Quantum Online Algorithms with Respect to Space Complexity

Khadiev, Kamil; Khadieva, Aliya; Mannapov, Ilnaz

doi:10.48550/arxiv.1709.08409

Cited by 1 publication

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“…This measure is an analog of a number of states for finite automaton and OBDDs can be seen as nonuniform finite automata (see for example [4]). As for many other computation models, it is possible to consider quantum OBDDs, and during the last decade they have been studied vividly [1,5,43,48,47,2,24,23,20,19,21,31,30].…”

Section: Introductionmentioning

confidence: 99%

Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies

Khadiev¹,

Khadieva²,

Knop³

2022

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In this paper, we study quantum Ordered Binary Decision Diagrams(OBDD) model; it is a restricted version of read-once quantum branching programs, with respect to "width" complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique ("reordering") for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function REQ such that the deterministic OBDD complexity of it is at least 2 Ω(n/ log n) , and the quantum OBDD complexity of it is at most O(n 2 / log n). It is the biggest known gap for explicit functions not representable by OBDDs of a linear width. Another function(shifted equality function) allows us to obtain a gap 2 Ω(n) vs O(n 2 ). Moreover, we prove the bounded error quantum and probabilistic OBDD width hierarchies for complexity classes of Boolean functions. Additionally, using "reordering" method we extend a hierarchy for read-k-times Ordered Binary Decision Diagrams (k-OBDD) of polynomial width, for k = o(n/ log 3 n). We prove a similar hierarchy for bounded error probabilistic k-OBDDs of polynomial, superpolynomial and subexponential width.

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