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

Jin-Yi Cai, Artem Govorov
2020 arXiv   pre-print
It is shown that 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.  ... 
arXiv:1909.03179v2 fatcat:j6by6lyb6bbbvgeykkewsgm3xq

From the Ising and Potts models to the general graph homomorphism polynomial [article]

Klas Markström
2015 arXiv   pre-print
In this note we study some 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.  ... 
arXiv:1401.6335v2 fatcat:3mzln32dwvcljmhwvq6gyvjb4q

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

Jin-Yi Cai, Xi Chen, Pinyan Lu
2011 arXiv   pre-print
We study 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.  ... 
arXiv:0903.4728v2 fatcat:6ssjnabttneblhx2xwakhlyhla

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  ...  Since the running time of simulating each exponential term is assumed to be polynomial in n, the overall running time is polynomial in n.  ... 
arXiv:0808.0369v1 fatcat:gsiyvpw7mnd2hlmki5tvgjwgvu

The Quantum Frontier [article]

Joseph F. Fitzsimons, Eleanor G. Rieffel, Valerio Scarani
2013 arXiv   pre-print
We start by describing aspects 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  ... 
arXiv:1206.0785v2 fatcat:mkd7qulj55fvteq5pb5dty6rbm

Quantum Complexity: restrictions on algorithms and architectures [article]

Daniel James Shepherd
2010 arXiv   pre-print
A dissertation submitted to 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' ?  ... 
arXiv:1005.1425v1 fatcat:iusngjiypfgxzlezpnz5yiyarq

Entropy estimates for simplicial quantum gravity

M. Carfora, A. Marzuoli
1995 Journal of Geometry and Physics  
They have an 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.  ... 
doi:10.1016/0393-0440(94)00022-v fatcat:mssxaamr4ve7pk3o35csulo7ki

Two paradigms for topological quantum computation [article]

Eric C. Rowell
2008 arXiv   pre-print
In particular we suggest correspondences between 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.  ... 
arXiv:0803.1258v1 fatcat:xajxm6kyfvhbrgwkuhwvtubsvi

Quantum Cryptanalysis (Dagstuhl Seminar 19421)

Michele Mosca, Maria Naya-Plasencia, Rainer Steinwandt, Michael Wagner
2020 Dagstuhl Reports  
Identifying new cryptanalytic improvements that make use 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.  ... 
doi:10.4230/dagrep.9.10.47 dblp:journals/dagstuhl-reports/MoscaNS19 fatcat:b4fbhk267zhsdch2lotalrzwoa

Counting Graph Homomorphisms [chapter]

Christian Borgs, Jennifer Chayes, László Lovász, Vera T. Sós, Katalin Vesztergombi
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").  ... 
doi:10.1007/3-540-33700-8_18 fatcat:2uf4y2p66ndqrcedsiu52scgja

Non-abelian Quantum Statistics on Graphs

Tomasz Maciążek, Adam Sawicki
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  ... 
doi:10.1007/s00220-019-03583-5 fatcat:makcyc3hevesrpiu6jwzpykujy

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

Scott Aaronson
2016 arXiv   pre-print
Those topics include 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!  ... 
arXiv:1607.05256v1 fatcat:mnpmspgwlrdk5pm3fcsthl3lui

Normal subgroup reconstruction and quantum computation using group representations

Sean Hallgren, Alexander Russell, Amnon Ta-Shma
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.  ... 
doi:10.1145/335305.335392 dblp:conf/stoc/HallgrenRT00 fatcat:kxwbcoyltzhn7byzki4f2kkzym

Topspin networks in loop quantum gravity

Christopher L Duston
2012 Classical and quantum gravity  
This extends 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.  ... 
doi:10.1088/0264-9381/29/20/205015 fatcat:wlhywxusfjd53o7xnwtks2mzwa

Adiabatic Quantum Computation [article]

Friederike Anna Dziemba
2016 arXiv   pre-print
Yet 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.  ... 
arXiv:1610.04708v1 fatcat:n2dasjyqe5afrmk5omvolksfje
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