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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Vydané v:	Natural computing Ročník 22; číslo 4; s. 723 - 736
Hlavní autori:	Khadiev, Kamil, Khadieva, Aliya, Knop, Alexander
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Dordrecht Springer Netherlands 01.12.2023
Predmet:	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
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Abstract	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 )
AbstractList	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 )
Author	Knop, Alexander Khadiev, Kamil Khadieva, Aliya
Author_xml	– sequence: 1 givenname: Kamil orcidid: 0000-0002-5151-9908 surname: Khadiev fullname: Khadiev, Kamil email: kamilhadi@gmail.com organization: Institute of Computational Mathematics and IT, Kazan Federal University – sequence: 2 givenname: Aliya surname: Khadieva fullname: Khadieva, Aliya organization: Institute of Computational Mathematics and IT, Kazan Federal University, Faculty of Computing, University of Latvia – sequence: 3 givenname: Alexander surname: Knop fullname: Knop, Alexander organization: Department of Mathematics, University of California, San Diego
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Snippet	In this paper, we study quantum Ordered Binary Decision Diagrams( OBDD ) model; it is a restricted version of read-once quantum branching programs, with...
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SubjectTerms	Artificial Intelligence Complex Systems Computer Science Evolutionary Biology Processor Architectures Theory of Computation
Title	Exponential separation between quantum and classical ordered binary decision diagrams, reordering method and hierarchies
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