prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What does the paper aim to solve?

Read the following text and answer briefly.

Context 1: Paper Info

Title: Generalized Pole-Residue Method for Dynamic Analysis of Nonlinear Systems based on Volterra Series
Publish Date: March 7, 2023
Author List: Qianying Cao (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology), Anteng Chang (from College of Engineering, Ocean University of China), Junfeng Du (from College of Engineering, Ocean University of China), Lin Lu (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology)

Figure

Fig  1: Procedure to compute the response by a combination of Volterra series and Laguerre polynomials
Fig  2: Linear frequency response function: (a) Modulus of H 1 (ω), (b) phase angle of H 1 (ω)
Fig  6: Comparison of h 1 (t) based on the analytical and reconstructed by Laguerre polynomials
Fig  11: Response for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  18: Comparison of original excitations and reconstructed results: (a) Case 1, (b) Case 2 (c) Case 3
Fig  19: Response to irregular excitation for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  23: Input-output dataset used to identify Volterra series: (a) input excitation, (b) output response
Fig  26: Comparison of responses between the predicted and numerical results: (a) response to regular excitation, (b) response to irregular excitation
Parameter values of the irregular excitation

abstract

Dynamic systems characterized by second-order nonlinear ordinary differential equations appear in many fields of physics and engineering  To solve these kinds of problems, time-consuming stepby-step numerical integration methods and convolution methods based on Volterra series in the time domain have been widely used 
In contrast, this work develops an efficient generalized pole-residue method based on the Volterra series performed in the Laplace domain  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the pole-residue method in this paper is how to deal with the higher-order poles and their corresponding coefficients  Because the proposed method derives an explicit, continuous response function of time, it is much more efficient than traditional numerical methods 
Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the natural response, forced response and cross response are naturally obtained in the solution procedure, meaningful mathematical and physical insights are gained  In numerical studies, systems with a known equation of motion and an unknown equation of motion are investigated 
For each system, regular excitations and complex irregular excitations with different parameters are studied  Numerical studies validate the good accuracy and high efficiency of the proposed method by comparing it with the fourth-order Runge-Kutta method 

Introduction

Most real dynamic systems, as encountered in mechanical and civil engineering, are inherently nonlinear and include geometric nonlinearities, nonlinear constitutive relations in material or nonlinear resistances, etc    Nonlinear problems are attracting increasing attention from engineers and scientists 
This work focuses on solving nonlinear system vibration problems, i e , computing transient responses of nonlinear oscillators under arbitrary irregular excitations based on a combination of a pole-residue operation and Volterra series  Because Volterra series are single-valued, the scope of the present study is restricted to nonlinear behaviours without bifurcations  
To analyse nonlinear vibration problems, researchers have performed extensive studies and developed various mathematical methods  Popular methods include step-by-step numerical integration methods in the time domain, such as the Runge-Kutta method  This kind of method not only requires a small time-step resolution for obtaining high-precision solutions but also is prone to numerical instability  
For a long response with small time steps, the time domain methods are very costly in computational time  Volterra series is another widely used method, which is the extension of the Duhamel integral for linear systems   Volterra series can reproduce many nonlinear phenomena, but they are very complex due to higher-dimensional convolution integrals  
Since 1980's, significant progress has been made in the general area of the Volterra series  The reader is referred to Ref  for a quite thorough literature review on the relevant topics  After 2017, most papers focus on Volterra series identification  De Paula and Marques proposed a method for the identification of Volterra kernels, which was based on time-delay neural networks 
Son and Kim presented a method for a direct estimation of the Volterra kernel coefficients  Dalla Libera et al  introduced two new kernels for Volterra series identification  Peng et al  used the measured response to identify the kernel function and performed the nonlinear structural damage detection  Only a few papers concentrated on simplifying the computation of convolution integrals 
Traditional methods for computing convolution integrals involved in the Volterra series have been performed in three distinct domains: time, frequency and Laplace  The time domain method based on Volterra series refers to discrete time convolution methods, which also suffer computational cost problems  
Both the frequency domain method and the Laplace domain method based on the Volterra series consist of three steps: (1) Volterra series are transformed into an algebraic equation in the frequency domain or Laplace domain; the algebraic equation is solved by purely algebraic manipulations; and (3) the solution in Step ( ) is transformed back to the time domain 
Many researchers have used the frequency domain method to compute the responses of nonlinear systems  Billings et al  developed a new method for identifying the generalized frequency response function (GFRF) of nonlinear systems and then predicted the nonlinear response based on these GFRFs  Carassale et al  introduced a frequency domain approach for nonlinear bridge aerodynamics and aeroelasticity 
Ho et al  computed an output frequency domain function of a nonlinear damped duffing system modelled by a Volterra series under a sinusoidal input  Kim et al  identified the higher order frequency response functions by using the nonlinear autoregressive with exogenous input technique and the harmonic probing method 
This type of frequency domain method is much more efficient than the time domain method due to the fast Fourier transform algorithm  However, the frequency domain method not only is limited by frequency resolutions but also suffers from leakage problems due to the use of discrete Fourier transforms  In addition, the frequency domain method calculates only a steady-state response 
A natural response generated by initial conditions and a cross response caused by interactions between a system and an excitation are ignored  In contrast, the Laplace domain method can calculate all response components because initial conditions are considered in the computational procedure  However, it has been restricted to analytical operations for simple excitations, such as sinusoidal excitations and exponential excitations  
The proposed method falls into the category of the Volterra series method computed in the Laplace domain  Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the proposed method follows a similar path as a pole-residue method for linear systems , the proposed method to solve nonlinear system vibration problems is called the generalized pole-residue method 
The main concept of the pole-residue method developed by Hu et al  was that the poles and residues of the response could be easily obtained from those of the input and system transfer function to obtain the closed-form response solution of linear systems  This method included three steps: (1) writing the system transfer function into pole-residue form; (2) writing the excitation into pole-residue form by the Prony-SS method; (3) computing the poles and residues of the response by an algebraic operation based on those from system and excitation 
Compared to Hu et al  , which was regarded as an efficient tool to compute responses of linear systems, the generalized pole-residue method in this paper is introduced to compute responses of nonlinear systems  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the generalized pole-residue method is how to deal with the higher-order poles and their corresponding coefficients  Similar to the Taylor series, the Volterra series representation is an infinite series, and convergence conditions are needed to assure that the representation is meaningful 
Because the proposed method is based on the Volterra series, only the system with convergent Volterra series representation can be treated by the proposed method  The paper is organized as follows  In Section 2, the nonlinear response is modelled by a Volterra series, and Volterra kernel functions are decoupled by Laguerre polynomials 
Then, the pole-residue method for computing explicit responses is developed in Section 3  Numerical studies and discussions are given in Section 4  Finally, the conclusions are drawn in Section 5 

Response calculation based on Volterra series

A nonlinear oscillator, whose governing equation of motion is given by where z(t, y, ẏ) represents an arbitrary nonlinear term; m, c, and k are the mass, damping and linear stiffness, respectively; y(t), ẏ(t) and ÿ(t) are the displacement, velocity and acceleration, respectively; and f (t) is the time-dependent excitation 
If the energy of excitation f (t) is limited, the nonlinear response under zero initial conditions (i e , zero displacement and zero velocity) can be represented by the Volterra series : where N is the order of Volterra series and In Eq  3, h 1 (τ ) is called the first-order Volterra kernel function, which represents the linear behaviour of the system; h n (τ 1 ,      
, τ n ) for n > 1 are the higher-order Volterra kernel functions, which describe the nonlinear behaviour of the system  The complete formulation of y(t) includes infinite series where the labour of calculating the n th term increases quickly with the growth of n  Fortunately, the response accuracy may be ensured by the first several order Volterra series 
This is proved here in numerical studies  The commonly known Laguerre polynomials are represented as : where p i is the order of the Laguerre polynomials and a i is the damping rate  The Laguerre polynomials satisfy the orthogonal relationship expressed as: By using Laguerre polynomials, the Volterra kernel function h n (t 1 ,      
, t n ) in Eq  3 can be decoupled as follows : where the coefficient is computed resorting to the orthogonal relationship in Eq  5: Substituting Eq  6 into Eq  3 yields       The above operation that uses the Laguerre polynomials to decouple Volterra higher order kernel functions has been well-developed 
The reader is referred to Refs  for details about the adopted technique  After decoupling Volterra higher order kernel functions in time, one can regroup Eq  8 into:       By denoting Eq  9 becomes The above procedure to compute the nonlinear response by a combination of Volterra series and Laguerre polynomials is schematically shown in Fig   
Volterra kernel functions h n (t 1 ,       , t n ) can be obtained by either an equation of motion or measured input-output signals  To derive a closedform solution of the response, we must obtain a closed-form solution of x i (t) first  In the following presentation, a closed-form solution of the aforementioned x i (t) and y n (t) is derived by using the pole-residue method 
3  Pole-residue method for calculating x i (t) and y n (t) Performing the Laplace transform of x i (t) in Eq  10 yields where in which Eq  13 includes a single pole and several higher-order poles  For k = 0, −a i is a single pole, and b p i (0) is a corresponding coefficient, namely, the residue  For k > 0, −a i are higher-order poles, and b p i (k) are corresponding coefficients 
For an irregular excitation signal f (t) of a finite duration of T , it can always be approximated into a pole-residue form by using the complex exponential signal decomposition method-Prony-SS : where N ℓ is the number of components; α ℓ and λ ℓ are constant coefficients, which either are real numbers or occur in complex conjugate pairs 
We define λ ℓ = −δ ℓ + iΩ ℓ , where Ω ℓ is the excitation frequency and δ ℓ is the damping factor of the ℓ th component  We denote α ℓ = A ℓ e iθ ℓ , where A ℓ is the amplitude and θ ℓ is the sinusoidal initial phase in radians  Taking the Laplace transform of Eq  15 yields Note that the concept of the Prony-SS method is similar to that of a principal component method 
A smooth excitation usually requires just several terms to achieve a good approximation  For high irregular loadings, including more terms would achieve a better approximation  Substituting Eqs  13 and 16 into Eq  12 yields Expressing xi (s) in its pole-residue form yields where λ ℓ are simple poles, and the corresponding residues are easily obtained by
and −a i are higher-order poles, and the corresponding coefficients are firstly derived as: By taking the inverse Laplace transform of Eq  18, a closed-form solution is obtained: Substituting Eqs  11 and 21 into Eq  2 yields Theoretically speaking, the proposed method for deriving the closed-form solution of the nonlinear response is applicable to any order of the Volterra series 
For practical engineering, usually only the first several order responses dominate  By setting up N = 2, Eq  22 can be simplified into three components: where the natural response, which is only related to system poles, is given by and the cross response, which is related to both system poles and excitation poles, is given by
and the forced response, which is related only to excitation poles, is given by The first term in Eq  26 is the first-order forced response governed by the excitation frequency, i e , the imaginary part of the pole λ ℓ   The second term corresponds to the second-order nonlinear forced response, which includes the sum frequency and difference frequency responses governed by λ ℓ + λ j  
Eq  26 straightforwardly offers visible information about the possible nonlinear vibrations by the cooperation of excitation frequencies  Particularly, consider a sinusoidal excitation f (t) = sin ω r t, which can be expressed as f (t) = γe λt + γ * e λ * t , where γ = −0 5i and λ = iω r   Substituting these values into Eq 
26, the second term of Eq  26 is simplified as where the first term is the difference frequency response, and the second term is the sum frequency response 

