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

J. A. Gregor Lagodzinski, Andreas Göbel, Katrin Casel, Tobias Friedrich, Nikhil Bansal, Emanuela Merelli, James Worrell
2021
We study the problem 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  ... 
doi:10.4230/lipics.icalp.2021.91 fatcat:hqmttj7zuzapxnkhvrkxgmziki

On the refined counting of graphs on surfaces

Robert de Mello Koch, Sanjaye Ramgoolam, Congkao Wen
2013 Nuclear Physics B  
We review and extend relevant mathematical literature and present results 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.  ... 
doi:10.1016/j.nuclphysb.2013.01.023 fatcat:uhsvpgozx5cbvk6ascaddc4vty

New mathematical structures in renormalizable quantum field theories

Dirk Kreimer
2003 Annals of Physics  
Computations 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.  ... 
doi:10.1016/s0003-4916(02)00023-4 fatcat:gic66hpzwfclzioicvdryrbwgu

Shorter quantum circuits [article]

Vadym Kliuchnikov, Kristin Lauter, Romy Minko, Adam Paetznick, Christophe Petit
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] .  ... 
arXiv:2203.10064v1 fatcat:awisa2gi2jcqzfnymvlixrr37q

Quantum Algorithms [article]

Michele Mosca
2008 arXiv   pre-print
This article surveys 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  ...  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  ... 
arXiv:0808.0369v1 fatcat:gsiyvpw7mnd2hlmki5tvgjwgvu

Efficient quantum processing of ideals in finite rings [article]

Pawel M. Wocjan, Stephen P. Jordan, Hamed Ahmadi, Joseph P. Brennan
2009 arXiv   pre-print
Suppose we are given black-box access to a 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.  ... 
arXiv:0908.0022v1 fatcat:paabwibbvjb5dko5hkoondyr5u

Hall algebras and quantum Frobenius

Kevin Mcgerty
2010 Duke mathematical journal  
Lusztig has constructed a Frobenius morphism for 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  ... 
doi:10.1215/00127094-2010-036 fatcat:cxm3nibg3jgurihy6pzebi3u7y

Quantum field theory over

Dori Bejleri, Matilde Marcolli
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  ... 
doi:10.1016/j.geomphys.2013.03.002 fatcat:rqueqxjgendt3jmin4dddfepae

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

Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo
2022 arXiv   pre-print
The attention is focused 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).  ... 
arXiv:2205.00170v1 fatcat:4e2h4zs4bjhf7pwh6gths43foy

Quantum Statistical Mechanics of the Absolute Galois Group [article]

Yuri I. Manin, Matilde Marcolli
2020 arXiv   pre-print
We present possible extensions 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.  ... 
arXiv:1907.13545v2 fatcat:h3c5feeejzhfhj73lmfzy6uy24

Reversible quantum cellular automata [article]

B. Schumacher, R.F. Werner
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).  ... 
arXiv:quant-ph/0405174v1 fatcat:df3dx2zuezd65glrd7saosk2vy

Quantum Statistical Mechanics of the Absolute Galois Group

Yuri I. Manin, Max Planck Institute for Mathematics, Germany, Matilde Marcolli, California Institute of Technology, USA
2020 Symmetry, Integrability and Geometry: Methods and Applications  
We present possible extensions 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.  ... 
doi:10.3842/sigma.2020.038 fatcat:qbibph6e2nfb3f6d6nt7zk6344

Quantum algorithms for problems in number theory, algebraic geometry, and group theory [article]

Wim van Dam, Yoshitaka Sasaki
2012 arXiv   pre-print
Understanding what other computational problems can be solved significantly faster using 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  ... 
arXiv:1206.6126v1 fatcat:xux56wit4ncy5jtqiwqodel2iu

A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory [article]

Ali Shojaei-Fard
2020 arXiv   pre-print
This monograph aims to build some new mathematical structures originated from Dyson--Schwinger equations for the description 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.  ... 
arXiv:1811.05333v3 fatcat:zjrioqvturgc7ojutvs5wo47ce

Teichmüller Space (Classical and Quantum)

Shigeyuki Morita, Athanase Papadopoulos, Robert Penner
2006 Oberwolfach Reports  
This is a short report 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.  ... 
doi:10.4171/owr/2006/26 fatcat:iccoztwgyrgfnfiruryd25newu
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