Parameterized Complexity Classes Defined by Threshold Circuits: Using Sorting Networks to Show Collapses with W-hierarchy Classes

2021 ◽  
pp. 348-363
Author(s):  
Raffael M. Paranhos ◽  
Janio Carlos Nascimento Silva ◽  
Uéverton S. Souza ◽  
Luiz Satoru Ochi
Keyword(s):  
Parameterized Complexity ◽  
Complexity Classes ◽  
Sorting Networks ◽  
Threshold Circuits

scholarly journals Counting, fanout and the complexity of quantum ACC

2002 ◽  
Vol 2 (1) ◽  
pp. 35-65
Author(s):  
F. Green ◽  
S. Homer ◽  
C. Moore ◽  
C. Pollett
Keyword(s):  
Complexity Classes ◽  
Constant Depth ◽  
Classical Counterpart ◽  
Classical Class ◽  
Polynomial Size ◽  
Constant Depth Circuits ◽  
Language Classes ◽  
Threshold Circuits ◽  
Quantum Analog ◽  
Quantum Complexity

We propose definitions of QAC^0, the quantum analog of the classical class AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC[q], the analog of the class ACC[q] where Mod_q gates are also allowed. We prove that parity or fanout allows us to construct quantum MOD_q gates in constant depth for any q, so QACC[2] = QACC. More generally, we show that for any q,p > 1, MOD_q is equivalent to MOD_p (up to constant depth and polynomial size). This implies that QAC^0 with unbounded fanout gates, denoted QACwf^0, is the same as QACC[q] and QACC for all q. Since \ACC[p] \ne ACC[q] whenever p and q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC^0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomial-size circuits. With further restrictions, language classes related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth.

scholarly journals Describing parameterized complexity classes

2003 ◽  
Vol 187 (2) ◽  
pp. 291-319 ◽  
Author(s):  
Jörg Flum ◽  
Martin Grohe
Keyword(s):  
Parameterized Complexity ◽  
Complexity Classes

Computational Power of Threshold Circuits of Energy at most Two

2018 ◽  
Vol E101.A (9) ◽  
pp. 1431-1439
Author(s):  
Hiroki MANIWA ◽  
Takayuki OKI ◽  
Akira SUZUKI ◽  
Kei UCHIZAWA ◽  
Xiao ZHOU
Keyword(s):  
Computational Power ◽  
Threshold Circuits
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