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Perfect Matchings, Rank of Connection Tensors and Graph Homomorphisms
[article]

2020
*
arXiv
*
pre-print

It is shown that

arXiv:1909.03179v2
fatcat:j6by6lyb6bbbvgeykkewsgm3xq
*counting*perfect matchings, and a host*of*other*graph*properties naturally defined as Holant problems (edge models), cannot be expressed by*graph**homomorphism*functions with both*complex*... This is modeled after*the*theory developed by Freedman, Lov\'asz and Schrijver in [22] for connection matrices, in*the*study*of**graph**homomorphism*functions over real edge weight and positive vertex weight ... To do that, in addition to*the*algebras*of**quantum**graphs*G(S), we define a second type*of*algebras*of**quantum**graphs*G ⊆ (S), where S ⊆ Z + is a finite set*of*labels. ...##
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From the Ising and Potts models to the general graph homomorphism polynomial
[article]

2015
*
arXiv
*
pre-print

In this note we study some

arXiv:1401.6335v2
fatcat:3mzln32dwvcljmhwvq6gyvjb4q
*of**the*properties*of**the*generating polynomial for*homomorphisms*from a*graph*to at complete weighted*graph*on $q$ vertices. ... We discuss how this polynomial relates to a long list*of*other well known*graph*polynomials and*the*partition functions for different spin models, many*of*which are specialisations*of**the**homomorphism*... polynomial expected running*time*for several 2-CSP which in general are NP-hard, and for larger p*the*same algorithms have an*exponential*expected running*time*. ...##
###
Graph Homomorphisms with Complex Values: A Dichotomy Theorem
[article]

2011
*
arXiv
*
pre-print

We study

arXiv:0903.4728v2
fatcat:6ssjnabttneblhx2xwakhlyhla
*the*computational*complexity**of*Z_A for arbitrary symmetric matrices A with algebraic*complex*values. ...*The*function Z_A can encode many interesting*graph*properties, including*counting*vertex covers and k-colorings. ...*The**counting*problem for*graph**homomorphism*is to compute*the*number*of**homomorphisms*from G to H. For a fixed*graph*H, this problem is also known as*the*#H-coloring problem. ...##
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Quantum Algorithms
[article]

2008
*
arXiv
*
pre-print

This article surveys

arXiv:0808.0369v1
fatcat:gsiyvpw7mnd2hlmki5tvgjwgvu
*the*state*of**the*art in*quantum*computer algorithms, including both black-box and non-black-box results. ... This includes a summary*of**the*early*quantum*algorithms, a description*of**the*Abelian Hidden Subgroup algorithms (including Shor's factoring and discrete logarithm algorithms),*quantum*searching and amplitude ... Since*the*running*time**of*simulating each*exponential*term is assumed to be polynomial in n,*the*overall running*time*is polynomial in n. ...##
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The Quantum Frontier
[article]

2013
*
arXiv
*
pre-print

We start by describing aspects

arXiv:1206.0785v2
fatcat:mkd7qulj55fvteq5pb5dty6rbm
*of**quantum*mechanics that are at*the*heart*of*a*quantum*view*of*information processing. ... A number*of**the*phenomena described were initially viewed as oddities*of**quantum*mechanics. ... As analysis*of*Grover's algorithm focuses on query*complexity*,*counting*only*the*number*of**times*a database or function must be queried in order to find a match rather than considering*the*computational ...##
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Quantum Complexity: restrictions on algorithms and architectures
[article]

2010
*
arXiv
*
pre-print

A dissertation submitted to

arXiv:1005.1425v1
fatcat:iusngjiypfgxzlezpnz5yiyarq
*the*University*of*Bristol in accordance with*the*requirements*of**the*degree*of*Doctor*of*Philosophy (PhD) in*the*Faculty*of*Engineering, Department*of*Computer Science, July ... RFW acknowledges support from*the*QUPRODIS network*of**the*EU. ...*Complexity*What*counts*as (*quantum*) information, and how do we decide whether*the*processing that it has been subject to is '*quantum*' ? ...##
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Entropy estimates for simplicial quantum gravity

1995
*
Journal of Geometry and Physics
*

They have an

doi:10.1016/0393-0440(94)00022-v
fatcat:mssxaamr4ve7pk3o35csulo7ki
*exponential*leading behavior determined by*the*Reidemeister-Franz torsion associated with orthogonal representations*of**the*fundamental group*of**the*manifold. ... Such results are either consistent with*the*known asymptotics*of*dynamically triangulated two-dimensional surfaces, or with*the*numerical evidence supporting an*exponential*leading behavior for*the*number ...*The*authors are also much indebted to*the*referee, who carefully read a preliminary version*of**the*manuscript and made many constructive suggestions. ...##
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Two paradigms for topological quantum computation
[article]

