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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| Vydáno v: | Natural computing Ročník 22; číslo 4; s. 723 - 736 |
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01.12.2023
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| 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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| Copyright | The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Issue | 4 |
| Keywords | Branching programs Hierarchy Quantum computing Quantum vs classical OBDD Probabilistic OBDD Quantum OBDD Quantum models Computational complexity |
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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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