Numerical studies

In practical engineering, some systems have an accurate equation of motion  Additionally, some systems have difficulty constructing their equations of motion because of complex nonlinear dynamic behaviours and uncertain system parameters  In this article, a system with a known equation of motion is called a known system, and a system with an unknown equation of motion is called an unknown system for simplicity 
In this section, two numerical studies are presented  The first study verifies the proposed method using a known nonlinear oscillator, and the second study demonstrates the applicability of the proposed method to an unknown system  Throughout the numerical studies, the unit system is the metre-kilogramme-second (MKS) system; for conciseness, explicit units for quantities are omitted 

A known nonlinear system

This study chooses a nonlinear oscillator written as: where mass m = 1, damping c = 1, linear stiffness k 1 = 10, quadratic stiffness k 2 = 20 and cubic stiffness k 3 = 20  It is a case that has been studied in a previously published article   The linear natural frequency of the system ω 0 = k 1 /m = 3 16 and the damping ratio ζ = c/(2mω 0 ) = 15 8% 
This kind of oscillator occurs in many engineering problems, such as a model of fluid resonance in a narrow gap between large vessels   In the model, k 1 y represents the linear restoring force of the fluid, and k 2 y 2 and k 3 y 3 are respectively the quadratic and cubic nonlinear restoring forces of the fluid 

Volterra kernel functions

Generally, the first several order responses dominate the total response of a system  Hence, the order of the Volterra series in Eq  22 is chosen to be 3, namely, N = 3  For computing the first three order responses from Eq  22, the first three order Volterra kernel functions need to be known  Since Volterra kernel functions and corresponding frequency response functions are related by a specific Fourier transform pair, we can first write the first three orders of frequency response functions directly from Eq  28 
Then, Volterra kernel functions are obtained by the inverse Fourier transform  Based on the harmonic probing algorithm , the linear frequency response function (LFRF) H 1 (ω), the quadratic frequency response function (QFRF) H 2 (ω 1 , ω 2 ) and the cubic frequency response function (CFRF) H 3 (ω 1 , ω 2 , ω 3 ) are analytically given by:
and Figures show H 1 (ω), H 2 (ω 1 , ω 2 ) and H 3 (ω 1 , ω 2 , ω 3 ), respectively, which agree well with those reported in Ref    As expected, the modulus of H 1 (ω) in Fig  peaks near the linear natural frequency ω 0 , and the phase angle decreases monotonically from 0 to -π with increasing frequency 
Figure shows the sum frequency QFRF, where the energy converges along the line of ω 1 +ω 2 ≈ ω 0   Therefore, when the sum frequency of a two-tone excitation equals the linear resonant frequency, the second-order response may reach its maximum  Additionally, those pairs of excitations in line ω 1 + ω 2 ≈ ω 0 may produce non-negligible vibration magnitudes due to second-order nonlinear effects 
For the difference frequency QFRF in Fig  (b), the energy converges along two main lines, i e , ω 1 ≈ ω 0 and ω 2 ≈ ω 0   Figures show moduli of H 3 (ω, ω, ω) and H 3 (ω, ω, −ω), which are diagonal terms of the sum frequency CFRF and the difference frequency CFRF, respectively  While the modulus of H 3 (ω, ω, ω) peaks near ω ≈ ω 0 /3 and ω 0 , that of H 3 (ω, ω, −ω) peaks near ω ≈ ω 0 with a small hump around ω ≈ ω 0 /2 
Values at ω ≈ ω 0 /3 and ω 0 /2 may be magnified by higher-order stiffness terms in Eq  28  By performing the inverse fast Fourier transform to Eqs  29-31, the corresponding linear impulse response function h 1 (t), quadratic impulse response function h 2 (t 1 , t 2 ) and cubic impulse response function h 3 (t 1 , t 2 , t 3 ) are obtained 
Here, h 1 (t) and h 2 (t 1 , t 2 ) are plotted in Figs  , respectively, and h 3 (t, t, t) is shown in Fig    In the numerical implementation, Eqs  29-31 have been utilized with the frequency interval ∆ω = 0 1, number of frequency components N n = 1025, and cut-off frequencies 102 4 and −102 4  For decoupling Volterra kernel functions by using Laguerre polynomials, the damping rate and number of Laguerre polynomials for each order Volterra kernel function need to be determined (see Eqs  4 and 6) 
In this example, we set a 1 = a 2 = a 3 = 2 and R 1 = R 2 = R 3 = 24 because coefficients c p 1    pn become very small when R n > 24, n = 1, 2, 3  According to Eq  7, the coefficients of the first three order Volterra kernel functions are calculated, which are shown in Figs  9 and 10  For convenience, Fig  plots only c p 1 p 2 p 3 for p 3 = 0 
With the increase of the order of Laguerre polynomials, coefficients in Figs  9 and 10 gradually decrease, which illustrates how the first several orders of Laguerre polynomials dominate all orders of the Volterra kernel function  With the known Laguerre polynomials and corresponding coefficients, Volterra kernel functions are reconstructed by Eq  6 
For comparison, reconstructed Volterra kernel functions are also plotted in Figs    The reconstructed results agree well with the analytical values, which verifies the accuracy of the decomposition 

Sinusoidal excitation

From Eq  28, we consider a sinusoidal excitation where A and Ω are the amplitude and the frequency, respectively  Five cases of A and Ω are shown in Table   Excitation frequencies in Cases 1 and 2 are larger than the linear natural frequency (ω 0 ≈ 3 16), those in Case 3 are very close to ω 0 , and those in Cases 4 and 5 are smaller than ω 0  
All cases have same amplitudes  The poles of a sinusoidal excitation are λ 1,2 = ±iΩ, and the residues are α 1,2 = ∓iA/2  Numerical values of excitation poles and residues for different cases are listed in Table   Table : Parameter values, poles and residues of the sinusoidal excitation Substituting poles and residues of the excitation, as well as those of the system into Eqs 
20 and 19, response coefficients β p i ,k corresponding to system poles −a i and response coefficients γ p i ,ℓ corresponding to excitation poles λ ℓ are calculated, respectively  According to Eq  22, the first three orders of responses for each case in Table are calculated  Figures )-15(a) show the comparison of responses obtained by the proposed method and the fourth-order Runge-Kutta method with ∆t = 10 −4  
For Cases 1 and 2, the first-order responses agree well with the total responses obtained by the Runge-Kutta method, and the higher-order responses only slightly improve the transient parts  For Cases 3-5, the sum of the first three orders of responses is in good agreement with the Runge-Kutta solution 
When the response nonlinearity increases, higher-order responses need to be considered  In other words, the proposed method can accurately compute the nonlinear responses by choosing a small number N of Volterra series terms  Figures )-15(b) show the contributions of the three response components for the five cases 
In each case, the first-order response is the most dominant component, and the contributions of secondand third-order responses are much less than those of the first-order response  Especially for Cases 1 and 2, whose excitation frequencies are far from the linear natural frequency, second-and thirdorder responses are close to zero 
This may be because the QFRF and CFRF approach zero when the frequency is larger than 4 rad/s (see Figs  )  Furthermore, the mean values of the first-order responses are approximately zero, and those of the second-order responses are always smaller than zero, which are the difference frequency components in Eq  27 
Moreover, it is clearly observed that second-order responses for Cases 3-5 exhibit a periodic oscillation with a period near half of that for the first-order response, which is excited by the sum frequency component of the excitation (see second part of Eq  27)  Compared with steady-state solutions of first-and second-order responses, those of third-order responses in Cases 3-5 are no longer single regular motions 
By performing the FFT, frequency spectra of these three third-order responses are shown in Fig    We find that these three third-order responses are all dominated by their own fundamental harmonic component and the third harmonic (triple frequency) component  Figure shows the computational time to calculate the response of the oscillator for Case 1 by the proposed method, the fourth-order Runge-Kutta method and the convolution method 
The proposed method, which has an explicit solution, is much more efficient in computational time than the latter two methods, which need small time steps to obtain high-precision solutions  In particular, the efficiency of the proposed method increases with the length of the response time  Computation time (sec )
t=0 02s Convolution, t=0 02s Convolution, t=0 001s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the proposed method, the fourth-fifth order Runge-Kutta method and the convolution method regular loading in Case 1

Irregular excitation

In Eq  28, considering an irregular excitation consisting of several cosine functions where N f is the number of cosine components; A n , Ω n and θ n are the amplitude, frequency and phase angle of the n th component, respectively  Table lists three cases of these parameters  In each case, the amplitudes of all components are the same, and phase angles θ n uniformly distributed between 0 and 2π are randomly generated 
To decompose the excitation into a pole-residue form, the Prony-SS method is used, whose concept is similar to that of a principal component method  The readers are referred to Ref  for details  The chosen rank of each case is also shown in Table   Figure shows the comparison of original excitations and reconstructed results of these three cases, which all have excellent agreement 
show the results computed by the fourth-order Runge-Kutta method  In all cases, the sums of the first three orders of responses agree well with those obtained by the Runge-Kutta method  The contributions of the first three orders of responses for each case are plotted in Figs  )-21(b)  Similarly, the system vibration is dominated by the first-order response 
However, the contributions of second-and third-order significantly grow with increasing excitation magnitude and frequency number  Furthermore, when the magnitude of the nonlinear response becomes large, sharp troughs are present  This phenomenon may be induced by the nonlinear stiffness  While the first-order response fails to capture these troughs, the higher-order responses successfully capture these troughs 
Figure plots the computational time to calculate the response of the oscillator for the irregular loading in Case 1 by the proposed method and the fourth-fifth order Runge-Kutta method, respectively  While the fourth-fifth order Runge-Kutta method is more efficient under a small response length, the proposed method becomes much more efficient when the response length is larger than about 130 s 
In addition, the proposed method obtains the explicit response solution, so one can directly obtain the response value at a specific time t p instead of integrating from 0 to t p for traditional numerical methods  Computation time (sec ) Proposed, t=0 02s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the method and the fourth-fifth order Runge-Kutta method for irregular loading in Case 1

