Semiclassical interpretation of Wei–Norman factorization for <i>SU</i>(1, 1) and its related integral transforms

Guerrero, Julio Becerra; Berrondo, M.

doi:10.1063/1.5143586

“…To solve the PDE (7) we shall use the Wei-Norman factorization method [33,34,17], which allows solving this kind of PDEs even in the case of time-dependent coefficients L S t . The price to be paid by such a method is that it can be only applied when L S t is a linear combination of the generators of a Lie algebra of constant differential operators, and this will impose severe restrictions on the functions appearing in L S t .…”

Section: Slv Model Local Volatility (Lv) Models First Proposed Bymentioning

confidence: 99%

Stochastic local volatility models and the Wei-Norman factorization method

Guerrero¹,

Orlando²

2022

DCDS-S

Self Cite

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<p style='text-indent:20px;'>In this paper, we show that a time-dependent local stochastic volatility (SLV) model can be reduced to a system of autonomous PDEs that can be solved using the heat kernel, by means of the Wei-Norman factorization method and Lie algebraic techniques. Then, we compare the results of traditional Monte Carlo simulations with the explicit solutions obtained by said techniques. This approach is new in the literature and, in addition to reducing a non-autonomous problem into an autonomous one, allows for shorter time in numerical computations.</p>

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“…Inverting this equation requires taking a smooth analytic continuation across any singular point of the inverse of the matrix, see, for example ref. [42] .…”

Section: An Explicit Lie-algebraic Solution For a Two Excited Levels ...mentioning

confidence: 99%

A Quantum Information Processing Machine for Computing by Observables

Remacle

¹

,

Levine

²

2022

Preprint

0

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A quantum machine that accepts an input and processes it in parallel is described. The logic variables of the machine are not wavefunctions (qubits) but observables (i.e., operators) and its operation is decribed in the Heisenberg picture. The active core is a solid-state assembly of small nanosized colloidal quantum dots(QDs) or dimers of dots. The size dispersion of the nanodots that causes fluctuations in their discrete electronic energies is a limiting factor. The input to the machine is provided by a train of very short in time laser pulses, at least four in number. The coherent band width of each ultrashort pulse needs to span at least several and preferably all the single electron excited states of the dots. The spectrum of the QD assembly is measured as a function of the time delays between the input laser pulses. The dependence of the spectrum on the time delays is Fourier transformed to a frequency spectrum. This spectrum of a finite range in time is made up of discrete pixels. These are the visible, raw, basic logic variables. The spectrum is analyzed to determine a possibly smaller number of principal components. A Lie-algebraic point of view is used to explore the use of the machine to emulate the dynamics of other quantum systems. An explicit example demonstrates the considerable quantum advantage of our scheme.

show abstract

“…Inverting this equation requires taking a smooth analytic continuation across any singular point of the inverse of the matrix, see, for example ref. (45) .…”

Section: An Explicit Lie-algebraic Solution For a Two Excited Levels ...mentioning

confidence: 99%

A Quantum Information Processing Machine for Computing by Observables

Remacle

¹

,

Levine

²

2022

Preprint

0

View full text Add to dashboard Cite

A quantum machine that accepts an input and processes it in parallel is described. The logic variables of the machine are not wavefunctions (qubits) but observables (i.e., operators) and its operation is decribed in the Heisenberg picture. The active core is a solid-state assembly of small nanosized colloidal quantum dots(QDs) or dimers of dots. The size dispersion of the nanodots that causes fluctuations in their discrete electronic energies is a limiting factor. The input to the machine is provided by a train of very short in time laser pulses, at least four in number. The coherent band width of each ultrashort pulse needs to span at least several and preferably all the single electron excited states of the dots. The spectrum of the QD assembly is measured as a function of the time delays between the input laser pulses. The dependence of the spectrum on the time delays is Fourier transformed to a frequency spectrum. This spectrum of a finite range in time is made up of discrete pixels. These are the visible, raw, basic logic variables. The spectrum is analyzed to determine a possibly smaller number of principal components. A Lie-algebraic point of view is used to explore the use of the machine to emulate the dynamics of other quantum systems. An explicit example demonstrates the considerable quantum advantage of our scheme.

show abstract

Semiclassical interpretation of Wei–Norman factorization for SU(1, 1) and its related integral transforms

Cited by 5 publications

References 34 publications

Stochastic local volatility models and the Wei-Norman factorization method

Stochastic local volatility models and the Wei-Norman factorization method

A Quantum Information Processing Machine for Computing by Observables

A Quantum Information Processing Machine for Computing by Observables

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