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On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order

2021

We study the problem

doi:10.4230/lipics.icalp.2021.91
fatcat:hqmttj7zuzapxnkhvrkxgmziki
*of**counting*the number*of**homomorphisms*from an input*graph*G to a fixed (*quantum*)*graph*̄{H}*in*any*finite**field**of**prime**order*ℤ_p. ... First, we introduce the study*of**quantum**graphs*to the study*of*modular*counting**homomorphisms*. ... Focusing*on*these "reduced"*quantum**graphs*we obtain the following inheritance theorem. 91:4*On**Counting*(*Quantum*-)*Graph**Homomorphisms**in**Finite**Fields**of**Prime**Order*This shows that the complexity ...##
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On the refined counting of graphs on surfaces

2013
*
Nuclear Physics B
*

We review and extend relevant mathematical literature and present results

doi:10.1016/j.nuclphysb.2013.01.023
fatcat:uhsvpgozx5cbvk6ascaddc4vty
*on*the*counting**of*some infinite classes*of*bi-partite*graphs*. ... These*counting*problems can be expressed*in*terms*of*observables*in*three-dimensional topological*field*theory with S_d gauge group which gives them a topological membrane interpretation. ... CW would like to thank Peking University, Shanghai Jiaotong University and Zhejiang University for the hospitality where part*of*the work was done. ...##
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New mathematical structures in renormalizable quantum field theories

2003
*
Annals of Physics
*

Computations

doi:10.1016/s0003-4916(02)00023-4
fatcat:gic66hpzwfclzioicvdryrbwgu
*in*renormalizable perturbative*quantum**field*theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying*quantum**field*theory. ... We review these new structures and the role they can play*in*future developments. ...*In*section*one*we summarize the basic notions*of*perturbative*quantum**field*theory using the pre-Lie structure*of**graph*insertions. ...##
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Shorter quantum circuits
[article]

2022
*
arXiv
*
pre-print

*In*particular, over the Clifford+√(T) gate set we achieve an average non-Clifford gate

*count*

*of*0.23log_2(1/ε)+2.13 and T-

*count*0.56log_2(1/ε)+5.3 with mixed fallback approximations for diamond norm accuracy ...

*in*sequence length by a factor

*of*7/9. ... Sarnak first observed the connection between LPS

*graphs*, quaternion

*orders*and

*quantum*gate sets

*in*his letter to Aaronson and Pollington

*on*the Solvay-Kitaev Theorem and golden gates [Sar] . ...

##
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Quantum Algorithms
[article]

2008
*
arXiv
*
pre-print

This article surveys the state

arXiv:0808.0369v1
fatcat:gsiyvpw7mnd2hlmki5tvgjwgvu
*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 ...*One*can also define representations for*finite*non-Abelian groups G, except*in**order*to fully capture the structure*of*G, we allow*homomorphisms*ρ to invertible matrices over C (*in*the Abelian case, we ...##
###
Efficient quantum processing of ideals in finite rings
[article]

2009
*
arXiv
*
pre-print

Suppose we are given black-box access to a

arXiv:0908.0022v1
fatcat:paabwibbvjb5dko5hkoondyr5u
*finite*ring R, and a list*of*generators for an ideal I*in*R. We show how to find an additive basis representation for I*in*poly(log |R|) time. ... , and test the injectivity and surjectivity*of*ring*homomorphisms*. ... As shown*in*[6] , both integer factorization and*graph*isomorphism reduce to the problem*of**counting*automorphisms*of*rings. This*counting*problem is contained*in*AM∩coAM. ...##
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Hall algebras and quantum Frobenius

2010
*
Duke mathematical journal
*

Lusztig has constructed a Frobenius morphism for

doi:10.1215/00127094-2010-036
fatcat:cxm3nibg3jgurihy6pzebi3u7y
*quantum*groups at an -th root*of*unity, which gives an integral lift*of*the Frobenius map*on*universal enveloping algebras*in*positive characteristic. ... Using the Hall algebra we give a simple construction*of*this map for the positive part*of*the*quantum*group attached to an arbitrary Cartan datum. To George Lusztig*on*his 60th birthday. ... Our construction hinges*on*the (trivial) observation that the q-Schur algebra construction works for all*finite**fields*-and so*in*particular, it works both for the*field*F q and the*field*F q -and*on*the ...##
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Quantum field theory over

2013
*
Journal of Geometry and Physics
*

*In*this paper we discuss some questions about geometry over the

*field*with

*one*element, motivated by the properties

*of*algebraic varieties that arise

*in*perturbative

*quantum*

*field*theory. ... compactifications

*of*the

*graph*configuration spaces, that arise

*in*the computation

*of*Feynman integrals

*in*position space, admit an F1-structure. ... The second author thanks Paolo Aluffi for many useful discussions and for a careful reading

*of*the manuscript, Javier López-Peña for reading an earlier draft

*of*the paper and offering comments and suggestions ...

