Nuclear Computing Part 3

  

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1. Quantum Many-Body Hamiltonian with Nuclear Shell Model: $�={\sum }_{�=1}^{�}(-\frac{{\mathrm{\hslash }}^2}{2{�}_{�}}{\mathrm{\nabla }}_{�}^2+{�}_{\text{ext}}({\mathbf{�}}_{�}))+{\sum }_{�,�=1}^{�}{�}_{\text{int}}({\mathbf{�}}_{�}-{\mathbf{�}}_{�})+{\sum }_{�=1}^{�}{\sum }_{\text{shells}}{�}_{\text{shell}}({�}_{�},{�}_{�},{�}_{�})$ Many-body Hamiltonian for a nuclear system, combining kinetic energy, external potential ${�}_{\text{ext}}$, nucleon-nucleon interaction potential ${�}_{\text{int}}$, and energy contributions from nuclear shell model.

2. Entropy of Black Hole Information using Bekenstein-Hawking Formula: $�=\frac{{�}_{\text{horizon}}{�}_{\text{B}}}{4\mathrm{\hslash }�}$ Entropy ($�$) associated with the information content of a black hole, where ${�}_{\text{horizon}}$ is the area of the event horizon, ${�}_{\text{B}}$ is the Boltzmann constant, $\mathrm{\hslash }$ is the reduced Planck constant, and $�$ is the gravitational constant.

3. Quantum Error Correction using Steane Code: ${\mid �⟩}_{\text{encoded}}=\frac{1}{\sqrt{8}}{(\mid 0_{�}⟩+\mid 1_{�}⟩)}^{\otimes 7}$ Quantum state encoded using the Steane code, a quantum error-correcting code involving seven physical qubits ($\mid 0_{�}⟩$ and $\mid 1_{�}⟩$).

4. Digital Image Reconstruction in Gamma-Ray Tomography: $�(�,�,�)={\sum }_{�=1}^{{�}_{\text{det}}}{�}_{�}(�,�,�)\times {�}^{-�{\int }_{{�}_{�}}�(\mathbf{�})��}$ Equation for reconstructing three-dimensional images in gamma-ray tomography, considering measured projections ${�}_{�}(�,�,�)$, attenuation coefficient $�$, and the path length ${�}_{�}$ through the object.

5. Quantum Teleportation in Quantum Information Processing: ${\mid �⟩}_{\text{teleported}}={�}_{\text{CNOT}}(�\otimes �){�}_{\text{CNOT}}(\text{CNOT}\otimes �)(\mid �⟩\otimes \mid {\mathrm{\Phi }}^{+}⟩)$ Quantum teleportation circuit, where ${�}_{\text{CNOT}}$ represents controlled-NOT gates and $�$ represents the Hadamard gate, enabling quantum state teleportation using entangled states $\mid {\mathrm{\Phi }}^{+}⟩$.

6. Digital Signal Processing in Nuclear Pulse Spectroscopy: $�(�)={\sum }_{�=1}^{{�}_{\text{peaks}}}{�}_{�}{�}^{-\frac{(�-{�}_{�}{)}^2}{2{�}^2}}$ Signal representation in nuclear pulse spectroscopy, involving the summation of Gaussian-shaped peaks (${�}_{\text{peaks}}$) with amplitudes ${�}_{�}$, positions ${�}_{�}$, and widths $�$.

1. Quantum Entanglement in Nuclear Spin States: $\mid �⟩=\frac{1}{\sqrt{2}}(\mid \uparrow \downarrow ⟩-\mid \downarrow \uparrow ⟩)$ Quantum entangled state of two nuclear spins ($\mid \uparrow ⟩$ and $\mid \downarrow ⟩$) demonstrating spin singlet configuration.

2. Astrophysical Neutrino Flux Calculation: $\mathrm{\Phi }(�)=\frac{�}{4�{�}^2}\times �(�)$ Calculation of astrophysical neutrino flux ($\mathrm{\Phi }(�)$) arriving at Earth from a distant source, where $�$ is the luminosity of the source, $�$ is the distance from the source to Earth, and $�(�)$ represents the neutrino energy spectrum.

3. Quantum Bayesian Inference in Nuclear Decay: $�(�\mathrm{\mid }�,�)\propto {�}^{�}{�}^{-��}\times �(�\mathrm{\mid }�)$ Bayesian inference for estimating the decay constant ($�$) in nuclear decay, given $�$ decays in time $�$, incorporating the Poisson likelihood and prior probability $�(�\mathrm{\mid }�)$.