2008
*
arXiv
*
pre-print

In particular we suggest correspondences between

arXiv:0803.1258v1
fatcat:xajxm6kyfvhbrgwkuhwvtubsvi
*the*computational power*of*topological*quantum*computers, computational*complexity**of*link invariants and images*of*braid group representations. ... We present two paradigms relating algebraic, topological and*quantum*computational statistics for*the*topological model for*quantum*computation. ...*The*class*of**counting*functions*of**complexity*#P are related to decision problems*of**complexity*N P , where instead*of*asking if there exists a "yes" answer one*counts**the*number*of*"yes" answers. ...##
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Quantum Cryptanalysis (Dagstuhl Seminar 19421)

2020
*
Dagstuhl Reports
*

Identifying new cryptanalytic improvements that make use

doi:10.4230/dagrep.9.10.47
dblp:journals/dagstuhl-reports/MoscaNS19
fatcat:b4fbhk267zhsdch2lotalrzwoa
*of**quantum*algorithms and expanding*the*applicability*of**the*best known cryptanalytic attacks by means*of**quantum*technology. ... This seminar report documents*the*program and*the*outcomes*of*Dagstuhl Seminar 19421*Quantum*Cryptanalysis, which took place in October 2019. ...*The*strong*exponential*-*time*hypothesis (SETH) is a commonly used conjecture in*the*field*of**complexity*theory. ...##
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Counting Graph Homomorphisms
[chapter]

*
Algorithms and Combinatorics
*

*Counting*

*homomorphisms*between

*graphs*(often with weights) comes up in a wide variety

*of*areas, including extremal

*graph*theory, properties

*of*

*graph*products, partition functions in statistical physics ... We define a distance

*of*two

*graphs*in terms

*of*similarity

*of*their global structure, which also reflects

*the*closeness

*of*(appropriately scaled)

*homomorphism*numbers into

*the*two

*graphs*. ... Let twr(ε) denote

*the*1/ε 2

*times*iterated

*exponential*function (

*the*"tower"). ...

##
###
Non-abelian Quantum Statistics on Graphs

2019
*
Communications in Mathematical Physics
*

*The*framework involves a study

*of*isomorphism classes

*of*flat

*complex*vector bundles over

*the*configuration space

*of*X which can be achieved by determining its homology groups. ... These conclusions are based on our solution

*of*

*the*so-called universal presentation problem for homology groups

*of*

*graph*configuration spaces for certain families

*of*

*graphs*. ... They are relevant in

*the*context

*of*

*quantum*computing where one is interested mainly in universality

*of*unitary representations

*of*braid groups and

*the*dimension

*of*

*the*representations grow

*exponentially*...

##
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The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes
[article]

2016
*
arXiv
*
pre-print

Those topics include

arXiv:1607.05256v1
fatcat:mnpmspgwlrdk5pm3fcsthl3lui
*the*power*of**quantum*proofs and advice states; how to construct*quantum*money schemes secure against counterfeiting; and*the*role*of**complexity*in*the*black-hole information paradox ...*The*focus is*quantum*circuit*complexity*---i.e.,*the*minimum number*of*gates needed to prepare a given*quantum*state or apply a given unitary transformation---as a unifying theme tying together several ... Fourth, it's probably easy to prepare*quantum*states that have*exponential*circuit*complexity*, using only polynomial*time*! ...##
###
Normal subgroup reconstruction and quantum computation using group representations

2000
*
Proceedings of the thirty-second annual ACM symposium on Theory of computing - STOC '00
*

*The*Hidden Subgroup Problem is

*the*foundation

*of*many

*quantum*algorithms. ...

*The*non-Abelian case is open; an efficient solution would give rise to an efficient

*quantum*algorithm for

*Graph*Isomorphism. ...

*Counting*dimensions, IGI = ~-'~d 2. (1) i

*The*main tool in

*quantum*polynomial

*time*algorithms is

*the*Fourier transform. DEFINITION 2. Let f : G --+ C. ...

##
###
Topspin networks in loop quantum gravity

2012
*
Classical and quantum gravity
*

This extends

doi:10.1088/0264-9381/29/20/205015
fatcat:wlhywxusfjd53o7xnwtks2mzwa
*the*idea*of*"background independence" in loop*quantum*gravity to include topology as well as geometry. ... It is hoped this work will confirm*the*usefulness*of**the*topspin network formalism and open up several new avenues for research into*quantum*gravity. ...*Of*course, since*the*dynamics*of*LQG occur only on*the**graphs*, they will contribute directly to*the**quantum*theory. ...##
###
Adiabatic Quantum Computation
[article]

2016
*
arXiv
*
pre-print

Yet

arXiv:1610.04708v1
fatcat:n2dasjyqe5afrmk5omvolksfje
*the*main focus*of*this work lies in studying*the*efficiency*of**quantum*circuit simulations by adabatic*quantum*computation. ...*The*standard Hamiltonian construction by Kitaev is based on a path*graph*reflecting*the*$L$ computation steps and influencing*the*scaling*of**the*necessary evolution*time*by its spectral gap*of*$\mathcal ... A standard way to formulate*the*evolution*complexity*is to*count**the*numbers*of*gates and oracle queries for*the*matrix entries a*quantum*circuit needs to simulate this evolution. ...
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