Function theory of antilinear operators

Huhtanen, Marko; Perämäki, Allan

doi:10.48550/arxiv.1212.0360

Cited by 3 publications

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“…Quaternions. In this paper, we studied the relationship between computing using real and com- 21 Coined by John Smolin [45].…”

Section: Discussion and Open Questionsmentioning

confidence: 99%

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Quantum simulation from the bottom up: the case of rebits

Koh,

Niu,

Yoder

2017

Preprint

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Typically, quantum mechanics is thought of as a linear theory with unitary evolution governed by the Schrödinger equation. While this is technically true and useful for a physicist, with regards to computation it is an unfortunately narrow point of view. Just as a classical computer can simulate highly nonlinear functions of classical states, so too can the more general quantum computer simulate nonlinear evolutions of quantum states. We detail one particular simulation of nonlinearity on a quantum computer, showing how the entire class of R-unitary evolutions (on n qubits) can be simulated using a unitary, real-amplitude quantum computer (consisting of n + 1 qubits in total). These operators can be represented as the sum of a linear and antilinear operator, and add an intriguing new set of nonlinear quantum gates to the toolbox of the quantum algorithm designer. Furthermore, a subgroup of these nonlinear evolutions, called the R-Cliffords, can be efficiently classically simulated, by making use of the fact that Clifford operators can simulate non-Clifford (in fact, non-linear) operators. This perspective of using the physical operators that we have to simulate non-physical ones that we do not is what we call bottom-up simulation, and we give some examples of its broader implications.

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“…Quaternions. In this paper, we studied the relationship between computing using real and com- 21 Coined by John Smolin [45].…”

Section: Discussion and Open Questionsmentioning

confidence: 99%

“…See, for example,[20][21][22]. Note that our definition of R-linearity coincides with their definition of real linearity.…”

mentioning

confidence: 91%

Quantum simulation from the bottom up: the case of rebits

Koh,

Niu,

Yoder

2017

Preprint

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“…The intersection of two ϑ-normal subspaces is a ϑ-normal subspace. If one of the two subspaces is minimal, then their intersection is either the minimal ϑ-normal subspace itself or it consists of the zero vector only, see also [43]. Hence: Proposition 4.6 Any ϑ-normal subspace can be decomposed into mutually orthogonal minimal ϑ-normal subspaces.…”

Section: Decompositions Of Normal Antilinear Operatorsmentioning

confidence: 98%

“…Hence their direct sum is a 2-graded algebra. An active domain of research are operator functions, for instance power series of elements of this algebra, see M. Huhtanen and A. Peramaki in [43,44] for example. Please notice: This rich theory is not under consideration here.…”

Section: Introductionmentioning

confidence: 99%

Anti- (conjugate) linearity

Uhlmann

2016

Sci. China Phys. Mech. Astron.

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This is an introduction to antilinear operators. In following E. P. Wigner the terminus "antilinear" is used as it is standard in Physics. Mathematicians prefer to say "conjugate linear".By restricting to finite-dimensional complex-linear spaces, the exposition becomes elementary in the functional analytic sense. Nevertheless it shows the amazing differences to the linear case.Basics of antilinearity is explained in sections 2, 3, 4, 7 and in subsection 1.2: Spectrum, canonical Hermitian form, antilinear rank one and two operators, the Hermitian adjoint, classification of antilinear normal operators, (skew) conjugations, involutions, and acq-lines, the antilinear counterparts of 1-parameter operator groups. Applications include the representation of the Lagrangian Grassmannian by conjugations, its covering by acq-lines. as well as results on equivalence relations. After remembering elementary Tomita-Takesaki theory, antilinear maps, associated to a vector of a two-partite quantum system, are defined. By allowing to write modular objects as twisted products of pairs of them, they open some new ways to express EPR and teleportation tasks. The appendix presents a look onto the rich structure of antilinear operator spaces. 1 2

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“…in contradiction with theorem 40 and in particular equation (66). 21 Gate set here refers to a set of gates, which may be either finite or infinite.…”

Section: Simulating Partial Antiunitary Operatorsmentioning

confidence: 99%

Quantum simulation from the bottom up: the case of rebits

Koh

Niu

Yoder

2018

J. Phys. A: Math. Theor.

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Typically, quantum mechanics is thought of as a linear theory with unitary evolution governed by the Schrödinger equation. While this is technically true and useful for a physicist, with regards to computation it is an unfortunately narrow point of view. Just as a classical computer can simulate highly nonlinear functions of classical states, so too can the more general quantum computer simulate nonlinear evolutions of quantum states. We detail one particular simulation of nonlinearity on a quantum computer, showing how the entire class of -unitary evolutions (on n qubits) can be simulated using a unitary, real-amplitude quantum computer (consisting of n + 1 qubits in total). These operators can be represented as the sum of a linear and antilinear operator, and add an intriguing new set of nonlinear quantum gates to the toolbox of the quantum algorithm designer. Furthermore, a subgroup of these nonlinear evolutions, called the -Cliffords, can be efficiently classically simulated, by making use of the fact that Clifford operators can simulate non-Clifford (in fact, non-linear) operators. This perspective of using the physical operators that we have to simulate non-physical ones that we do not is what we call bottom-up simulation, and we give some examples of its broader implications.

show abstract

Function theory of antilinear operators

Cited by 3 publications

References 0 publications

Quantum simulation from the bottom up: the case of rebits

Quantum simulation from the bottom up: the case of rebits

Anti- (conjugate) linearity

Quantum simulation from the bottom up: the case of rebits

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