An unknown nonlinear

To check the applicability of the proposed method to an unknown nonlinear system, a known input excitation and its corresponding response are used to identify its Volterra kernel functions  When the Volterra kernel functions are known, we can follow the procedure in Section 4 1 to predict system responses 
In this study, the input excitation is white noise with a constant power spectrum S 0 = 0 001, and the corresponding response is obtained by solving Eq  28 by the fourth-order Runge-Kutta method, which is shown in Fig    From Section 4 1, we determine that the sum of the first two orders of responses agrees well with the total response 
In this study, the order of Volterra series N is chosen to be 2, damping rates of Laguerre polynomials are a 1 = a 2 = 2, and numbers of Laguerre polynomials are R 1 = R 2 = 24  To estimate the first two orders of Volterra kernel functions, a matrix equation is constructed using excitation data and response data 
By using the least square method to solve this matrix equation, coefficients c p 1 and c p 1 p 2 in Eq  8 are identified  Figure plots c p 1 and c p 1 p 2 , respectively, which have good agreement with the exact results in Fig    Then, the first two order Volterra kernel functions are constructed by Eq  6 
Compared with the exact results in Figs  , the identified Volterra kernel functions in Fig  completely agree well with the exact solutions  Note that the white noise excitation, which can excite more frequency components of the response, is chosen to obtain good Volterra kernel functions  A regular excitation f (t) = sin(πt) and an irregular excitation f (t) = N f n=1 A n cos(Ω n t + θ n ) with A n = 0 3 and Ω n varying from 0 to 40 with equal interval 1 are chosen as input excitations 
The predicted responses, along with results obtained by the fourth-order Runge-Kutta method, are shown in Fig    In both cases, the proposed method accurately predicts system responses  As presented in Eq  23, a nonlinear response is the sum of three terms: natural response y s (t), forced response y f (t) and cross response y c (t) 
These individual terms, as well as their sum to two excitations, are shown in Figs  27 and 28, respectively  As shown in Figs  and 28, both first-and second-order responses include the natural response y s (t) and the forced response y f (t), but the cross response y c (t) only exists in second-order responses 
When t becomes larger, both y s (t) and y c (t) diminish due to the presence of system damping, and the total response is entirely governed by y f (t)  Moreover, we notice some features at t = 0 for these components, including y s (0) = −y f (0) for the first-order response and y s (0) + y f (0) = −y c (0) for the second-order response, which are due to imposed zero initial conditions 