4. Digital Filtering in Gamma Spectroscopy: ${�}_{\text{out}}(�)={\sum }_{�=0}^{�-1}ℎ(�)\times {�}_{\text{in}}(�-�)$ Digital filtering equation for processing gamma spectroscopy data, where ${�}_{\text{in}}(�)$ is the input signal at sample $�$, $ℎ(�)$ is the filter coefficients, and $�$ is the filter length.

5. Quantum Circuit for Grover's Algorithm in Nuclear Isomer State Search: $\text{Grover}({�}_{\text{isomer}})={�}_{\text{oracle}}\times {�}_{\text{diffusion}}$ Quantum circuit for Grover's algorithm, where ${�}_{\text{oracle}}$ marks the isomer state, and ${�}_{\text{diffusion}}$ performs amplitude amplification.

6. Nuclear Magnetic Resonance (NMR) Signal Frequency: $�=�{�}_0$ Nuclear magnetic resonance (NMR) signal frequency ($�$) in a magnetic field (${�}_0$) with gyromagnetic ratio ($�$).

7. Quantum Channel Capacity for Quantum Communication in Nuclear Systems: $�={\max }_{�}�(�)$ Quantum channel capacity ($�$) representing the maximum amount of quantum information that can be transmitted through a noisy nuclear channel, where $�(�)$ is the quantum mutual information and $�$ is the density matrix of the channel.

1. Quantum Many-Body Wave Function for Nuclear Structure: $\mathrm{\Psi }({\mathbf{�}}_1,{\mathbf{�}}_2,\dots ,{\mathbf{�}}_{�})=�\cdot \text{det}[{�}_{�}({\mathbf{�}}_{�})]$ Many-body quantum wave function ($\mathrm{\Psi }$) describing the spatial configuration of $�$ nucleons, where ${�}_{�}({\mathbf{�}}_{�})$ represents single-particle wave functions, and $�$ is the antisymmetrization operator.

2. Astrophysical Nuclear Reaction Rate: $�={�}_{�}⟨��⟩$ Nuclear reaction rate ($�$) in astrophysical environments, where ${�}_{�}$ is Avogadro's number, $�$ is the cross-section of the nuclear reaction, and $⟨�⟩$ is the average relative velocity of the colliding particles.

3. Quantum Bayesian Learning in Nuclear Data Analysis: $�(�\mathrm{\mid }�,�)\propto �(�\mathrm{\mid }�,�)\times �(�\mathrm{\mid }�)$ Bayesian posterior probability distribution for a parameter $�$ given data $�$ and prior information $�$, combining the likelihood $�(�\mathrm{\mid }�,�)$ and the prior $�(�\mathrm{\mid }�)$.

4. Digital Filtering for Pulse Shape Analysis in Detectors: ${�}_{\text{out}}(�)={\sum }_{�=0}^{�-1}ℎ(�)\times {�}_{\text{in}}(�-�)$ Digital filtering equation for processing signals from particle detectors, where ${�}_{\text{in}}(�)$ is the input signal at sample $�$, $ℎ(�)$ is the filter coefficients, and $�$ is the filter length.

5. Quantum Circuit for Quantum Error Correction in Quantum Computing: $\text{Code}={\text{CNOT}}_{12}\times {\text{CNOT}}_{23}\times {\text{CNOT}}_{34}\times {�}_1\times {�}_2\times {�}_3$ Quantum error correction code using CNOT gates (${\text{CNOT}}_{��}$) and Hadamard gates (${�}_{�}$) to protect quantum information from errors.

6. Electromagnetic Radiation Power from Accelerating Nuclear Particles: $�=\frac{2}{3}\frac{{�}^2{�}^2}{{�}^3}$ Power ($�$) radiated electromagnetically by an accelerating nuclear particle with charge $�$, acceleration $�$, and the speed of light $�$.

7. Quantum Circuit for Variational Quantum Eigensolver (VQE) in Nuclear Hamiltonian Simulation: $\mid �⟩=�(\mathit{�})\times \mid \text{trial}⟩$ Quantum circuit for VQE, where $�(\mathit{�})$ represents a parameterized unitary operator and $\mid \text{trial}⟩$ is an initial trial state, used to find the ground state energy of a nuclear Hamiltonian.

