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A new Holant dichotomy inspired by quantum computation [article]

Miriam Backens
2017 arXiv   pre-print
The Holant framework is based on the theory of holographic algorithms, which was originally inspired by concepts from quantum computation, but this connection appears not to have been explored before.  ...  Here, we employ quantum information theory to explain existing results in a concise way and to derive a dichotomy for a new family of problems, which we call Holant^+.  ...  Acknowledgements I would like to thank Ashley Montanaro, both for introducing me to Holant problems and for helpful comments on earlier drafts of this paper.  ... 
arXiv:1702.00767v1 fatcat:r53p5qomafb6nklgs7htd34xfe

A New Holant Dichotomy Inspired by Quantum Computation

Miriam Backens, Marc Herbstritt
2017 International Colloquium on Automata, Languages and Programming  
The Holant framework is based on the theory of holographic algorithms, which was originally inspired by concepts from quantum computation, but this connection appears not to have been explored before.  ...  Here, we employ quantum information theory to explain existing results in a concise way and to derive a dichotomy for a new family of problems, which we call Holant + .  ...  I would like to thank Ashley Montanaro, both for introducing me to Holant problems and for helpful comments on earlier drafts of this paper.  ... 
doi:10.4230/lipics.icalp.2017.16 dblp:conf/icalp/Backens17 fatcat:f2ya5apjgbdtvnhbluaerqnunq

A New Holant Dichotomy Inspired by Quantum Computation * †

Miriam Backens
unpublished
The Holant framework is based on the theory of holographic algorithms, which was originally inspired by concepts from quantum computation, but this connection appears not to have been explored before.  ...  Here, we employ quantum information theory to explain existing results in a concise way and to derive a dichotomy for a new family of problems, which we call Holant +.  ...  I would like to thank Ashley Montanaro, both for introducing me to Holant problems and for helpful comments on earlier drafts of this paper.  ... 
fatcat:pnnkqtyr2vborfxvbslodexbdm

From Holant to Quantum Entanglement and Back

Jin-Yi Cai, Zhiguo Fu, Shuai Shao, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
Then we use entanglement properties of constraint functions to derive a new complexity dichotomy for all real-valued Holant problems containing a signature of odd arity.  ...  Holant problems are intimately connected with quantum theory as tensor networks. We first use techniques from Holant theory to derive new and improved results for quantum entanglement theory.  ...  Existing dichotomies inspired by entanglement theory There are many natural connections between Holant problems and quantum theory.  ... 
doi:10.4230/lipics.icalp.2020.22 dblp:conf/icalp/CaiF020 fatcat:ywcyv7focvgv5abuejepviw37y

A Full Dichotomy for $\hol^{c}$, Inspired by Quantum Computation

Miriam Backens
2021 SIAM journal on computing (Print)  
They arise from the theory of holographic algorithms, which was originally inspired by concepts from quantum computation.  ...  Here, we employ quantum information theory to explain existing results about holant problems in a concise way and to derive two new dichotomies: one for a new family of problems, which we call Holant^+  ...  Acknowledgements I would like to thank Pinyan Lu for pointing out a flaw in the original statement of the main theorem, Jin-Yi Cai for interesting discussions about extending the Holant + result to the  ... 
doi:10.1137/20m1311557 fatcat:nh3bbgfnevh2ldif272mzdqasy

From Holant to Quantum Entanglement and Back [article]

Jin-Yi Cai, Zhiguo Fu, Shuai Shao
2020 arXiv   pre-print
Then we use entanglement properties of constraint functions to derive a new complexity dichotomy for all real-valued Holant problems containing an odd-arity signature.  ...  Holant problems are intimately connected with quantum theory as tensor networks. We first use techniques from Holant theory to derive new and improved results for quantum entanglement theory.  ...  Existing dichotomies inspired by entanglement theory There are many natural connections between Holant problems and quantum theory.  ... 
arXiv:2004.05706v1 fatcat:b4mzukj765c4tpvyhcxc3ygq7a

Computational Counting (Dagstuhl Seminar 18341)

Ivona Bezáková, Leslie Ann Goldberg, Mark R. Jerrum, Marc Herbstritt
2018 Dagstuhl Reports  
A total of 43 researchers from all over the world, with interests and expertise in different aspects of computational counting, actively participated in the meeting.  ...  This report documents the program and the outcomes of Dagstuhl Seminar 17341 "Computational Counting".  ...  While the Holant framework was originally inspired by ideas from quantum computation, this connection has not been exploited before to analyse their complexity.  ... 
doi:10.4230/dagrep.7.8.23 dblp:journals/dagstuhl-reports/BezakovaGJ17 fatcat:yp3oqvgo4fal5lbio5yeqt4uje

Preliminary study for developing instantaneous quantum computing algorithms (IQCA)

