Exponential separation between quantum and classical ordered binary decision diagrams, reordering method and hierarchies

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 ex...

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Bibliographic Details
Published in:Natural computing Vol. 22; no. 4; pp. 723 - 736
Main Authors: Khadiev, Kamil, Khadieva, Aliya, Knop, Alexander
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.12.2023
Subjects:
Artificial Intelligence
Complex Systems
Computer Science
Evolutionary Biology
Processor Architectures
Theory of Computation
Branching programs
Hierarchy
Quantum computing
Quantum vs classical
OBDD
Probabilistic OBDD
Quantum OBDD
Quantum models
Computational complexity
ISSN:1567-7818, 1572-9796
Online Access:Get full text
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Summary: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 OBDD s 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 - OBDD s of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 – 12, 2017 Khadiev and Khadieva ( 2017 )
ISSN:1567-7818
1572-9796
DOI:10.1007/s11047-022-09904-3