Conclusions

Considering arbitrary irregular excitations, an efficient generalized pole-residue method to compute the nonlinear dynamic response modelled by the Volterra series was developed  A core of the proposed method was obtaining poles and corresponding coefficients of Volterra kernel functions, then those of each order response modelled by each order Volterra series 
Once the poles and corresponding coefficients of Volterra kernel functions and excitations were both available, the remaining derivation could follow a similar pole-residue method that had been developed for ordinary linear oscillators  To obtain the poles and corresponding coefficients of Volterra kernel functions, two steps were included: (1) using Laguerre polynomials to decouple higher-order Volterra kernel functions with respect to time and (2) obtaining poles and corresponding coefficients of Laguerre polynomials in the Laplace domain 
Because the proposed method gave an explicit, continuous response function of time, it was much more efficient than traditional numerical methods  Moreover, many meaningful physical and mathematical insights were gained because not only each order response but also the natural response, the forced response and the cross response of each order were obtained in the solution procedure 
To demonstrate that the proposed method was not only suitable for a system with a known equation of motion but also applicable to a system with an unknown equation of motion, two numerical studies were conducted  For each study, regular excitations and complex irregular excitations with different parameters were investigated 
The efficiency of the proposed method was verified by the fourth-order Runge-Kutta method  This paper only computes the response under zero initial conditions  The response under non-zero initial conditions will be investigated in our future work 
Context 2: Vitamin K - Wikipedia
(Redirected from Vitamin k)
This article needs more medical references for verification or relies too heavily on primary sources  Please review the contents of the article and add the appropriate references if you can  Unsourced or poorly sourced material may be challenged and removed  (November 2015)
This article is about the family of vitamers  For vitamin K1 the form usually used as a supplement, see Phytomenadione 
Vitamin K structures  MK-4 and MK-7 are both subtypes of K2 
Vitamin K deficiency, Warfarin overdose
Vitamin K is a group of structurally similar, fat-soluble vitamins the human body requires for complete synthesis of certain proteins that are prerequisites for blood coagulation and which the body also needs for controlling binding of calcium in bones and other tissues  The vitamin K-related modification of the proteins allows them to bind calcium ions, which they cannot do otherwise  Without vitamin K, blood coagulation is seriously impaired, and uncontrolled bleeding occurs  Low levels of vitamin K also weaken bones and promote calcification of arteries and other soft tissues[citation needed] 
Chemically, the vitamin K family comprises 2-methyl-1,4-naphthoquinone (3-) derivatives  Vitamin K includes two natural vitamers: vitamin K1 and vitamin K2 [1] Vitamin K2, in turn, consists of a number of related chemical subtypes, with differing lengths of carbon side chains made of isoprenoid groups of atoms 
Vitamin K1, also known as phylloquinone, is made by plants, and is found in highest amounts in green leafy vegetables because it is directly involved in photosynthesis  It may be thought of as the plant form of vitamin K  It is active as a vitamin in animals and performs the classic functions of vitamin K, including its activity in the production of blood-clotting proteins  Animals may also convert it to vitamin K2 
Bacteria in the gut flora can also convert K1 into vitamin K2  In addition, bacteria typically lengthen the isoprenoid side chain of vitamin K2 to produce a range of vitamin K2 forms, most notably the MK-7 to MK-11 homologues of vitamin K2  All forms of K2 other than MK-4 can only be produced by bacteria, which use these forms in anaerobic respiration  The MK-7 and other bacterially derived forms of vitamin K2 exhibit vitamin K activity in animals, but MK-7's extra utility over MK-4, if any, is unclear and is a matter of investigation 
Three synthetic types of vitamin K are known: vitamins K3, K4, and K5  Although the natural K1 and all K2 homologues and synthetic K4 and K5 have proven nontoxic, the synthetic form K3 (menadione) has shown toxicity [2]
1 2 Cardiovascular health
1 4 Coumarin poisoning
4 1 Conversion of vitamin K1 to vitamin K2
4 2 Vitamin K2
6 Absorption and dietary need
7 Dietary reference intake
10 Biochemistry
10 1 Function in animals
10 2 Gamma-carboxyglutamate proteins
10 3 Methods of assessment
10 4 Function in bacteria
11 Injection in newborns
11 3 Controversy
A review of 2014 concluded that there is positive evidence that monotherapy using MK-4, one of the forms of Vitamin K2, reduces fracture incidence in post-menopausal women with osteoporosis, and suggested further research on the combined use of MK-4 with bisphosphonates  In contrast, an earlier review article of 2013 concluded that there is no good evidence that vitamin K supplementation helps prevent osteoporosis or fractures in postmenopausal women [3]
A Cochrane systematic review of 2006 suggested that supplementation with Vitamin K1 and with MK4 reduces bone loss; in particular, a strong effect of MK-4 on incident fractures among Japanese patients was emphasized [4]
A review article of 2016 suggested to consider, as one of several measures for bone health, increasing the intake of foods rich in vitamins K1 and K2 [5]
Cardiovascular health[edit]
Adequate intake of vitamin K is associated with the inhibition of arterial calcification and stiffening,[6] but there have been few interventional studies and no good evidence that vitamin K supplementation is of any benefit in the primary prevention of cardiovascular disease [7]
One 10-year population study, the Rotterdam Study, did show a clear and significant inverse relationship between the highest intake levels of menaquinone (mainly MK-4 from eggs and meat, and MK-8 and MK-9 from cheese) and cardiovascular disease and all-cause mortality in older men and women [8]
Vitamin K has been promoted in supplement form with claims it can slow tumor growth; there is however no good medical evidence that supports such claims [9]
Coumarin poisoning[edit]
Vitamin K is part of the suggested treatment regime for poisoning by rodenticide (coumarin poisoning) [10]
Although allergic reaction from supplementation is possible, no known toxicity is associated with high doses of the phylloquinone (vitamin K1) or menaquinone (vitamin K2) forms of vitamin K, so no tolerable upper intake level (UL) has been set [11]
Blood clotting (coagulation) studies in humans using 45 mg per day of vitamin K2 (as MK-4)[12] and even up to 135 mg per day (45 mg three times daily) of K2 (as MK-4),[13] showed no increase in blood clot risk  Even doses in rats as high as 250 mg/kg, body weight did not alter the tendency for blood-clot formation to occur [14]
Unlike the safe natural forms of vitamin K1 and vitamin K2 and their various isomers, a synthetic form of vitamin K, vitamin K3 (menadione), is demonstrably toxic at high levels  The U S  FDA has banned this form from over-the-counter sale in the United States because large doses have been shown to cause allergic reactions, hemolytic anemia, and cytotoxicity in liver cells [2]
Phylloquinone (K1)[15][16] or menaquinone (K2) are capable of reversing the anticoagulant activity of the anticoagulant warfarin (tradename Coumadin)  Warfarin works by blocking recycling of vitamin K, so that the body and tissues have lower levels of active vitamin K, and thus a deficiency of vitamin K 
Supplemental vitamin K (for which oral dosing is often more active than injectable dosing in human adults) reverses the vitamin K deficiency caused by warfarin, and therefore reduces the intended anticoagulant action of warfarin and related drugs [17] Sometimes small amounts of vitamin K are given orally to patients taking warfarin so that the action of the drug is more predictable [17] The proper anticoagulant action of the drug is a function of vitamin K intake and drug dose, and due to differing absorption must be individualized for each patient [citation needed] The action of warfarin and vitamin K both require two to five days after dosing to have maximum effect, and neither warfarin or vitamin K shows much effect in the first 24 hours after they are given [18]
The newer anticoagulants dabigatran and rivaroxaban have different mechanisms of action that do not interact with vitamin K, and may be taken with supplemental vitamin K [19][20]
Vitamin K2 (menaquinone)  In menaquinone, the side chain is composed of a varying number of isoprenoid residues  The most common number of these residues is four, since animal enzymes normally produce menaquinone-4 from plant phylloquinone 
A sample of phytomenadione for injection, also called phylloquinone
The three synthetic forms of vitamin K are vitamins K3 (menadione), K4, and K5, which are used in many areas, including the pet food industry (vitamin K3) and to inhibit fungal growth (vitamin K5) [21]
Conversion of vitamin K1 to vitamin K2[edit]
Vitamin K1 (phylloquinone) – both forms of the vitamin contain a functional naphthoquinone ring and an aliphatic side chain  Phylloquinone has a phytyl side chain 
The MK-4 form of vitamin K2 is produced by conversion of vitamin K1 in the testes, pancreas, and arterial walls [22] While major questions still surround the biochemical pathway for this transformation, the conversion is not dependent on gut bacteria, as it occurs in germ-free rats[23][24] and in parenterally-administered K1 in rats [25][26] In fact, tissues that accumulate high amounts of MK-4 have a remarkable capacity to convert up to 90% of the available K1 into MK-4 [27][28] There is evidence that the conversion proceeds by removal of the phytyl tail of K1 to produce menadione as an intermediate, which is then condensed with an activated geranylgeranyl moiety (see also prenylation) to produce vitamin K2 in the MK-4 (menatetrione) form [29]
Vitamin K2[edit]
Main article: Vitamin K2
Vitamin K2 (menaquinone) includes several subtypes  The two subtypes most studied are menaquinone-4 (menatetrenone, MK-4) and menaquinone-7 (MK-7) 
Vitamin K1, the precursor of most vitamin K in nature, is a stereoisomer of phylloquinone, an important chemical in green plants, where it functions as an electron acceptor in photosystem I during photosynthesis  For this reason, vitamin K1 is found in large quantities in the photosynthetic tissues of plants (green leaves, and dark green leafy vegetables such as romaine lettuce, kale and spinach), but it occurs in far smaller quantities in other plant tissues (roots, fruits, etc )  Iceberg lettuce contains relatively little  The function of phylloquinone in plants appears to have no resemblance to its later metabolic and biochemical function (as "vitamin K") in animals, where it performs a completely different biochemical reaction 
Vitamin K (in animals) is involved in the carboxylation of certain glutamate residues in proteins to form gamma-carboxyglutamate (Gla) residues  The modified residues are often (but not always) situated within specific protein domains called Gla domains  Gla residues are usually involved in binding calcium, and are essential for the biological activity of all known Gla proteins [30]
At this time[update], 17 human proteins with Gla domains have been discovered, and they play key roles in the regulation of three physiological processes:
Blood coagulation: prothrombin (factor II), factors VII, IX, and X, and proteins C, S, and Z[31]
Bone metabolism: osteocalcin, also called bone Gla protein (BGP), matrix Gla protein (MGP),[32] periostin,[33] and the recently discovered Gla-rich protein (GRP) [34][35]
Vascular biology: growth arrest-specific protein 6 (Gas6)[36]
Unknown function: proline-rich γ-carboxyglutamyl proteins (PRGPs) 1 and 2, and transmembrane γ-carboxy glutamyl proteins (TMGs) 3 and 4 [37]
Like other lipid-soluble vitamins (A, D and E), vitamin K is stored in the fatty tissue of the human body 
Absorption and dietary need[edit]