Richard L Amoroso
2019 Journal of Physics, Conference Series  
In this work a penultimate form of QC speedup -Instantaneous Quantum Computing Algorithms (IQCA) is proposed.  ...  This is a naturally occurring freebie Feynman talked of the necessity of a synchronization backbone requirement for implementing universal quantum computing (UQC) [5, [13] [14] [15].  ...  For Holant problems Eq (4) is called the Holant after a related exponential sum introduced by Valiant [63] . To further clarify Holant is a framework of counting characterized by local constraints.  ... 
doi:10.1088/1742-6596/1251/1/012005 fatcat:coezjez5end35hllvdhrsic2mq

A Dichotomy for Real Boolean Holant Problems [article]

Shuai Shao, Jin-Yi Cai
2020 arXiv   pre-print
We prove a complexity dichotomy for Holant problems on the boolean domain with arbitrary sets of real-valued constraint functions.  ...  It is proved that for every set F of real-valued constraint functions, Holant(F) is either P-time computable or #P-hard. The classification has an explicit criterion.  ...  Acknowledgement We would like to thank Professor Mingji Xia for pointing out that the strong Bell property can be used to prove Lemma 8.9, which led to the inspiration to prove Lemma 8.10.  ... 
arXiv:2005.07906v1 fatcat:nib2ti4iazgclbfd7umsxodgk4

On the Complexity of #CSP^d

Jiabao Lin, James R. Lee
2021
The Holant is a framework that generalizes counting CSP.  ...  the complexity of Holant problems.  ...  Theorem 3 and Theorem 6 give a unified complexity dichotomy for the whole #CSP d family. Being abstract enough, they reveal that essentially there is no new tractable structure for d > 1.  ... 
doi:10.4230/lipics.itcs.2021.40 fatcat:7q5s4pzw4ra7fexly4ir2tyf5e

Counting Homomorphisms and Partition Functions [article]

Martin Grohe, Marc Thurley
2011 arXiv   pre-print
In the second part, we give a proof of a theorem due to Bulatov and the first author of this paper [BulGro05], which classifies the complexity of partition functions described by matrices with non-negative  ...  As we shall see, homomorphisms, or more precisely the numbers of homomorphisms between two structures, are also related to a fundamental computational problem of statistical physics.  ...  In the context of holant functions complexity dichotomies have been obtained by Cai, Lu, and Xia [CLX08, CLX09] .  ... 
arXiv:1104.0185v3 fatcat:n633g6nfsveubideomdpcu5u4y

Graph Homomorphisms with Complex Values: A Dichotomy Theorem [article]

Jin-Yi Cai, Xi Chen, Pinyan Lu
2011 arXiv   pre-print
Building on work by Dyer and Greenhill, Bulatov and Grohe, and especially the recent beautiful work by Goldberg, Grohe, Jerrum and Thurley, we prove a complete dichotomy theorem for this problem.  ...  We study the computational complexity of Z_A for arbitrary symmetric matrices A with algebraic complex values.  ...  The counting problem on an input instance Ω is to compute Holant(Ω) = edge assignments σ v∈V F v (σ | E(v) ).  ... 
arXiv:0903.4728v2 fatcat:6ssjnabttneblhx2xwakhlyhla

Dagstuhl Reports, Volume 7, Issue 8, August 2017, Complete Issue [article]

2018
Acknowledgments Schloss Dagstuhl was the perfect place for hosting a seminar like this.  ...  While the Holant framework was originally inspired by ideas from quantum computation, this connection has not been exploited before to analyse their complexity.  ...  One of the new results, the full dichotomy for Holant c , solves a long-standing open problem.  ... 
doi:10.4230/dagrep.7.8 fatcat:gksmijgk5ff6reblxsqnt33aze

Counting Problems on Quantum Graphs: Parameterized and Exact Complexity Classifications [article]

Marc Roth, Universität Des Saarlandes, Universität Des Saarlandes
2019
parameterized counting problems: The framework of parameterized counting complexity was introduced by Flum and Grohe, and McCartin in 2002 and forms a hybrid between the classical field of computational  ...  Quantum graphs, as defined by Lovász in the late 60s, are formal linear combinations of simple graphs with finite support.  ...  without a new major insight.  ... 
doi:10.22028/d291-28348 fatcat:o2mlimd7fjeptaz7y2h5ffiake

2013 Jahresbericht Annual Report

Gesellschafter
unpublished
New quantum algorithms to attack potential new hardness assumptions: For instance, can quantum algorithms be used to improve on classical solutions for computational problems in lattices or for the decoding  ...  Quantum computational assumptions: Which problems are currently considered as intractable, even for a quantum computer, and possibly might have the potential to be of cryptographic interest?  ...  providing a seminar Veranstaltungen zu unterstützen. center for academic events.  ... 
fatcat:xixyfnmw65ehzlozcfxl55r5uq
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