1. Quantum Density Matrix for Nuclear Spin Ensemble: $�={\sum }_{�=1}^{�}{�}_{�}\mid {�}_{�}⟩⟨{�}_{�}\mid$ Quantum density matrix ($�$) describing a nuclear spin ensemble, where $\mid {�}_{�}⟩$ represents the individual spin states and ${�}_{�}$ are the probabilities associated with each state.

2. Nuclear Fusion Cross Section in Stellar Interiors: $�(�)=�(�){�}^{-\frac{2�{�}_1{�}_2{�}^2}{\mathrm{\hslash }�}}$ Nuclear fusion cross-section ($�(�)$) in the stellar environment, where $�(�)$ is the astrophysical S-factor, ${�}_1$ and ${�}_2$ are atomic numbers of the reacting nuclei, $�$ is the elementary charge, $\mathrm{\hslash }$ is the reduced Planck constant, and $�$ is the relative velocity of the nuclei.

3. Quantum Entropy for Nuclear Energy Levels: $�=-{�}_{\text{B}}{\sum }_{�}{�}_{�}\log ({�}_{�})$ Entropy ($�$) representing the information content associated with the probabilities (${�}_{�}$) of different nuclear energy levels, where ${�}_{\text{B}}$ is the Boltzmann constant.

4. Digital Signal Processing for Nuclear Magnetic Resonance (NMR) Spectroscopy: $�(�)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}�(�){�}^{-�2���}��$ Fourier transform equation for converting the time-domain signal ($�(�)$) obtained from NMR spectroscopy into the frequency domain ($�(�)$) to obtain the NMR spectrum.

5. Quantum Error Correction Code for Nuclear Qubits: ${\mid �⟩}_{\text{encoded}}=\frac{1}{\sqrt{2}}(\mid 0_{�}⟩+\mid 1_{�}⟩)$ Encoding of a logical qubit ($\mid �⟩$) into a nuclear qubit using a simple quantum error correction code, where $\mid 0_{�}⟩$ and $\mid 1_{�}⟩$ represent the logical basis states.

6. Digital Filter Design Equation (FIR Filter): $�(�)={\sum }_{�=0}^{�-1}ℎ(�){�}^{-���}$ Frequency response ($�(�)$) of a Finite Impulse Response (FIR) filter, where $ℎ(�)$ are the filter coefficients and $�$ is the angular frequency.

7. Quantum Circuit for Quantum Phase Estimation Algorithm in Nuclear Physics: ${�}_{\text{QPE}}={�}^{\otimes �}\times {\text{QFT}}^{-1}\times {�}_{\text{Nuclear}}\times \text{QFT}\times {�}^{\otimes �}$ Quantum circuit for the Quantum Phase Estimation (QPE) algorithm, where ${�}_{\text{Nuclear}}$ represents the unitary operator encoding nuclear physics information and QFT denotes the Quantum Fourier Transform gate.

1. Quantum Monte Carlo Method for Nuclear Structure Calculation: $�=\frac{\int {\mathrm{\Psi }}^{\ast }(\mathbf{�})\widehat{�}\mathrm{\Psi }(\mathbf{�})�\mathbf{�}}{\int {\mathrm{\Psi }}^{\ast }(\mathbf{�})\mathrm{\Psi }(\mathbf{�})�\mathbf{�}}$ Energy calculation in nuclear physics using the Quantum Monte Carlo method, where $\mathrm{\Psi }(\mathbf{�})$ is the trial wave function, $\widehat{�}$ is the Hamiltonian operator, and $\mathbf{�}$ represents the nuclear coordinates.

2. Blackbody Radiation Spectrum in Astrophysics: $�(�,�)=\frac{8�{�}^2}{{�}^3}\frac{1}{{�}^{\frac{ℎ�}{{�}_{\text{B}}�}}-1}$ Blackbody radiation intensity ($�(�,�)$) as a function of frequency ($�$) and temperature ($�$), where $ℎ$ is the Planck constant, ${�}_{\text{B}}$ is the Boltzmann constant, and $�$ is the speed of light.

3. Quantum Mutual Information in Nuclear Entanglement: $�(�:�)=�(�)+�(�)-�(�\cup �)$ Quantum mutual information ($�(�:�)$) quantifying the entanglement between subsystems $�$ and $�$, where $�(�)$ and $�(�)$ are the von Neumann entropies of subsystems $�$ and $�$, and $�(�\cup �)$ is their joint entropy.

