; 1g 4n ! f0; 1g and g n : f0; 1g n ! f0; 1g such that...

We investigate the relationship between probabilistic and nondeterministic complexity classes PP , BPP , NP and coNP for the ordered read-once branching programs (OBDDs) . We exhibit two explicit boolean functions q n ; r n such that: 1. q n : f0; 1g n ! f0; 1g belongs to BPP n (NP [ coNP ) in the context of OBDDs; 2. r n : f0; 1g n ! f0; 1g belong...

The superposition (or composition) problem is a problem of representation of a function f by a superposition of "simpler" (in a different meanings) set Ω of functions. In terms of circuits theory this means a possibility of computing f by a finite circuit with 1 fan-out gates Ω of functions. Using a discrete approximation and communication approach...

We prove an exponential lower bound (2OmegaGamma n= log n) ) on the size of any randomized ordered read-once branching program computing integer multiplication. Our proof depends on proving a new lower bound on Yao's randomized one-way communication complexity of certain boolean functions. It generalizes to some other common models of randomized br...

We define the notion of a randomized branching program in the natural way similar to the definition of a randomized circuit. We exhibit an explicit boolean function fn : f0; 1g n ! f0; 1g for which we prove that: 1) fn can be computed by polynomial size randomized read-once ordered branching program with a small one-sided error; 2) fn cannot be com...

In [3] we exhibited a simple boolean functions f n in n variables such that: 1) f n can be computed by polynomial size randomized ordered readonce branching program with one sided small error; 2) any nondeterministic ordered read-once branching program that computes f n has exponential size. In this paper we present a simple boolean function g n in...

We prove three different types of complexity lower bounds for the one-way unbounded-error and bounded-error error probabilistic communication protocols for boolean functions. The lower bounds are proved in terms of the deterministic communication complexity of functions and in terms of the notion “probabilistic communication characteristic” that we...

We define the notion of a randomized branching program in the natural way similar to the definition of a randomized circuit. We exhibit an explicit function fn for which we prove that: 1) f n can be computed by polynomial size randomized read-once ordered branching program with a small one-sided error; 2) fn cannot be computed in polynomial size by...

We prove two different types of complexity lower bounds for the one-way bounded-error error probabilistic space complexity. The lower bounds are proved for arbitrary languges in the common way in terms of the deterministic communication dimension of languages and in terms of the notion “probabilistic communication characteristic” of language that w...

The lower bound 0(n log log n) has been proved for the time for recognizing a non-regular languages by one-tape off-line probabilistic machine with bound error probability. This lower bound proves the correctness of Freivald’s longstanding hypothesis, first announced more than ten years ago. Secondly, the lower bound 2logD(L)-th(1/2+e) has been pro...

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New lower bound of complexity for probabilistic automata with error bounded probability was proved. It depends on the language structure and on error probability of recognition. It is shown that for languages which are "rich" with formulated property a new lower bound of probabilistic complexity is more precise than that of Rabin's lower bound. In...

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For several problems there exist probabilistic algorithms which are more effective than any deterministic algorithms solving these problems. For other problems probabilistic algorithms do not have such advantages. We are interested in understanding, why it is so and how to tell one kind of the problems from another. Of course, we are not able to pr...

In this paper we investigate a well known sequential model of computation: one-way LOG-SPACE Turing machines. We analyze a different known method for constructing an effective probabilistic algorithm. We prove a lower bound for probabilistic space complexity, which is good enough for understanding the above problem for the one-way LOG-SPACE Turing...