##
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Quantum Tomography and Schwinger's Picture of Quantum Mechanics
[article]

2022
*
arXiv
*
pre-print

The attention is focused

arXiv:2205.00170v1
fatcat:4e2h4zs4bjhf7pwh6gths43foy
*on*spin tomography:*In*this context the groupoid*of*interest is the groupoid*of*pairs over a*finite*set. ...*In*this paper the problem*of*tomographic reconstruction*of*states is investigated within the so-called Schwinger's picture*of**Quantum*Mechanics*in*which a groupoid is associated with every*quantum*system ... " (C\&QIG-BG-CM-UC3M), and*in*the context*of*the V PRICIT (Regional Programme*of*Research and Technological Innovation). ...##
###
Quantum Statistical Mechanics of the Absolute Galois Group
[article]

2020
*
arXiv
*
pre-print

We present possible extensions

arXiv:1907.13545v2
fatcat:h3c5feeejzhfhj73lmfzy6uy24
*of*the*quantum*statistical mechanical formulation*of*class*field*theory to the non-abelian case, based*on*the action*of*the absolute Galois group*on*Grothendieck's dessins ... d'enfant, the embedding*in*the Grothendieck-Teichm\"uller group, and the Drinfeld-Ihara involution. ... We thank the anonymous referees for several very useful comments that significantly improve the paper, and Lieven Le Bruyn for his suggestions*in*a series*of*mails that helped us to avoid ambiguities. ...##
###
Reversible quantum cellular automata
[article]

2004
*
arXiv
*
pre-print

*quantum*circuits, and

*on*Clifford transformations with respect to a description

*of*the single cells by

*finite*Weyl systems. ... We present several construction methods for

*quantum*cellular automata, based

*on*unitaries commuting with their translates,

*on*the quantization

*of*(arbitrary) reversible classical cellular automata,

*on*... Such systems have been proposed for purposes

*of*

*quantum*computation,

*in*particular for search problems

*on*

*graphs*(see [35] for a review). ...

##
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Quantum Statistical Mechanics of the Absolute Galois Group

2020
*
Symmetry, Integrability and Geometry: Methods and Applications
*

We present possible extensions

doi:10.3842/sigma.2020.038
fatcat:qbibph6e2nfb3f6d6nt7zk6344
*of*the*quantum*statistical mechanical formulation*of*class*field*theory to the non-abelian case, based*on*the action*of*the absolute Galois group*on*Grothendieck's dessins ...*of*the*quantum*statistical mechanical system with the Galois action*on*K ab . ... We thank the anonymous referees for several very useful comments that significantly improve the paper, and Lieven Le Bruyn for his suggestions*in*a series*of*mails that helped us to avoid ambiguities. ...##
###
Quantum algorithms for problems in number theory, algebraic geometry, and group theory
[article]

2012
*
arXiv
*
pre-print

Understanding what other computational problems can be solved significantly faster using

arXiv:1206.6126v1
fatcat:xux56wit4ncy5jtqiwqodel2iu
*quantum*algorithms is*one**of*the major challenges*in*the theory*of**quantum*computation, and such algorithms motivate ... Undoubtedly the best-known example*of*this is Shor's discovery*of*an efficient*quantum*algorithm for factoring integers, whereas the same problem appears to be intractable*on*classical computers. ...*Counting*Points*of**Finite**Field*Equations As*in*the case*of*elliptic curve over*finite**fields*, for f ∈ F q [x, y] a polynomial*in*two variables with coefficients*in*F q , the*finite*set*of*zeros*of*f make ...##
###
A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory
[article]

2020
*
arXiv
*
pre-print

This monograph aims to build some new mathematical structures originated from Dyson--Schwinger equations for the description

arXiv:1811.05333v3
fatcat:zjrioqvturgc7ojutvs5wo47ce
*of*non-perturbative aspects*of*gauge*field*theories whenever bare or running ... Now the Tutte polynomial*of*the totally*ordered**graph*G can be defined (independent*of*the chosen total*order*) by the formal expansion T (G; x, y) = i,j t ij x i y j (3.8) such that t ij*counts*spanning ... Thanks to Theorem 5.2.2,*homomorphism*densities*on*large Feynman diagrams can be computed*in*terms*of**homomorphism*densities*of**finite*partial sums. ...##
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Teichmüller Space (Classical and Quantum)

2006
*
Oberwolfach Reports
*

This is a short report

doi:10.4171/owr/2006/26
fatcat:iccoztwgyrgfnfiruryd25newu
*on*the conference "Teichmüller Space (Classical and*Quantum*) " held*in*... Non-trivial homogeneous quasi-*homomorphisms*are never bounded, and they vanish*on*elements*of**finite**order*. ... geometry, dynamical systems, topological*quantum**field*theory, string theory, and many others. ...
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