Jordan-Wigner formalism for arbitrary 2-input 2-output matchgates and their classical simulation

2015 ◽  
Vol 15 (7&8) ◽  
pp. 541-556
Author(s):  
Richard Jozsa ◽  
Akimasa Miyake ◽  
Sergii Strelchuk
Keyword(s):  
Lie Algebra ◽  
First Line ◽  
Linear Combinations ◽  
Efficient Simulation ◽  
Classical Simulation ◽  
Topological Closure ◽  
Algebra Theory ◽  
Simulation Results

In Valiant's matchgate theory, 2-input 2-output matchgates are $4\times 4$ matrices that satisfy ten so-called matchgate identities. We prove that the set of all such matchgates (including non-unitary and non-invertible ones) coincides with the topological closure of the set of all matrices obtained as exponentials of linear combinations of the 2-qubit Jordan-Wigner (JW) operators and their quadratic products, extending a previous result of Knill. In Valiant's theory, outputs of matchgate circuits can be classically computed in poly-time. Via the JW formalism, Terhal \& DiVincenzo and Knill established a relation of a unitary class of these circuits to the efficient simulation of non-interacting fermions. We describe how the JW formalism may be used to give an efficient simulation for all cases in Valiant's simulation theorem, which in particular includes the case of non-interacting fermions generalised to allow arbitrary 1-qubit gates on the first line at any stage in the circuit. Finally we give an exposition of how these simulation results can be alternatively understood from some basic Lie algebra theory, in terms of a formalism introduced by Somma et al.

scholarly journals Combinatorial bases of modules for affine Lie algebra B 2(1)

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Mirko Primc
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Intertwining Operators ◽  
Type B ◽  
Highest Weight Module ◽  
Highest Weight ◽  
Affine Lie Algebra ◽  
Algebra Theory ◽  
Simple Currents

AbstractWe construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases and the same presentation P/I.

scholarly journals A note on the zeroth products of Frenkel–Jing operators

2017 ◽  
Vol 16 (03) ◽  
pp. 1750053 ◽  
Author(s):  
Slaven Kožić
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Quantum Vertex ◽  
Algebra Theory

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

An Efficient Simulation Method for Distributed Lumped Fluid Networks

1977 ◽  
Vol 99 (1) ◽  
pp. 34-40 ◽  
Author(s):  
R. S. Sidell ◽  
D. N. Wormley
Keyword(s):  
Finite Difference Methods ◽  
Computation Time ◽  
Simulation Method ◽  
Transmission Network ◽  
Efficient Simulation ◽  
Fluid Networks ◽  
Lumped Elements ◽  
Simulation Results ◽  
Core Storage ◽  
Nonlinear Resistance

A method is presented for the simulation of fluid networks consisting of uniform distributed elements and lumped dynamic, nonlinear elements. The uniform transmission elements may be lossy and dispersive. General relationships are derived for their terminations in junctions with other elements and/or with dynamic, nonlinear lumped elements. The basic computer simulation method is efficient in terms of computation time and core storage requirements in comparison to direct finite difference methods and may be implemented on a minicomputer. Simulation results are compared with experimental data for a pneumatic transmission line terminated with a nonlinear resistance and for a pneumatic transmission network consisting of three lines of incommensurate lengths.

scholarly journals Lie Algebras with BCL Algebras

2018 ◽  
Vol 11 (2) ◽  
pp. 444-448
Author(s):  
Yonghong Liu
Keyword(s):  
Abelian Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Subject Matter ◽  
The Subject ◽  
Algebra Theory

The subject matter of this work is hoping for a new relationship between the Lie algebras and the algebra of logic, which will constitute an important part of our study of "pure'' algebra theory. $BCL$ algebras as a class of logical algebras is can be generated by a Lie algebra. The opposite is also true that when special conditions occur. The aim of this paper is to prove several theorems on Lie algebras with $BCL$ algebras. I introduce the notion of a "pseudo-association'' which I propose as the adjoint notion of $BCL$ algebra in the abelian group.

scholarly journals Marginal Posterior Sampling for Slate Bandits

2019 ◽  
Author(s):  
Maria Dimakopoulou ◽  
Nikos Vlassis ◽  
Tony Jebara
Keyword(s):  
Web Services ◽  
Monotone Function ◽  
Posterior Distributions ◽  
Computationally Efficient ◽  
Thompson Sampling ◽  
Efficient Simulation ◽  
Posterior Sampling ◽  
Simulation Results

We introduce a new Thompson sampling-based algorithm, called marginal posterior sampling, for online slate bandits, that is characterized by three key ideas. First, it postulates that the slate-level reward is a monotone function of the marginal unobserved rewards of the base actions selected in the slates's slots, but it does not attempt to estimate this function. Second, instead of maintaining a slate-level reward posterior, the algorithm maintains posterior distributions for the marginal reward of each slot's base actions and uses the samples from these marginal posteriors to select the next slate. Third, marginal posterior sampling optimizes at the slot-level rather than the slate-level, which makes the approach computationally efficient. Simulation results establish substantial advantages of marginal posterior sampling over alternative Thompson sampling-based approaches that are widely used in the domain of web services.

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