4. Digital Signal Processing for Gamma-Ray Spectroscopy (Pulse Height Analysis): $�=�\times {\sum }_{�=1}^{�}{�}_{�}$ Energy calculation in gamma-ray spectroscopy using pulse height analysis, where $�$ is the total energy, $�$ is the gain factor, $�$ is the number of detected pulses, and ${�}_{�}$ is the amplitude of the ${�}^{�ℎ}$ pulse.

5. Quantum Teleportation Equation for Nuclear Quantum States: $\mid {�}_{�}⟩={�}_{\text{CNOT}}(\text{Hadamard}\otimes �){�}_{\text{CNOT}}(�\otimes \text{CNOT})(\mid {�}_{�}⟩\otimes \mid {\mathrm{\Phi }}^{+}⟩)$ Quantum teleportation equation, where ${�}_{\text{CNOT}}$ represents controlled-NOT gates, Hadamard gate is denoted as $\text{Hadamard}$, and $\mid {\mathrm{\Phi }}^{+}⟩$ represents the Bell state.

6. Digital Filter Design Equation (IIR Filter): $�(�)=\frac{{\sum }_{�=0}^{�}{�}_{�}{�}^{-�}}{1+{\sum }_{�=1}^{�}{�}_{�}{�}^{-�}}$ Transfer function of an Infinite Impulse Response (IIR) filter, where ${�}_{�}$ and ${�}_{�}$ are the filter coefficients and ${�}^{-�}$ represents the ${�}^{�ℎ}$ delay element.

1. Quantum Chromodynamics (QCD) Lagrangian: ${\mathcal{�}}_{\text{QCD}}={\sum }_{�}{\overline{�}}_{�}(�{�}^{�}{�}_{�}-{�}_{�}){�}_{�}-\frac{1}{4}{�}_{��}^{�}{�}^{���}$ QCD Lagrangian describing the interactions of quarks (${�}_{�}$) and gluons (${�}_{��}^{�}$) with the strong force, where ${�}_{�}$ is the covariant derivative, ${�}_{�}$ is the quark mass, and $�$ is the color index.

2. Nuclear Fusion Cross Section in Plasmas: $�(�)=\frac{�(�)}{�}{�}^{-2��(�)}$ Nuclear fusion cross-section ($�(�)$) in a plasma, where $�(�)$ is the astrophysical factor, $�$ is the energy of the colliding particles, and $�(�)$ is the Sommerfeld parameter.

3. Quantum Relative Entropy in Nuclear State Analysis: $�(�\mathrm{\mid }\mathrm{\mid }�)=\text{Tr}(�(\log �-\log �))$ Quantum relative entropy ($�(�\mathrm{\mid }\mathrm{\mid }�)$) quantifying the distinguishability between quantum states $�$ and $�$, where $\text{Tr}$ represents the trace operator.

4. Digital Pulse Shape Analysis for Nuclear Detectors: $�={\sum }_{�=1}^{�}{�}_{�}\times �(�-{�}_{�})$ Energy calculation using pulse shape analysis, where ${�}_{�}$ represents the amplitude of the ${�}^{�ℎ}$ pulse, ${�}_{�}$ is the arrival time of the ${�}^{�ℎ}$ pulse, and $�(�)$ represents the detector response function.

5. Quantum Circuit for Shor's Algorithm in Factorization: ${�}_{\text{Shor}}={\text{QFT}}^{†}\times {�}_{�}\times \text{QFT}$ Quantum circuit for Shor's algorithm, where $\text{QFT}$ represents the Quantum Fourier Transform, and ${�}_{�}$ represents the function evaluating unitary operator for period finding.

6. Digital Filter Design Equation (FIR Filter with Windowing): $ℎ(�)=�(�)\times {ℎ}_{\text{desired}}(�)$ Digital filter design using Finite Impulse Response (FIR) filter coefficients $ℎ(�)$ obtained by windowing a desired impulse response ${ℎ}_{\text{desired}}(�)$ with a window function $�(�)$.

