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Rational Hadamard products via Quantum Diagonal Operators [article]

Gérard Henry Edmond Duchamp, Karol A. Penson
2008 arXiv   pre-print
fonctions) of the Hadamard product by a rational fraction.  ...  We use the remark that, through Bargmann-Fock representation, diagonal operators of the Heisenberg-Weyl algebra are scalars for the Hadamard product to give some properties (like the stability of periodic  ...  In order to compute the Hadamard products of the elements of the basis (5) , one remarks that the diagonal operators are scalars for the Hadamard product.  ... 
arXiv:0810.3641v1 fatcat:767m56lk35b7bnrp3nq2exci7y

Generating functions for effective Hamiltonians via the symmetrized Hadamard product

Boris Zhilinskii
2008 Journal of Physics A: Mathematical and Theoretical  
The method is based on the application of the Molien generating function technique and the Hadamard product of rational functions.  ...  The latter leads to the representation of the answer in a form of a rational function.  ...  n z n . (2) The following fact is known: the Hadamard product of two rational functions is a rational function [8, 11, 15, 25] .  ... 
doi:10.1088/1751-8113/41/38/382004 fatcat:qxwfewqw4nbolizzggdaiqkcr4

Quantum approximate optimization is computationally universal [article]

Seth Lloyd
2018 arXiv   pre-print
The quantum approximate optimization algorithm (QAOA) applies two Hamiltonians to a quantum system in alternation.  ...  This paper shows that the same alternating procedure can be used to perform universal quantum computation: the times for which the Hamiltonians are applied can be programmed to give a computationally universal  ...  He −it 2 H Z He −it 1 H Z , (9) where H = H 1 ⊗ . . . ⊗ H j . . . is the tensor product of single qubit Hadamard operations, is also computationally universal.  ... 
arXiv:1812.11075v1 fatcat:ogbcoqbnpzgb7npsjl74vfn7j4

What is random about a quantum random walk? [article]

Arul Lakshminarayan
2003 arXiv   pre-print
We use simple deterministic dynamical systems as coins in studying quantum walks.  ...  Introducing multi-Harper maps as deterministic models of random walks we study the effect of coin chaos on the quantum walk.  ...  It has solely been financed by the tax payers of India via the agency of the Department of Space, Govt. of India.  ... 
arXiv:quant-ph/0305026v1 fatcat:m4kaf6ubtjbudastsplmcuoxau

Quantum random walks in one dimension via generating functions

Andrew Bressler, Robin Pemantle
2007 Discrete Mathematics & Theoretical Computer Science  
International audience We analyze nearest neighbor one-dimensional quantum random walks with arbitrary unitary coin-flip matrices.  ...  This unitary composition of operators represents one step of the Hadamard walk.  ...  Ambainis et al. focus on the Hadamard walk. This is based on the Hadamard transformation, a unitary operator on l 2 (Σ) whose matrix with respect to the standard basis is U √ 1 2 := 1 √ 2 1 1 1 −1 .  ... 
doi:10.46298/dmtcs.3533 fatcat:72ex56vaynfxpopsj6ens6tbkq

Bounds on the number of time steps for simulating arbitrary interaction graphs [article]

Dominik Janzing, Pawel Wocjan, Thomas Beth
2002 arXiv   pre-print
To express the effect of the control operations on the coupling matrix J note that any unitary operation u ∈ SU (d) corresponds to a rotation on the m-dimensional sphere via the relation u † α c α σ α  ...  Here a quantum computer can be any quantum system provided that its time evolution can be controlled in a universal way.  ... 
arXiv:quant-ph/0203061v2 fatcat:44iubkv2pngylgpbzpoexzbiri

Quantum information processing by weaving quantum Talbot carpets

Osvaldo Jiménez Farías, Fernando de Melo, Pérola Milman, Stephen P. Walborn
2015 Physical Review A. Atomic, Molecular, and Optical Physics  
Interpretation of the Talbot effect as a quantum logic operation provides a beautiful and interesting way to visualize quantum computation through wave propagation and interference.  ...  Adding a diagonal phase gate, we show that a complete set of single quDit gates can be implemented.  ...  Diagonal gates and the quantum Fourier transform While free space propagation naturally implements a whole family of single quDit transformations via the Talbot effect, some commonly used linear optics  ... 
doi:10.1103/physreva.91.062328 fatcat:xgizpx3my5bxfmwuzogfthqzvq