Previous theory held that dietary deficiency is extremely rare unless the small intestine was heavily damaged, resulting in malabsorption of the molecule  Another at-risk group for deficiency were those subject to decreased production of K2 by normal intestinal microbiota, as seen in broad spectrum antibiotic use [38] Taking broad-spectrum antibiotics can reduce vitamin K production in the gut by nearly 74% in people compared with those not taking these antibiotics [39] Diets low in vitamin K also decrease the body's vitamin K concentration [40] Those with chronic kidney disease are at risk for vitamin K deficiency, as well as vitamin D deficiency, and particularly those with the apoE4 genotype [41] Additionally, in the elderly there is a reduction in vitamin K2 production [42]
The National Academy of Medicine (NAM) updated an estimate of what constitutes an adequate intake (AI) for vitamin K in 2001  The NAM does not distinguish between K1 and K2 – both are counted as vitamin K  At that time there was not sufficient evidence to set the more rigorous estimated average requirement (EAR) or recommended dietary allowance (RDA) given for most of the essential vitamins and minerals  The current daily AIs for vitamin K for adult women and men are 90 μg and 120 μg respectively  The AI for pregnancy and lactation is 90 μg  For infants up to 12 months the AI is 2–2 5 μg, and for children aged 1 to 18 years the AI increases with age from 30 to 75 μg  As for safety, the FNB also sets tolerable upper intake levels (known as ULs) for vitamins and minerals when evidence is sufficient  In the case of vitamin K no UL is set, as evidence for adverse effects is not sufficient  Collectively EARs, RDAs, AIs and ULs are referred to as dietary reference intakes [43] The European Food Safety Authority reviewed the same safety question and did not set an UL [44]
For U S  food and dietary supplement labeling purposes, the amount in a serving is expressed as a percentage of daily value (%DV)  For vitamin K labeling purposes the daily value was 80 μg, but as of May 2016 it has been revised upwards to 120 μg  A table of the pre-change adult daily values is provided at reference daily intake  Food and supplement companies have until 28 July 2018 to comply with the change 
See also: Vitamin K2 § Dietary sources
K1 (μg)[45]
Kale, cooked
Collards, cooked
Collards, raw
Swiss chard, cooked
Swiss chard, raw
Turnip greens, raw
Romaine lettuce, raw
Table from "Important information to know when you are taking: Warfarin (Coumadin) and Vitamin K", Clinical Center, National Institutes of Health Drug Nutrient Interaction Task Force [46]
Vitamin K1 is found chiefly in leafy green vegetables such as dandelion greens (which contain 778 4 μg per 100 g, or 741% of the recommended daily amount), spinach, swiss chard, lettuce and Brassica vegetables (such as cabbage, kale, cauliflower, broccoli, and brussels sprouts) and often the absorption is greater when accompanied by fats such as butter or oils; some fruits, such as avocados, kiwifruit and grapes, are also high in vitamin K  By way of reference, two tablespoons of parsley contains 153% of the recommended daily amount of vitamin K [47] Some vegetable oils, notably soybean oil, contain vitamin K, but at levels that would require relatively large calorie consumption to meet the USDA-recommended levels [48] colonic bacteria synthesize a significant portion of humans' vitamin K needs; newborns often receive a vitamin K shot at birth to tide them over until their colons become colonized at five to seven days of age from the consumption of breast milk 
The tight binding of vitamin K1 to thylakoid membranes in chloroplasts makes it less bioavailable  For example, cooked spinach has a 5% bioavailability of phylloquinone, however, fat added to it increases bioavailability to 13% due to the increased solubility of vitamin K in fat [49]
Main article: Vitamin K deficiency
Average diets are usually not lacking in vitamin K, and primary deficiency is rare in healthy adults  Newborn infants are at an increased risk of deficiency  Other populations with an increased prevalence of vitamin K deficiency include those who suffer from liver damage or disease (e g  alcoholics), cystic fibrosis, or inflammatory bowel diseases, or have recently had abdominal surgeries  Secondary vitamin K deficiency can occur in people with bulimia, those on stringent diets, and those taking anticoagulants  Other drugs associated with vitamin K deficiency include salicylates, barbiturates, and cefamandole, although the mechanisms are still unknown  Vitamin K1 deficiency can result in coagulopathy, a bleeding disorder [50]Symptoms of K1 deficiency include anemia, bruising, nosebleeds and bleeding of the gums in both sexes, and heavy menstrual bleeding in women 
Osteoporosis[51][52] and coronary heart disease[53][54] are strongly associated with lower levels of K2 (menaquinone)  Vitamin K2 (as menaquinones MK-4 through MK-10) intake level is inversely related to severe aortic calcification and all-cause mortality [8]
Function in animals[edit]
Mechanism of action of vitamin K1 
The function of vitamin K2 in the animal cell is to add a carboxylic acid functional group to a glutamate (Glu) amino acid residue in a protein, to form a gamma-carboxyglutamate (Gla) residue  This is a somewhat uncommon posttranslational modification of the protein, which is then known as a "Gla protein"  The presence of two −COOH (carboxylic acid) groups on the same carbon in the gamma-carboxyglutamate residue allows it to chelate calcium ions  The binding of calcium ions in this way very often triggers the function or binding of Gla-protein enzymes, such as the so-called vitamin K-dependent clotting factors discussed below 
Within the cell, vitamin K undergoes electron reduction to a reduced form called vitamin K hydroquinone, catalyzed by the enzyme vitamin K epoxide reductase (VKOR) [55] Another enzyme then oxidizes vitamin K hydroquinone to allow carboxylation of Glu to Gla; this enzyme is called gamma-glutamyl carboxylase[56][57] or the vitamin K-dependent carboxylase  The carboxylation reaction only proceeds if the carboxylase enzyme is able to oxidize vitamin K hydroquinone to vitamin K epoxide at the same time  The carboxylation and epoxidation reactions are said to be coupled  Vitamin K epoxide is then reconverted to vitamin K by VKOR  The reduction and subsequent reoxidation of vitamin K coupled with carboxylation of Glu is called the vitamin K cycle [58] Humans are rarely deficient in vitamin K1 because, in part, vitamin K1 is continuously recycled in cells [59]
Warfarin and other 4-hydroxycoumarins block the action of VKOR [60] This results in decreased concentrations of vitamin K and vitamin K hydroquinone in tissues, such that the carboxylation reaction catalyzed by the glutamyl carboxylase is inefficient  This results in the production of clotting factors with inadequate Gla  Without Gla on the amino termini of these factors, they no longer bind stably to the blood vessel endothelium and cannot activate clotting to allow formation of a clot during tissue injury  As it is impossible to predict what dose of warfarin will give the desired degree of clotting suppression, warfarin treatment must be carefully monitored to avoid overdose 
Gamma-carboxyglutamate proteins[edit]
Main article: Gla domain
The following human Gla-containing proteins ("Gla proteins") have been characterized to the level of primary structure: blood coagulation factors II (prothrombin), VII, IX, and X, anticoagulant proteins C and S, and the factor X-targeting protein Z  The bone Gla protein osteocalcin, the calcification-inhibiting matrix Gla protein (MGP), the cell growth regulating growth arrest specific gene 6 protein (Gas6), and the four transmembrane Gla proteins (TMGPs), the function of which is at present unknown  Gas6 can function as a growth factor to activate the Axl receptor tyrosine kinase and stimulate cell proliferation or prevent apoptosis in some cells  In all cases in which their function was known, the presence of the Gla residues in these proteins turned out to be essential for functional activity 
Gla proteins are known to occur in a wide variety of vertebrates: mammals, birds, reptiles, and fish  The venom of a number of Australian snakes acts by activating the human blood-clotting system  In some cases, activation is accomplished by snake Gla-containing enzymes that bind to the endothelium of human blood vessels and catalyze the conversion of procoagulant clotting factors into activated ones, leading to unwanted and potentially deadly clotting 
Another interesting class of invertebrate Gla-containing proteins is synthesized by the fish-hunting snail Conus geographus [61] These snails produce a venom containing hundreds of neuroactive peptides, or conotoxins, which is sufficiently toxic to kill an adult human  Several of the conotoxins contain two to five Gla residues [62]
Methods of assessment[edit]
Vitamin K status can be assessed by:
The prothrombin time (PT) test measures the time required for blood to clot  A blood sample is mixed with citric acid and put in a fibrometer; delayed clot formation indicates a deficiency  This test is insensitive to mild deficiency, as the values do not change until the concentration of prothrombin in the blood has declined by at least 50% [63]
Undercarboxylated prothrombin (PIVKA-II); in a study of 53 newborns, found "PT (prothrombin time) is a less sensitive marker than PIVKA II",[64] and as indicated above, PT is unable to detect subclinical deficiencies that can be detected with PIVKA-II testing 
Plasma phylloquinone was found to be positively correlated with phylloquinone intake in elderly British women, but not men,[65] but an article by Schurgers et al  reported no correlation between FFQ[further explanation needed] and plasma phylloquinone [66]
Urinary γ-carboxyglutamic acid responds to changes in dietary vitamin K intake  Several days are required before any change can be observed  In a study by Booth et al , increases of phylloquinone intakes from 100 μg to between 377 and 417 μg for five days did not induce a significant change  Response may be age-specific [67]
Undercarboxylated osteocalcin (UcOc) levels have been inversely correlated with stores of vitamin K[68] and bone strength in developing rat tibiae  Another study following 78 post-menopausal Korean women found a supplement regimen of vitamins K and D, and calcium, but not a regimen of vitamin D and calcium, was inversely correlated with reduced UcOc levels [69]
Function in bacteria[edit]
Many bacteria, such as Escherichia coli found in the large intestine, can synthesize vitamin K2 (menaquinone-7 or MK-7, up to MK-11),[70] but not vitamin K1 (phylloquinone)  In these bacteria, menaquinone transfers two electrons between two different small molecules, during oxygen-independent metabolic energy production processes (anaerobic respiration) [71] For example, a small molecule with an excess of electrons (also called an electron donor) such as lactate, formate, or NADH, with the help of an enzyme, passes two electrons to menaquinone  The menaquinone, with the help of another enzyme, then transfers these two electrons to a suitable oxidant, such fumarate or nitrate (also called an electron acceptor)  Adding two electrons to fumarate or nitrate converts the molecule to succinate or nitrite plus water, respectively 
Some of these reactions generate a cellular energy source, ATP, in a manner similar to eukaryotic cell aerobic respiration, except the final electron acceptor is not molecular oxygen, but fumarate or nitrate  In aerobic respiration, the final oxidant is molecular oxygen (O2), which accepts four electrons from an electron donor such as NADH to be converted to water  E  coli, as facultative anaerobes, can carry out both aerobic respiration and menaquinone-mediated anaerobic respiration 
Injection in newborns[edit]
The blood clotting factors of newborn babies are roughly 30–60% that of adult values; this may be due to the reduced synthesis of precursor proteins and the sterility of their guts  Human milk contains 1–4 μg/L of vitamin K1, while formula-derived milk can contain up to 100 μg/L in supplemented formulas  Vitamin K2 concentrations in human milk appear to be much lower than those of vitamin K1  Occurrence of vitamin K deficiency bleeding in the first week of the infant's life is estimated at 0 25–1 7%, with a prevalence of 2–10 cases per 100,000 births [72] Premature babies have even lower levels of the vitamin, so they are at a higher risk from this deficiency 