1. Quantum Electrodynamics (QED) Interaction Lagrangian: ${\mathcal{�}}_{\text{QED}}=-\frac{1}{4}{�}_{��}{�}^{��}-\overline{�}(�{�}^{�}{�}_{�}-�)�$ QED Lagrangian describing the interactions of electrons ($�$) and photons (${�}_{�}$) with the electromagnetic force, where ${�}_{��}$ is the electromagnetic field tensor, ${�}_{�}$ is the covariant derivative, and $�$ is the electron mass.

2. Friedmann Equation in Cosmology: ${�}^2=\frac{8��}{3}�-\frac{�}{{�}^2}+\frac{\mathrm{\Lambda }}{3}$ Friedmann equation describing the expansion of the universe, where $�$ is the Hubble parameter, $�$ is the energy density, $�$ represents the curvature of space, $�$ is the scale factor, $�$ is the gravitational constant, and $\mathrm{\Lambda }$ is the cosmological constant.

3. Quantum Fisher Information for Nuclear Parameter Estimation: $�(�)=4{\sum }_{�,�}\frac{{\mid ⟨{�}_{�}\mathrm{\mid }\frac{\mathrm{\partial }\widehat{�}}{\mathrm{\partial }�}\mathrm{\mid }{�}_{�}⟩\mid }^2}{{�}_{��}^2}$ Quantum Fisher information ($�(�)$) quantifying the sensitivity of a quantum state $\mid {�}_{�}⟩$ to a parameter $�$ in a Hamiltonian $\widehat{�}$, where ${�}_{��}$ is the energy difference between states $\mid {�}_{�}⟩$ and $\mid {�}_{�}⟩$.

4. Digital Pulse Processing in Nuclear Spectroscopy (Pulse-Height Analysis): $�={\sum }_{�=1}^{�}{�}_{�}$ Energy calculation in nuclear spectroscopy using pulse-height analysis, where ${�}_{�}$ represents the amplitude of the ${�}^{�ℎ}$ pulse corresponding to detected radiation.

5. Quantum Circuit for Quantum Phase Estimation in Nuclear Hamiltonian Simulation: ${�}_{\text{QPE}}={�}_{\text{Hadamard}}\times {�}_{{\text{U}}_{\text{Nuclear}}}\times {�}_{\text{QFT}}\times {�}_{{\text{CU}}_{\text{Nuclear}}}\times {�}_{\text{QFT}}^{†}\times {�}_{\text{Hadamard}}^{†}$ Quantum circuit for Quantum Phase Estimation (QPE) algorithm, where ${�}_{{\text{U}}_{\text{Nuclear}}}$ represents the unitary operator encoding nuclear physics information, ${�}_{{\text{CU}}_{\text{Nuclear}}}$ is the controlled-${�}_{\text{Nuclear}}$ operator, ${�}_{\text{QFT}}$ is the Quantum Fourier Transform gate, and ${�}_{\text{Hadamard}}$ is the Hadamard gate.

6. Digital Filter Design Equation (IIR Filter with Bilinear Transformation): $�(�)=\frac{{\sum }_{�=0}^{�}{�}_{�}{�}^{-�}}{1+{\sum }_{�=1}^{�}{�}_{�}{�}^{-�}}$ Transfer function of an Infinite Impulse Response (IIR) filter obtained using the bilinear transformation method, where ${�}_{�}$ and ${�}_{�}$ are the filter coefficients and ${�}^{-�}$ represents the ${�}^{�ℎ}$ delay element.

1. Quantum Field Theory (QFT) Lagrangian for Electroweak Interaction: ${\mathcal{�}}_{\text{EW}}=-\frac{1}{4}{�}_{��}{�}^{��}-\frac{1}{4}{�}_{��}^{�}{�}^{���}+{\overline{�}}_{�}�{�}^{�}({�}_{�}-{�}^{\mathrm{\prime }}\frac{�}{2}{�}_{�}-�\frac{{�}^{�}}{2}{�}_{�}^{�}){�}_{�}$ Lagrangian describing the electroweak interaction in the framework of QFT, where ${�}_{��}$ and ${�}_{��}^{�}$ are the field tensors, ${�}_{�}$ is the covariant derivative, $�$ and ${�}^{\mathrm{\prime }}$ are coupling constants, ${�}^{�}$ are Pauli matrices, and $�$ is the weak hypercharge.

