Quantum Online Algorithms with Respect to Space and Advice Complexity

Abstract

Online algorithm is a well-known computational model. We introduce quantum online algorithms and investigate them with respect to a competitive ratio in two points of view: space complexity and advice complexity. We start with exploring a model with restricted memory and show that quantum online algorithms can be better than classical ones (deterministic or randomized) for sublogarithmic space (memory), and they can be better than deterministic online algorithms without restriction for memory. Additionally, we consider polylogarithmic space case and show that in this case, quantum online algorithms can be better than deterministic ones as well. Another point of view to the online algorithms model is advice complexity. So, we introduce quantum online algorithms with a quantum channel with an adviser. Firstly, we show that quantum algorithms have at least the same computational power as classical ones have. And we give some examples of quantum online algorithms with advice. Secondly, we show that if we allow to use shared entangled qubits (EPR-pairs), then quantum online algorithm can use two times less advise qubits comparing to a classical one. We apply this approach to the well-known Paging Problem.

Quantum Online Streaming Algorithms with Logarithmic Memory

“…In that case, we can consider automata (streaming algorithm) as online algorithms. This model was explored in [9,17,10,20]. We are interested in quantum online algorithms.…”

“…Here we claim, that we can construct algorithm with single qubit, but not two qubits as in Theorem 4. The idea of the algorithm is based on ides from [5,4,7,20]. Let us describe an algorithm B for BH t k,r,w (f ), f = P artialM OD β m .…”

Classical and Quantum Computations with Restricted Memory

“…In this paper, we consider streaming algorithms as online algorithms. This classical model was considered in [18,24,30,47]. Automata for online minimization problems were considered in [43].…”

“…We are interested in the investigation of a new model of online algorithms, the quantum online algorithms, that use the power of quantum computing for solving online minimization problems. This model was introduced in [47]. Here, we focus on quantum online streaming algorithms.…”

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

“…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].…”