Bleeding in infants due to vitamin K deficiency can be severe, leading to hospitalization, blood transfusions, brain damage, and death  Supplementation can prevent most cases of vitamin K deficiency bleeding in the newborn  Intramuscular administration is more effective in preventing late vitamin K deficiency bleeding than oral administration [73][74]
As a result of the occurrences of vitamin K deficiency bleeding, the Committee on Nutrition of the American Academy of Pediatrics has recommended 0 5–1 mg of vitamin K1 be administered to all newborns shortly after birth [74]
In the UK vitamin K supplementation is recommended for all newborns within the first 24 hours [75] This is usually given as a single intramuscular injection of 1 mg shortly after birth but as a second-line option can be given by three oral doses over the first month [76]
Controversy arose in the early 1990s regarding this practice, when two studies suggested a relationship between parenteral administration of vitamin K and childhood cancer,[77] however, poor methods and small sample sizes led to the discrediting of these studies, and a review of the evidence published in 2000 by Ross and Davies found no link between the two [78] Doctors reported emerging concerns in 2013,[79] after treating children for serious bleeding problems  They cited lack-of newborn vitamin K administration, as the reason that the problems occurred, and recommended that breastfed babies could have an increased risk unless they receive a preventative dose 
In the early 1930s, Danish scientist Henrik Dam investigated the role of cholesterol by feeding chickens a cholesterol-depleted diet [80] He initially replicated experiments reported by scientists at the Ontario Agricultural College (OAC) [81] McFarlane, Graham and Richardson, working on the chick feed program at OAC, had used chloroform to remove all fat from chick chow  They noticed that chicks fed only fat-depleted chow developed hemorrhages and started bleeding from tag sites [82] Dam found that these defects could not be restored by adding purified cholesterol to the diet  It appeared that – together with the cholesterol – a second compound had been extracted from the food, and this compound was called the coagulation vitamin  The new vitamin received the letter K because the initial discoveries were reported in a German journal, in which it was designated as Koagulationsvitamin  Edward Adelbert Doisy of Saint Louis University did much of the research that led to the discovery of the structure and chemical nature of vitamin K [83] Dam and Doisy shared the 1943 Nobel Prize for medicine for their work on vitamin K (K1 and K2) published in 1939  Several laboratories synthesized the compound(s) in 1939 [84]
For several decades, the vitamin K-deficient chick model was the only method of quantifying vitamin K in various foods: the chicks were made vitamin K-deficient and subsequently fed with known amounts of vitamin K-containing food  The extent to which blood coagulation was restored by the diet was taken as a measure for its vitamin K content  Three groups of physicians independently found this: Biochemical Institute, University of Copenhagen (Dam and Johannes Glavind), University of Iowa Department of Pathology (Emory Warner, Kenneth Brinkhous, and Harry Pratt Smith), and the Mayo Clinic (Hugh Butt, Albert Snell, and Arnold Osterberg) [85]
The first published report of successful treatment with vitamin K of life-threatening hemorrhage in a jaundiced patient with prothrombin deficiency was made in 1938 by Smith, Warner, and Brinkhous [86]
The precise function of vitamin K was not discovered until 1974, when three laboratories (Stenflo et al ,[87] Nelsestuen et al ,[88] and Magnusson et al [89]) isolated the vitamin K-dependent coagulation factor prothrombin (factor II) from cows that received a high dose of a vitamin K antagonist, warfarin  It was shown that, while warfarin-treated cows had a form of prothrombin that contained 10 glutamate (Glu) amino acid residues near the amino terminus of this protein, the normal (untreated) cows contained 10 unusual residues that were chemically identified as γ-carboxyglutamate (Gla)  The extra carboxyl group in Gla made clear that vitamin K plays a role in a carboxylation reaction during which Glu is converted into Gla 
The biochemistry of how vitamin K is used to convert Glu to Gla has been elucidated over the past thirty years in academic laboratories throughout the world 
^ "Vitamin K Overview"  University of Maryland Medical Center  ^ a b Higdon, Jane (Feb 2008)  "Vitamin K"  Linus Pauling Institute, Oregon State University  Retrieved 12 Apr 2008  ^ Hamidi, M  S ; Gajic-Veljanoski, O ; Cheung, A  M  (2013)  "Vitamin K and bone health"  Journal of Clinical Densitometry (Review)  16 (4): 409–413  doi:10 1016/j jocd 2013 08 017  PMID 24090644  ^ Cockayne, S ; Adamson, J ; Lanham-New, S ; Shearer, M  J ; Gilbody, S; Torgerson, D  J  (Jun 2006)  "Vitamin K and the prevention of fractures: systematic review and meta-analysis of randomized controlled trials"  Archives of Internal Medicine (Review)  166 (12): 1256–1261  doi:10 1001/archinte 166 12 1256  PMID 16801507  ^ O'Keefe, J  H ; Bergman, N ; Carrera Bastos, P ; Fontes Villalba, M ; Di Nicolantonio, J  J ; Cordain, L  (2016)  "Nutritional strategies for skeletal and cardiovascular health: hard bones, soft arteries, rather than vice versa"  Open Heart (Review)  3 (1): e000325  doi:10 1136/openhrt-2015-000325  PMC 4809188  PMID 27042317  ^ Maresz, K  (Feb 2015)  "Proper Calcium Use: Vitamin K2 as a Promoter of Bone and Cardiovascular Health"  Integrative Medicine (Review)  14 (1): 34–39  PMC 4566462  PMID 26770129  ^ Hartley, L ; Clar, C ; Ghannam, O ; Flowers, N ; Stranges, S ; Rees, K  (Sep 2015)  "Vitamin K for the primary prevention of cardiovascular disease"  The Cochrane Database of Systematic Reviews (Systematic review)  9 (9): CD011148  doi:10 1002/14651858 CD011148 pub2  PMID 26389791  ^ a b Geleijnse, J  M ; Vermeer, C ; Grobbee, D  E ; Schurgers, L  J ; Knapen, M  H ; van der Meer, I  M ; Hofman, A ; Witteman, J  C  (Nov 2004)  "Dietary intake of menaquinone is associated with a reduced risk of coronary heart disease: the Rotterdam Study"  Journal of Nutrition  134 (11): 3100–3105  PMID 15514282  ^ Ades, T  B , ed  (2009)  "Vitamin K"  American Cancer Society Complete Guide to Complementary and Alternative Cancer Therapies (2nd ed )  American Cancer Society  pp  558–563  ISBN 978-0-944235-71-3  ^ Lung, D  (Dec 2015)  Tarabar, A , ed  "Rodenticide Toxicity Treatment & Management"  Medscape  WebMD  ^ Rasmussen, S  E ; Andersen, N  L ; Dragsted, L  O ; Larsen, J  C  (Mar 2006)  "A safe strategy for addition of vitamins and minerals to foods"  European Journal of Nutrition  45 (3): 123–135  doi:10 1007/s00394-005-0580-9  PMID 16200467  ^ Ushiroyama, T ; Ikeda, A ; Ueki, M (Mar 2002)  "Effect of continuous combined therapy with vitamin K2 and vitamin D3 on bone mineral density and coagulofibrinolysis function in postmenopausal women"  Maturitas  41 (3): 211–221  doi:10 1016/S0378-5122(01)00275-4  PMID 11886767  ^ Asakura, H ; Myou, S ; Ontachi, Y ; Mizutani, T ; Kato, M ; Saito, M ; Morishita, E ; Yamazaki, M ; Nakao, S  (Dec 2001)  "Vitamin K administration to elderly patients with osteoporosis induces no hemostatic activation, even in those with suspected vitamin K deficiency"  Osteoporosis International  12 (12): 996–1000  doi:10 1007/s001980170007  PMID 11846334  ^ Ronden, J  E ; Groenen-van Dooren, M  M ; Hornstra, G ; Vermeer, C  (Jul 1997)  "Modulation of arterial thrombosis tendency in rats by vitamin K and its side chains"  Atherosclerosis  132 (1): 61–67  doi:10 1016/S0021-9150(97)00087-7  PMID 9247360  ^ Ansell, J ; Hirsh, J ; Poller, L ; Bussey, H ; Jacobson, A ; Hylek, E (Sep 2004)  "The pharmacology and management of the vitamin K antagonists: the Seventh ACCP Conference on Antithrombotic and Thrombolytic Therapy"  Chest  126 (3 Suppl ): 204S–233S  doi:10 1378/chest 126 3_suppl 204S  PMID 15383473  ^ Crowther, M  A ; Douketis, J  D ; Schnurr, T ; Steidl, L ; Mera, V ; Ultori, C ; Venco, A ; Ageno, W  (Aug 2002)  "Oral vitamin K lowers the international normalized ratio more rapidly than subcutaneous vitamin K in the treatment of warfarin-associated coagulopathy  A randomized, controlled trial"  Annals of Internal Medicine  137 (4): 251–254  doi:10 7326/0003-4819-137-4-200208200-00009  PMID 12186515  ^ a b "Important Information to Know When You Are Taking: Warfarin (Coumadin) and Vitamin K" (PDF)  National Institute of Health Clinical Center Drug-Nutrient Interaction Task Force  Retrieved 17 Apr 2015  ^ "Guidelines For Warfarin Reversal With Vitamin K" (PDF)  American Society of Health-System Pharmacists  Retrieved 17 Apr 2015  ^ "Pradaxa Drug Interactions"  Pradaxapro com  19 Mar 2012  Retrieved 21 Apr 2013  ^ Bauersachs, R ; Berkowitz, S  D ; Brenner, B ; Buller, H  R ; Decousus, H ; Gallus, A  S ; Lensing, A  W ; Misselwitz, F ; Prins, M  H ; Raskob, G  E ; Segers, A ; Verhamme, P ; Wells, P ; Agnelli, G ; Bounameaux, H ; Cohen, A ; Davidson, B  L ; Piovella, F ; Schellong, S  (Dec 2010)  "Oral rivaroxaban for symptomatic venous thromboembolism"  New England Journal of Medicine  363 (26): 2499–2510  doi:10 1056/NEJMoa1007903  PMID 21128814  ^ McGee, W  (1 Feb 2007)  "Vitamin K"  MedlinePlus  Retrieved 2 Apr 2009  ^ Shearer, M  J ; Newman, P  (Oct 2008)  "Metabolism and cell biology of vitamin K"  Thrombosis and Haemostasis  100 (4): 530–547  doi:10 1160/TH08-03-0147  PMID 18841274  ^ Davidson, R  T ; Foley, A  L ; Engelke, J  A ; Suttie, J  W  (Feb 1998)  "Conversion of dietary phylloquinone to tissue menaquinone-4 in rats is not dependent on gut bacteria"  Journal of Nutrition  128 (2): 220–223  PMID 9446847  ^ Ronden, J  E ; Drittij-Reijnders, M  J ; Vermeer, C ; Thijssen, H  H  (Jan 1998)  "Intestinal flora is not an intermediate in the phylloquinone–menaquinone-4 conversion in the rat"  Biochimica et Biophysica Acta  1379 (1): 69–75  doi:10 1016/S0304-4165(97)00089-5  PMID 9468334  ^ Thijssen, H   H ; Drittij-Reijnders, M  J  (Sep 1994)  "Vitamin K distribution in rat tissues: dietary phylloquinone is a source of tissue menaquinone-4"  The British Journal of Nutrition  72 (3): 415–425  doi:10 1079/BJN19940043  PMID 7947656  ^ Will, B  H ; Usui, Y ; Suttie, J  W  (Dec 1992)  "Comparative metabolism and requirement of vitamin K in chicks and rats"  Journal of Nutrition  122 (12): 2354–2360  PMID 1453219  ^ Davidson, R  T ; Foley, A  L ; Engelke, J  A ; Suttie, J  W  (Feb 1998)  "Conversion of dietary phylloquinone to tissue menaquinone-4 in rats is not dependent on gut bacteria"  Journal of Nutrition  128 (2): 220–223  PMID 9446847  ^ Ronden, J  E ; Drittij-Reijnders, M  J ; Vermeer, C ; Thijssen, H  H  (Jan 1998)  "Intestinal flora is not an intermediate in the phylloquinone-menaquinone-4 conversion in the rat"  Biochimica et Biophysica Acta  1379 (1): 69–75  doi:10 1016/S0304-4165(97)00089-5  PMID 9468334  ^ Al Rajabi, Ala (2011)  The Enzymatic Conversion of Phylloquinone to Menaquinone-4 (PhD thesis)  Tufts University, Friedman School of Nutrition Science and Policy  ^ Furie, B ; Bouchard, B  A ; Furie, B  C  (Mar 1999)  "Vitamin K-dependent biosynthesis of gamma-carboxyglutamic acid"  Blood  93 (6): 1798–1808  PMID 10068650  ^ Mann, K  G  (Aug 1999)  "Biochemistry and physiology of blood coagulation"  Thrombosis and Haemostasis  82 (2): 165–174  PMID 10605701  ^ Price, P  A  (1988)  "Role of vitamin-K-dependent proteins in bone metabolism"  Annual Review of Nutrition  8: 565–583  doi:10 1146/annurev nu 08 070188 003025  PMID 3060178  ^ Coutu, D  L ; Wu, J  H ; Monette, A ; Rivard, G  E ; Blostein, M  D ; Galipeau, J (Jun 2008)  "Periostin, a member of a novel family of vitamin K-dependent proteins, is expressed by mesenchymal stromal cells"  Journal of Biological Chemistry  283 (26): 17991–18001  doi:10 1074/jbc M708029200  PMID 18450759  ^ Viegas, C  S ; Simes, D  C ; Laizé, V ; Williamson, M  K ; Price, P  A ; Cancela, M  L  (Dec 2008)  "Gla-rich protein (GRP), a new vitamin K-dependent protein identified from sturgeon cartilage and highly conserved in vertebrates"  Journal of Biological Chemistry