2. Nuclear Magnetic Resonance (NMR) Signal Decay Equation: $�(�)={�}_0\cdot {�}^{-\frac{�}{{�}_2}}\cos (2�{�}_0�+�)$ NMR signal decay equation, where $�(�)$ is the signal strength at time $�$, ${�}_0$ is the initial signal strength, ${�}_2$ is the transverse relaxation time, ${�}_0$ is the Larmor frequency, and $�$ is the phase angle.

3. Quantum Entropy in Nuclear Quantum Computing: $�=-\text{Tr}(�\log �)$ Quantum entropy ($�$) describing the information content of a quantum state $�$, where $\text{Tr}$ represents the trace operator and $\log$ is the logarithm in base 2.

4. Digital Pulse Processing for Gamma-Ray Spectroscopy (Pulse Shape Discrimination): $�={\sum }_{�=1}^{�}{�}_{�}\cdot \text{PSD}({�}_{�})$ Energy calculation in gamma-ray spectroscopy using pulse shape discrimination, where ${�}_{�}$ is the amplitude of the ${�}^{�ℎ}$ pulse, and $\text{PSD}({�}_{�})$ is the pulse shape discrimination function.

5. Quantum Circuit for Quantum Key Distribution (QKD) Protocols: $\mid �⟩={�}_{\text{QKD}}(\text{Hadamard}\otimes \text{Hadamard})\mid 0⟩$ Quantum circuit for preparing quantum states in QKD protocols, where ${�}_{\text{QKD}}$ represents a series of quantum gates for QKD operations, and $\text{Hadamard}$ represents the Hadamard gate.

6. Digital Filter Design Equation (FIR Filter with Parks-McClellan Algorithm): $�(�)={\sum }_{�=0}^{�-1}ℎ(�){�}^{-���}$ Frequency response ($�(�)$) of a Finite Impulse Response (FIR) filter designed using the Parks-McClellan algorithm, where $ℎ(�)$ are the filter coefficients and $�$ is the angular frequency.

1. Quantum Chromodynamics (QCD) Beta Function: $�(�)=�\frac{��}{��}=-\frac{11}{3}{�}_{�}\frac{{�}^3}{(4�{)}^2}-\frac{4}{3}{�}_{�}{�}_{�}\frac{{�}^3}{(4�{)}^2}+\dots$ Beta function in QCD, describing the running of the strong coupling constant $�$ with respect to the energy scale $�$, where ${�}_{�}$ is a Casimir factor, ${�}_{�}$ is the trace normalization factor, and ${�}_{�}$ is the number of active quark flavors.

2. Schwarzschild Radius in General Relativity: ${�}_{�}=\frac{2��}{{�}^2}$ Schwarzschild radius (${�}_{�}$) representing the size of the event horizon of a non-rotating black hole with mass $�$, $�$ being the gravitational constant, and $�$ being the speed of light.

3. Quantum Information Density in Quantum Computing: $�=\frac{�\cdot �}{�}$ Quantum information density ($�$) representing the amount of quantum information that can be stored in a quantum system, where $�$ is the Boltzmann constant, $�$ is the number of qubits, and $�$ is the volume of the quantum system.

4. Digital Signal Processing Equation for Fast Fourier Transform (FFT): $�(�)={\sum }_{�=0}^{�-1}�(�)\cdot {�}^{-\frac{2��}{�}��}$ Equation for the Fast Fourier Transform (FFT), transforming a discrete signal $�(�)$ from the time domain to the frequency domain, where $�(�)$ represents the Fourier coefficients.

5. Quantum Error Correction Code for Quantum Communication: ${\mid �⟩}_{\text{encoded}}=\frac{1}{\sqrt{�}}{\sum }_{�=1}^{�}{\mid �⟩}_{\text{data}}\otimes {�}_{�}{\mid 0⟩}_{\text{ancilla}}$ Quantum error correction code, where $�$ represents the number of code words, ${\mid �⟩}_{\text{data}}$ are the data states, ${�}_{�}$ are unitary gates for error correction, and ${\mid 0⟩}_{\text{ancilla}}$ are ancilla qubits.

6. Digital Filter Design Equation (IIR Filter with Butterworth Response): $�(�)=\frac{1}{{\prod }_{�=1}^{�}(\frac{�}{{�}_{�}}+1)}$ Transfer function of an Infinite Impulse Response (IIR) filter designed with a Butterworth response, where $�$ is the complex frequency variable and ${�}_{�}$ are the poles of the filter.