What is the difference between exponential and quadratic functions?

Many people confuse quadratic and exponential functions because they both have an exponent in their equations. The key difference between quadratic and exponential functions is where the variable is located in the equation . If the variable is in the exponent, then the function is exponential.

What are the characteristics of exponential functions and how do the representations of exponential functions compare to linear and quadratic functions?

Linear function - has the form y = mx + b where the rate of change is constant m. Graph is a straight line. Exponential function - has the form y = a^x, where the rate of change is NOT constant and is different for different values of x. Graph is an exponential curve (not a straight line).

Is gravity exponential or quadratic?

Quadratic functions are used to model gravity problems where we want to find how long it takes an object to fall back to the ground after it's been thrown into the air, dropped from a certain height, or thrown down from a certain height.

Why do exponential functions overtake quadratic functions?

We also saw an explanation for why exponential growth eventually overtakes quadratic growth. When increases by 1, the exponential expression always increases by a factor of 2 . The quadratic expression increases by different factors, depending on , but these factors get smaller.

What is the difference between quadratic function and quadratic?

Quadratic equation is a mathematical statement which has equal sign and has value of zero, meanwhile the value of quadratic function can be zero and non-zero . c. Quadratic equation has equal sign, whereas quadratic function does not have.

How do you know if a function is exponential?

How can you tell if a function is an exponential function? If your function can be written in the form y = a b x , where and are constants, a ≠ 0 , b > 0 , and b ≠ 1 , then it must be exponential.

What is the difference between quadratic time and exponential time?

O(n²) — Quadratic Time: Given an input of size n, the number of steps it takes to accomplish a task is square of n. O(C^n) — Exponential Time: Given an input of size n, the number of steps it takes to accomplish a task is a constant to the n power (pretty large number).

How do you identify each function as linear quadratic or exponential?

Linear functions take the form y=mx+b. Quadratic functions take the form y=ax2+bx+c. Exponential functions take the form y=a⋅bx .

Quadratic-exponential functionals of Gaussian quantum processes (2024)

Abstract

This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen-Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.

Original language	English
Article number	2150024
Journal	Infinite Dimensional Analysis, Quantum Probability and Related Topics
Volume	24
Issue number	4
DOIs	https://doi.org/10.1142/S0219025721500247
Publication status	Published - 1 Dec 2021

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Vladimirov, I. G., Petersen, I. R., & James, M. R. (2021). Quadratic-exponential functionals of Gaussian quantum processes. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 24(4), Article 2150024. https://doi.org/10.1142/S0219025721500247

Vladimirov, Igor G. ; Petersen, Ian R. ; James, Matthew R. / Quadratic-exponential functionals of Gaussian quantum processes. In: Infinite Dimensional Analysis, Quantum Probability and Related Topics. 2021 ; Vol. 24, No. 4.

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Vladimirov, IG, Petersen, IR & James, MR 2021, 'Quadratic-exponential functionals of Gaussian quantum processes', Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 24, no. 4, 2150024. https://doi.org/10.1142/S0219025721500247

Quadratic-exponential functionals of Gaussian quantum processes. / Vladimirov, Igor G.; Petersen, Ian R.; James, Matthew R.
In: Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 24, No. 4, 2150024, 01.12.2021.

Research output: Contribution to journal › Article › peer-review

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AU - Petersen, Ian R.

AU - James, Matthew R.

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N2 - This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen-Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.

AB - This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen-Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.

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Vladimirov IG, Petersen IR, James MR. Quadratic-exponential functionals of Gaussian quantum processes. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 2021 Dec 1;24(4):2150024. doi: 10.1142/S0219025721500247

Quadratic-exponential functionals of Gaussian quantum processes (2024)

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