ground_truth: The paper aims to solve nonlinear system vibration problems efficiently.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are the symptoms of vitamin K deficiency?

Read the following text and answer briefly.

Context 2:  283 (52): 36655–36664  doi:10 1074/jbc M802761200  PMC 2605998  PMID 18836183  ^ Viegas, C  S ; Cavaco, S ; Neves, P  L ; Ferreira, A ; João, A ; Williamson, M  K ; Price, P  A ; Cancela, M  L ; Simes, D  C  (Dec 2009)  "Gla-rich protein is a novel vitamin K-dependent protein present in serum that accumulates at sites of pathological calcifications"  American Journal of Pathology  175 (6): 2288–2298  doi:10 2353/ajpath 2009 090474  PMC 2789615  PMID 19893032  ^ Hafizi, S ; Dahlbäck, B  (Dec 2006)  "Gas6 and protein S  Vitamin K-dependent ligands for the Axl receptor tyrosine kinase subfamily"  The FEBS Journal  273 (23): 5231–5244  doi:10 1111/j 1742-4658 2006 05529 x  PMID 17064312  ^ Kulman, J  D ; Harris, J  E ; Xie, L ; Davie, E  W  (May 2007)  "Proline-rich Gla protein 2 is a cell-surface vitamin K-dependent protein that binds to the transcriptional coactivator Yes-associated protein"  Proceedings of the National Academy of Sciences of the United States of America  104 (21): 8767–8772  doi:10 1073/pnas 0703195104  PMC 1885577  PMID 17502622  ^ "Vitamin K"  MedlinePlus  US National Library of Medicine, National Institutes of Health  Sep 2016  Retrieved 26 May 2009  ^ Conly, J; Stein, K  (Dec 1994)  "Reduction of vitamin K2 concentrations in human liver associated with the use of broad spectrum antimicrobials"  Clinical and Investigative Medicine  17 (6): 531–539  PMID 7895417  ^ Ferland, G ; Sadowski, J  A ; O'Brien, M  E  (Apr 1993)  "Dietary induced subclinical vitamin K deficiency in normal human subjects"  Journal of Clinical Investigation  91 (4): 1761–1768  doi:10 1172/JCI116386  PMC 288156  PMID 8473516  ^ Holden, R  M ; Morton, A  R ; Garland, J  S ; Pavlov, A ; Day, A  G ; Booth, S  L  (Apr 2010)  "Vitamins K and D status in stages 3-5 chronic kidney disease"  Clinical Journal of the American Society of Nephrology  5 (4): 590–597  doi:10 2215/CJN 06420909  PMC 2849681  PMID 20167683  ^ Hodges, S  J ; Pilkington, M  J ; Shearer, M  J ; Bitensky, L ; Chayen, J (Jan 1990)  "Age-related changes in the circulating levels of congeners of vitamin K2, menaquinone-7 and menaquinone-8"  Clinical Science  78 (1): 63–66  PMID 2153497  ^ "Vitamin K"  Dietary Reference Intakes for Vitamin A, Vitamin K, Arsenic, Boron, Chromium, Copper, Iodine, Iron, Manganese, Molybdenum, Nickel, Silicon, Vanadium, and Zinc (PDF)  National Academy Press  2001  p  162–196  ^ Tolerable Upper Intake Levels For Vitamins And Minerals (PDF), European Food Safety Authority, 2006 ^ a b Rhéaume-Bleue, p  42
^ "Important information to know when you are taking: Warfarin (Coumadin) and Vitamin K" (PDF)  National Institutes of Health Clinical Center  ^ "Nutrition Facts and Information for Parsley, raw"  Nutritiondata com  Retrieved 21 Apr 2013  ^ "Nutrition facts, calories in food, labels, nutritional information and analysis"  Nutritiondata com  13 Feb 2008  Retrieved 21 Apr 2013  ^ "Vitamin K"  Vivo colostate edu  2 Jul 1999  Retrieved 21 Apr 2013  ^ "Vitamin K"  Micronutrient Data Centre  ^ Ikeda, Y ; Iki, M ; Morita, A ; Kajita, E ; Kagamimori, S ; Kagawa, Y ; Yoneshima, H  (May 2006)  "Intake of fermented soybeans, natto, is associated with reduced bone loss in postmenopausal women: Japanese Population-Based Osteoporosis (JPOS) Study"  Journal of Nutrition  136 (5): 1323–1328  PMID 16614424  ^ Katsuyama, H ; Ideguchi, S ; Fukunaga, M ; Saijoh, K ; Sunami, S  (Jun 2002)  "Usual dietary intake of fermented soybeans (Natto) is associated with bone mineral density in premenopausal women"  Journal of Nutritional Science and Vitaminology  48 (3): 207–215  doi:10 3177/jnsv 48 207  PMID 12350079  ^ Sano, M ; Fujita, H ; Morita, I ; Uematsu, H ; Murota, S  (Dec 1999)  "Vitamin K2 (menatetrenone) induces iNOS in bovine vascular smooth muscle cells: no relationship between nitric oxide production and gamma-carboxylation"  Journal of Nutritional Science and Vitaminology  45 (6): 711–723  doi:10 3177/jnsv 45 711  PMID 10737225  ^ Gast, G  C ; de Roos, N  M ; Sluijs, I ; Bots, M  L ; Beulens, J  W ; Geleijnse, J  M ; Witteman, J  C ; Grobbee, D  E ; Peeters, P  H ; van der Schouw, Y  T  (Sep 2009)  "A high menaquinone intake reduces the incidence of coronary heart disease"  Nutrition, Metabolism, and Cardiovascular Diseases  19 (7): 504–510  doi:10 1016/j numecd 2008 10 004  PMID 19179058  ^ Oldenburg, J ; Bevans, C  G ; Müller, C  R ; Watzka, M  (2006)  "Vitamin K epoxide reductase complex subunit 1 (VKORC1): the key protein of the vitamin K cycle"  Antioxidants & Redox Signaling  8 (3–4): 347–353  doi:10 1089/ars 2006 8 347  PMID 16677080  ^ Suttie, J  W  (1985)  "Vitamin K-dependent carboxylase"  Annual Review of Biochemistry  54: 459–477  doi:10 1146/annurev bi 54 070185 002331  PMID 3896125  ^ Presnell, S  R ; Stafford, D  W  (Jun 2002)  "The vitamin K-dependent carboxylase"  Thrombosis and Haemostasis  87 (6): 937–946  PMID 12083499  ^ Stafford, D  W  (Aug 2005)  "The vitamin K cycle"  Journal of Thrombosis and Haemostasis  3 (8): 1873–1878  doi:10 1111/j 1538-7836 2005 01419 x  PMID 16102054  ^ Rhéaume-Bleue, p  79 
^ Whitlon, D  S ; Sadowski, J  A ; Suttie, J  W  (Apr 1978)  "Mechanism of coumarin action: significance of vitamin K epoxide reductase inhibition"  Biochemistry  17 (8): 1371–1377  doi:10 1021/bi00601a003  PMID 646989  ^ Terlau, H ; Olivera, B  M  (Jan 2004)  "Conus venoms: a rich source of novel ion channel-targeted peptides"  Physiological Reviews  84 (1): 41–68  doi:10 1152/physrev 00020 2003  PMID 14715910  ^ Buczek, O ; Bulaj, G ; Olivera, BM (Dec 2005)  "Conotoxins and the posttranslational modification of secreted gene products"  Cellular and Molecular Life Sciences  62 (24): 3067–3079  doi:10 1007/s00018-005-5283-0  PMID 16314929  ^ "Prothrombin Time"  WebMD  ^ Dituri, F ; Buonocore, G ; Pietravalle, A ; Naddeo, F ; Cortesi, M; Pasqualetti, P; Tataranno M  L ; R , Agostino (Sep 2012)  "PIVKA-II plasma levels as markers of subclinical vitamin K deficiency in term infants"  Journal of Maternal, Fetal & Neonatal Medicine  25 (9): 1660–1663  doi:10 3109/14767058 2012 657273  PMID 22280352  ^ Thane, C  W ; Bates, C  J ; Shearer, M  J ; Unadkat, N; Harrington, D  J ; Paul, A  A ; Prentice, A ; Bolton-Smith, C  (Jun 2002)  "Plasma phylloquinone (vitamin K1) concentration and its relationship to intake in a national sample of British elderly people"  British Journal of Nutrition  87 (6): 615–622  doi:10 1079/BJNBJN2002582  PMID 12067432  ^ McKeown, N  M ; Jacques, P  F ; Gundberg, C  M ; Peterson, J  W ; Tucker, K  L ; Kiel, D  P ; Wilson, P  W ; Booth, SL (Jun 2002)  "Dietary and nondietary determinants of vitamin K biochemical measures in men and women" (PDF)  Journal of Nutrition  132 (6): 1329–1334  PMID 12042454  ^ Yamano, M ; Yamanaka, Y ; Yasunaga, K ; Uchida, K  (Sep 1989)  "Effect of vitamin K deficiency on urinary gamma-carboxyglutamic acid excretion in rats"  Nihon Ketsueki Gakkai Zasshi  52 (6): 1078–1086  PMID 2588957  ^ Matsumoto, T ; Miyakawa, T ; Yamamoto, D  (Mar 2012)  "Effects of vitamin K on the morphometric and material properties of bone in the tibiae of growing rats"  Metabolism  61 (3): 407–414  doi:10 1016/j metabol 2011 07 018  PMID 21944271  ^ Je, S -H ; Joo, N -S ; Choi, B -H ; Kim, K -M ; Kim, B -T ; Park, S -B ; Cho, D -Y ; Kim, K -N ; Lee, D -J  (Aug 2011)  "Vitamin K supplement along with vitamin D and calcium reduced serum concentration of undercarboxylated osteocalcin while increasing bone mineral density in Korean postmenopausal women over sixty-years-old"  Journal of Korean Medical Science  26 (8): 1093–1098  doi:10 3346/jkms 2011 26 8 1093  PMC 3154347  PMID 21860562  ^ Bentley, R ; Meganathan, R  (Sep 1982)  "Biosynthesis of vitamin K (menaquinone) in bacteria" (PDF)  Microbiological Reviews  46 (3): 241–280  PMC 281544  PMID 6127606  ^ Haddock, B  A ; Jones, C  W  (Mar 1977)  "Bacterial respiration" (PDF)  Bacteriological Reviews  41 (1): 47–99  PMC 413996  PMID 140652  ^ Shearer, M  J  (Jan 1995)  "Vitamin K"  Lancet  345 (8944): 229–234  doi:10 1016/S0140-6736(95)90227-9  PMID 7823718  ^ Greer, J  P ; Foerster, J ; Lukens, J  N ; Rodgers, G  M ; Paraskevas, F ; Glader, B  (eds )  Wintrobe's Clinical Hematology (11th ed )  Philadelphia, Pennsylvania: Lippincott, Williams and Wilkens  ^ a b American Academy of Pediatrics Committee on Fetus Newborn  (Jul 2003)  "Controversies concerning vitamin K and the newborn  American Academy of Pediatrics Committee on Fetus and Newborn" (PDF)  Pediatrics  112 (1 1): 191–192  doi:10 1542/peds 112 1 191  PMID 12837888  ^ Logan, S ; Gilbert, R  (1998)  "Vitamin K For Newborn Babies" (PDF)  Department of Health  Retrieved 12 Oct 2014  ^ "Postnatal care: Routine postnatal care of women and their babies [CG37]"  www nice org uk  NICE  Jul 2006  Retrieved 12 Oct 2014  ^ Parker, L ; Cole, M ; Craft, A  W ; Hey, E  N  (1998)  "Neonatal vitamin K administration and childhood cancer in the north of England: retrospective case-control study"  BMJ (Clinical Research Edition)  316 (7126): 189–193  doi:10 1136/bmj 316 7126 189  PMC 2665412  PMID 9468683  ^ McMillan, D  D  (1997)  "Routine administration of vitamin K to newborns"  Paediatric Child Health  2 (6): 429–431  ^ "Newborns get rare disorder after parents refused shots"  Having four cases since February just at Vanderbilt was a little bit concerning to me ^ Dam, C  P  H  (1935)  "The Antihaemorrhagic Vitamin of the Chick: Occurrence And Chemical Nature"  Nature  135 (3417): 652–653  doi:10 1038/135652b0  ^ Dam, C  P  H  (1941)  "The discovery of vitamin K, its biological functions and therapeutical application" (PDF)  Nobel Prize Laureate Lecture  ^ McAlister, V  C  (2006)  "Control of coagulation: a gift of Canadian agriculture" (PDF)  Clinical and Investigative Medicine  29 (6): 373–377  ^ MacCorquodale, D  W ; Binkley, S  B ; Thayer, S  A ; Doisy, E  A  (1939)  "On the constitution of Vitamin K1"  Journal of the American Chemical Society  61 (7): 1928–1929  doi:10 1021/ja01876a510  ^ Fieser, L  F  (1939)  "Synthesis of Vitamin K1"  Journal of the American Chemical Society  61 (12): 3467–3475  doi:10 1021/ja01267a072  ^ Dam, C  P  H  (12 Dec 1946)  "The discovery of vitamin K, its biological functions and therapeutical application" (PDF)  Nobel Prize lecture  ^ Warner, E  D ; Brinkhous, K  M ; Smith, H  P  (1938)  "Bleeding Tendency of Obstructive Jaundice"  Proceedings of the Society of Experimental Biology and Medicine  37 (4): 628–630  doi:10 3181/00379727-37-9668P  ^ Stenflo, J; Fernlund, P ; Egan, W ; Roepstorff, P  (Jul 1974)  "Vitamin K dependent modifications of glutamic acid residues in prothrombin"  Proceedings of the National Academy of Sciences of the United States of America  71 (7): 2730–2733  doi:10 1073/pnas 71 7 2730  PMC 388542  PMID 4528109  ^ Nelsestuen, G  L ; Zytkovicz, T  H ; Howard, J  B  (Oct 1974)  "The mode of action of vitamin K  Identification of gamma-carboxyglutamic acid as a component of prothrombin" (PDF)  Journal of Biological Chemistry  249 (19): 6347–6350  PMID 4214105  ^ Magnusson, S ; Sottrup-Jensen, L ; Petersen, T  E ; Morris, H  R ; Dell, A  (Aug 1974)  "Primary structure of the vitamin K-dependent part of prothrombin"  FEBS Letters  44 (2): 189–193  doi:10 1016/0014-5793(74)80723-4  PMID 4472513  Bibliography[edit]
Rhéaume-Bleue, Kate (2012)  Vitamin K2 and the Calcium Paradox  John Wiley & Sons, Canada  ISBN 1-118-06572-7  External links[edit]
"Vitamin K: Another Reason to Eat Your Greens"  v
TPP / ThDP (B1)
FMN, FAD (B2)
NAD+, NADH, NADP+, NADPH (B3)
Coenzyme A (B5)
PLP / P5P (B6)
THFA / H4FA, DHFA / H2FA, MTHF (B9)
AdoCbl, MeCbl (B12)
Phylloquinone (K1), Menaquinone (K2)
non-vitamins
Coenzyme B
Heme / Haem (A, B, C, O)
Molybdopterin/Molybdenum cofactor
THMPT / H4MPT
Fe2+, Fe3+
vitamins: see vitamins
Antihemorrhagics (B02)
(coagulation)
Phytomenadione (K1)
Menadione (K3)
intrinsic: IX/Nonacog alfa
VIII/Moroctocog alfa/Turoctocog alfa
extrinsic: VII/Eptacog alfa
common: X
II/Thrombin
I/Fibrinogen
XIII/Catridecacog
combinations: Prothrombin complex concentrate (II, VII, IX, X, protein C and S)
Carbazochrome
thrombopoietin receptor agonist (Romiplostim
Eltrombopag)
Tetragalacturonic acid hydroxymethylester
Epinephrine/Adrenalone
amino acids (Aminocaproic acid
Aminomethylbenzoic acid)
serpins (Aprotinin
Alfa1 antitrypsin
Camostat) 
Context 3: Paper Info

Title: Compressed quantum error mitigation
Publish Date: 10 May 2023
Author List: Maurits Tepaske (from Physikalisches Institut, Universität Bonn), David Luitz (from Physikalisches Institut, Universität Bonn)

