Error-Free Affine, Unitary, and Probabilistic OBDDs

Adventures Between Lower Bounds and Higher Altitudes

et al. 2018

Self Cite

Automata and branching programs are known models of computation with restricted memory. These models of computation were in focus of a large number of researchers during the last decades. Streaming algorithms are a modern model of computation with restricted memory. In this paper, we present recent results on the comparative computational power of quantum and classical models of branching programs and streaming algorithms. In addition to comparative complexity results, we present a quantum branching program for computing a practically important quantum function (quantum hash function) and prove optimality of this algorithm.

Section: Las-vegas Modelsmentioning

confidence: 96%

“…Using this language, better gaps for the probabilistic and quantum cases were shown. Theorem 6 (Ibrahimov et al [38]). For each k > 0, we have:…”

Section: Las-vegas Modelsmentioning

confidence: 98%

See 1 more Smart Citation

Classical and Quantum Computations with Restricted Memory

Ablayev

Adventures Between Lower Bounds and Higher Altitudes

et al. 2018

Self Cite

“…Some underlying results are as follows: (i) they are more powerful than their probabilistic and quantum counterparts (PFAs and QFAs) with bounded and unbounded error; (ii) one-sided bounded-error AfAs and nondeterministic QFAs define the same class when using rational number transitions; and, (iii) AfAs can distinguish any given pair of strings by using two states with zero-error. Very recently, affine OBDD was introduced in [9] and it was shown that they can be exponentially narrower than bounded-error quantum and classical OBDDs.…”

Section: Introductionmentioning

confidence: 99%

Exact Affine Counter Automata

Nakanishi

Electron. Proc. Theor. Comput. Sci.

Prūsis

et al. 2017

Self Cite

We introduce an affine generalization of counter automata, and analyze their ability as well as affine finite automata. Our contributions are as follows. We show that there is a language that can be recognized by exact realtime affine counter automata but by neither 1-way deterministic pushdown automata nor realtime deterministic k-counter automata. We also show that a certain promise problem, which is conjectured not to be solved by two-way quantum finite automata in polynomial time, can be solved by Las Vegas affine finite automata. Lastly, we show that how a counter helps for affine finite automata by showing that the language MANYTWINS, which is conjectured not to be recognized by affine, quantum or classical finite state models in polynomial time, can be recognized by affine counter automata with one-sided bounded-error in realtime.Comment: In Proceedings AFL 2017, arXiv:1708.0622

“…There are many problems where quantum algorithms outperform the best known classical algorithms [18,28]. superior of quantum over classical was shown for different computing models like query model, streaming processing models, communication models and others [6,5,4,3,2,1,30,29,27,31].In this paper, we present the quantum algorithm for the class of problems on directed acyclic graphs (DAGs) that uses a dynamic programming approach. The dynamic programming approach is one of the most useful ways to solve problems in computer science [17].…”

mentioning

confidence: 99%

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

Unconventional Computation and Natural Computation

Safina

2019

Self Cite

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( √n m logn), and the running time of the best known deterministic algorithm is O(n + m), where n is the number of vertices,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.Quantum computing [32,9] is one of the hot topics in computer science of last decades. There are many problems where quantum algorithms outperform the best known classical algorithms [18,28]. superior of quantum over classical was shown for different computing models like query model, streaming processing models, communication models and others [6,5,4,3,2,1,30,29,27,31].In this paper, we present the quantum algorithm for the class of problems on directed acyclic graphs (DAGs) that uses a dynamic programming approach. The dynamic programming approach is one of the most useful ways to solve problems in computer science [17]. The main idea of the method is to solve a problem using pre-computed solutions of the same problem, but with smaller parameters. Examples of such problems for DAGs that are considered in this paper are the single source longest path search problem for weighted DAGs and the diameter search problem for unweighted DAGs.