Introduction to topological quantum computation with non-Abelian anyons

Bernard Field, Tapio Simula
2018 Quantum Science and Technology  
of braids to perform specific operations.  ...  Topological quantum computers promise a fault tolerant means to perform quantum computation.  ...  Acknowledgments 45 ACKNOWLEDGMENTS We acknowledge support from the Australian Research Council (ARC) via Discovery Project No. DP170104180.  ... 
doi:10.1088/2058-9565/aacad2 fatcat:53byvb5wjfhpfjukxxcwr5yvoa


Ebrahim Karimi, Farshad Nejadsattari, Yingwen Zhang, Frederic Bouchard, Hugo Larocque, Alicia Sit, Eliahu Cohen, Robert Fickler
2019 Figshare  
Hadamard coin.  ...  With the Hadamard coin operator, one can arrive at a matrix representation of the single-step time evolution operator in -space.  ...  Considering a Hadamard gate, one can write: The initial state of the walker is prepared as: | (0) = |0 ⊗ | .  ... 
doi:10.6084/m9.figshare.7588745.v1 fatcat:mkwoqyl665akzpl55t7btu7yy4

Perfect quantum state transfer using Hadamard diagonalizable graphs

Nathaniel Johnston, Steve Kirkland, Sarah Plosker, Rebecca Storey, Xiaohong Zhang
2017 Linear Algebra and its Applications  
Here, we focus on the corresponding Laplacian matrix, and those graphs for which the Laplacian can be diagonalized by a Hadamard matrix.  ...  Quantum state transfer within a quantum computer can be achieved by using a network of qubits, and such a network can be modelled mathematically by a graph.  ...  INTRODUCTION Accurate transmission of quantum states between processors and/or registers of a quantum computer is critical for short distance communication in a physical quantum computing scheme.  ... 
doi:10.1016/j.laa.2017.05.037 fatcat:ystcnwhxbzcxjn6sfls4gcddgm

Quantum circuits ofT-depth one

Peter Selinger
2013 Physical Review A. Atomic, Molecular, and Optical Physics  
This research was also supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior National Business Center contract number D11PC20168. The U.S.  ...  Let us call an operator almost classical if it can be written as a product of a classical operator and a diagonal operator.  ...  Then E |0 E |+ is a rational number. Proof.  ... 
doi:10.1103/physreva.87.042302 fatcat:hzhw6fxpsbc3pfcmmcozbwhxcq

Self-Testing of Universal and Fault-Tolerant Sets of Quantum Gates

Wim van Dam, Frédéric Magniez, Michele Mosca, Miklos Santha
2007 SIAM journal on computing (Print)  
The first universal quantum gate which operates on three qubits was identified by Deutsch [Deu89] .  ...  The intermediate quantum states of this process cannot be used by the program, which cannot perform any other quantum operations either.  ...  A quantum circuit operates on n quantum bits (qubits), where n is some integer.  ... 
doi:10.1137/s0097539702404377 fatcat:taebrrlf7fhejcagg267hqt7de

Towards topological quantum computer

D. Melnikov, A. Mironov, S. Mironov, A. Morozov, An. Morozov
2018 Nuclear Physics B  
One of the principal obstacles on the way to quantum computers is the lack of distinguished basis in the space of unitary evolutions and thus the lack of the commonly accepted set of basic operations (  ...  , integrable systems and modern theory of non-perturbative calculations in quantum field and string theory.  ...  This evolution matrix is diagonal, since we constructed it in the basis of irreps in the product of two spins.  ... 
doi:10.1016/j.nuclphysb.2017.11.016 fatcat:hhnisfmibbafrppx6gnea7pdyq

Quantum Automata and Quantum Grammars [article]

Cristopher Moore, James P. Crutchfield
1997 arXiv   pre-print
We find analogs of several classical theorems, including pumping lemmas, closure properties, rational and algebraic generating functions, and Greibach normal form.  ...  To study quantum computation, it might be helpful to generalize structures from language and automata theory to the quantum case.  ...  We hope that quantum grammars and automata will be fruitful areas of research and that they will be useful to people studying quantum computation.  ... 
arXiv:quant-ph/9707031v1 fatcat:xi7iemcd3vcolgaocn5py62qnq

Quantum automata and quantum grammars

Cristopher Moore, James P. Crutchfield
2000 Theoretical Computer Science  
We ÿnd analogs of several classical theorems, including pumping lemmas, closure properties, rational and algebraic generating functions, and Greibach normal form.  ...  To study quantum computation, it might be helpful to generalize structures from language and automata theory to the quantum case.  ...  Acknowledgements We are grateful to Bruce Litow, Philippe Flajolet, and Christophe Reutenauer for the proof that the Hadamard square of an algebraic series can be transcendental; Bruce Reznick, Jan-Christoph  ... 
doi:10.1016/s0304-3975(98)00191-1 fatcat:amjxp3twvzfmre4vtixfnxccna
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