Figure

FIG 3 The out-of-time-ordered correlator C otoc i=L/2,j (t) as a function of the operator position j and time t, for the infinite temperature initial state, for a denoised second-order Trotter supercircuit with Trotter depth Mtrot = 32 and denoiser depth M = 2 We consider evolution times t = 0 5, 1,    , 5, for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarizing noise with p = 0 01 
FIG  4  The complex eigenvalues λ of the noisy second-order Trotter supercircuit with Mtrot = 16 at time t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised Trotter supercircuit (right) The Trotter circuit is for a L = 6 Heisenberg model with PBC, and all twoqubit channels are affected by depolarizing noise with p = 0 0046 The unit circle, on which unitary eigenvalues must lie, is shown in black, and the noiseless eigenvalues are shown as blue bars It is evident that the denoiser recovers all the noiseless eigenvalues from the noisy circuit 
FIG  2  The complex eigenvalues λ of the noisy second-order Trotter supercircuit with Mtrot = 16 at time t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised Trotter supercircuit (right) The Trotter circuit is for a L = 6 Heisenberg model with PBC, and all twoqubit channels are affected by depolarizing noise with p = 0 036 The unit circle, on which unitary eigenvalues must lie, is shown in black, and the noiseless eigenvalues are shown as blue bars It is clear that the denoiser recovers with high accuracy the noiseless eigenvalues from the noisy circuit 
FIG  3  The half-chain channel entanglement entropy S at different two-qubit depolarizing noise strengths p, for a secondorder Trotter supercircuit with Mtrot = 16 and t = 2, for a M = 4 denoiser The Trotter circuit is for a Heisenberg model with PBC of size L = 6 The different curves correspond to the different supercircuits, i e  the noisy supercircuit, the denoiser, the corresponding denoised supercircuit, and the noiseless variant 
FIG  4  The out-of-time-ordered correlator C otoc i=L/2,j (t) as a function of the operator position j and stacked time t, for the infinite temperature initial state, for a denoised secondorder Trotter supercircuit with Trotter depth Mtrot = 32 and denoiser depth M = 2 It is optimized at t = 2 and stacked up to ten times The calculations are for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarization with p = 0 01 The denoiser is affected by the same noise 
FIG 6 The distribution of the ZZ angle α of M = 2 denoisers (top panels) and M = 8 denoisers (bottom panels), with the lightest color corresponding to the denoiser for the Trotter supercircuit with t = 0 5, and the darkest color with t = 5 As usual, we consider the Heisenberg model on a periodic chain, and second-order Trotter supercircuits with depths Mtrot = 8, 16, 32, 64, which together with the denoiser is affected by a two-qubit depolarizing noise with p = 0 01 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively 
FIG  7  The sampling overhead γ of the optimized denoisers from Fig  2 of the main text, with denoiser depths M = 1, 2, 4, 6, 8 and Trotter depths Mtrot = 8, 16, 32, 64 at times t = 0 5, 1,    , 5, for the Heisenberg model on a chain with PBC affected by two-qubit depolarizing noise with p = 0 01 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively 
FIG 8 The domain wall magnetization Z dw after evolving a periodic density wall |dw |dw * with the denoised second-order Trotter supercircuits D C from Fig 2of the main text These supercircuits have various Trotter depths Mtrot = 8, 16, 32, 64, denoiser depths M = 1, 2, 4, 6, 8, and evolution times t = 0 5, 1,    , 5, for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarizing noise of strength p = 0 01 The denoiser is affected by the same noise The non-denoised results are labelled with M = 0 and the noiseless results with p = 0 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively We see that the denoiser allows us to recover the noiseless behavior 

abstract

We introduce a quantum error mitigation technique based on probabilistic error cancellation to eliminate errors which have accumulated during the application of a quantum circuit  Our approach is based on applying an optimal "denoiser" after the action of a noisy circuit and can be performed with an arbitrary number of extra gates 
The denoiser is given by an ensemble of circuits distributed with a quasiprobability distribution  For a simple noise model, we show that efficient, local denoisers can be found, and we demonstrate their effectiveness for the digital quantum simulation of the time evolution of simple spin chains  Introduction 
-Quantum information processing has been theoretically shown to hold great promises, and quantum algorithms were developed which can in principle achieve an exponential speed-up over their classical counterparts, both for general purpose computing and quantum simulation   However, present day quantum computing prototypes still suffer from significant noise processes which hinder the execution of many potentially groundbreaking quantum algorithms  
Nontrivial quantum algorithms typically require large sequences of quantum gates, each of which introduces dissipation and hence an overall loss of coherence, eventually rendering the results useless  Until quantum error correction becomes practical, quantum error mitigation seems to be more feasible to increase the accuracy of expectation values 
Here the goal is to induce the (partial) cancellation of errors that stem from noisy quantum gates by extending the circuit corresponding to the desired algorithm with an ensemble of gates , sampled from a quasiprobability distribution  The traditional way to accomplish this is with the gatewise method from , where noise is mitigated by inverting the noise channel of each gate separately, i e  the cancellation of errors is performed for each gate on its own 
Here the local noise channel is approximated in a way such that it can be easily inverted analytically, e g  using Pauli twirling   Gates are then sampled from the inverted noise channel by interpreting it as a quasiprobability distribution  Because in this gate-wise approach every noisy gate has to be modified separately, the sign problem is exponentially large in the number of gates, limiting the practicality of the mitigation 
The success of the gate-wise approach resulted in a large body of work concerning these methods , including extensions for simultaneous mitigation of multiple gates by Pauli-twirling entire layers or variationally constructing a mitigating matrix product operator   In principle, errors during the execution of a circuit can propagate and accumulate 
These propagated errors * david luitz@uni-bonn de ≈ C

C

FIG  1  An example of the quantum error mitigation procedure used in this work for the time evolution of the wave function of a spin chain  The ideal second-order Trotter supercircuit C of depth Mtrot = 1 (light blue) is approximated by applying a denoiser D of depth M = 1 (red) to the noisy Trotter supercircuit C (dark blue) 
Because the denoiser is applied after fully executing the noisy Trotter supercircuit, it represents an approximate inverse of the global noise channel with a precision tunable by the depth of the denoiser  can potentially blow up and lead to large errors for the circuit as a whole   Here we introduce a mitigation technique that takes into account the propagation of errors, can be performed with a tunable number of extra gates, and works for non-Clifford local noise channels since the inversion of the accumulated global noise channel is implicit 
We first execute the targeted noisy circuit completely, letting the noise propagate and accumulate, and only afterwards we apply an extra random circuit sampled from a quasiprobability distribution  We call the corresponding ensemble of random circuits a denoiser, and we construct it such that upon averaging the accumulated errors cancel 
Essentially, the denoiser inverts a global noise channel  Since we will construct it as a local brickwall circuit, following the classical preprocessing approach from , we call this compressed quantum error mitigation  Method  -Due to the inevitable coupling of a quantum processor to its environment, every qubit operation is affected by noise 
Therefore, the simplest technique to minimize the impact of the resulting noise is to minimize the number of operations when performing a quantum algorithm  In we showed that many-body time evolution operators can be efficiently compressed into brick-wall circuits with high fidelity per gate  In this Letter, we consider the noise explicitly by treating quantum operations as (generally non-unitary) quantum channels, corresponding to completely positive and trace preserving (CPTP) maps  
For example, instead of a noiseless two-qubit gate G, which acts on a quantum state |ρ in superoperator form as G|ρ = G⊗G * |ρ , we get the noisy channel G = N G, where the noise channel N implements the two-qubit noise   These channels are used to construct a "supercircuit" C = N G i=1 Gi , consisting of N G channels, which is affected by multi-qubit accumulated noise 
This supercircuit encodes an ensemble of circuits   For simplicity, we assume that the noisy channels Gi in each half brickwall layer are lattice inversion and translation invariant, such that we can construct a denoiser with these properties, limiting the number of variational parameters  The purpose of quantum error mitigation is to modify the ensemble of circuits described by C in a way that we can use it to obtain the noiseless expectation values 
In superoperator language, we do this by following the supercircuit C with a denoiser supercircuit D, such that D C is as close to the noiseless supercircuit C = C ⊗ C * as possible  Here C is the target unitary circuit  Because the noise channel N is non-unitary, hence making the supercircuit C non-unitary, we need to use a non-unitary denoiser to retrieve the unitary C 
We illustrate the mitigation procedure in Fig  , where a denoiser with one layer is used to mitigate errors for a second-order Trotter supercircuit with one layer  This circuit architecture is commonly used to simulate the time evolution of a quantum many-body system, until some time t, with controllable precision , and we will use it to benchmark the denoiser 
In practice, we cannot directly implement a supercircuit, and so we have to utilize its interpretation as an ensemble of circuits  Essentially, after executing a shot of the noisy circuit we sample the denoiser and apply it  The goal is to construct the denoiser in a way that averaging over many of its samples cancels the accumulated errors and gives us a good approximation of the noiseless expectation values 
It should be noted that our approach requires more gate applications on the quantum processor than with the gate-wise scheme, since there each sample from the mitigation quasiprobability distribution can be absorbed into the original circuit, whereas our approach increases the circuit depth  We take this into account by imposing the same noise on the denoiser 
Furthermore, within our scheme, the dimensionality of the quasiprobabilistic mitigating ensemble can be controlled, in contrast to the gate-wise approach where it is equal to the gate count  To facilitate the stochastic interpretation we parameterize each two-qubit denoiser channel G i as a sum of CPTP maps, such that we can sample the terms in this sum and execute the sampled gate on the quantum processor 
Concretely, we use a trace preserv-ing sum of a unitary and a non-unitary channel  For the unitary part we take a two-qubit unitary channel U( φ i ) = U ( φ i ) ⊗ U * ( φ i ), with U ( φ i ) a two-qubit unitary gate parameterized by φ i   For this we take the two-qubit ZZ rotation exp(−iα(σ z ⊗ σ z )) with angle α, which can be obtained from native gates on current hardware , and dress it with four general one-qubit unitaries, only two of which are independent if we want a circuit that is space inversion symmetric around every bond 
The resulting gate has 7 real parameters φ i   For the non-unitary part, which is essential because D has to cancel the non-unitary accumulated noise to obtain the noiseless unitary circuit, we use a general onequbit measurement followed by conditional preparation channel M( , with V a general one-qubit unitary and each κ i a 3-dimensional vector, resulting in a real 9-dimensional ζ i  
This yields the two-qubit correlated measurement M( With these parts we construct the parameterization with coefficients η i ∈ R that satisfy η 0 + η 1 = 1 because G i is trace preserving  Note that here the tensor product symbol corresponds to combining two one-qubit channels to make a two-qubit channel, whereas in most of the paper it is used to link the column and row indices of a density matrix 
We construct the denoiser from the noisy channels Gi = N G i   With this parameterization one denoiser channel has 17 independent real parameters, such that a denoiser of depth M , i e  consisting of M brickwall layers, has 34M real parameters (we use one unique channel per half brickwall layer)  For reference, a general channel has 544M parameters 
To determine the mitigated expectation values we use the full expression where |ρ 0 is the initial state and |1 is the vectorized identity operator on the full Hilbert space  To evaluate this on a quantum processor, we use the stochastic interpretation of (1) to resample   In particular, from each channel (1) we get a unitary with probability p 0 = |η 0 |/γ and a measurement followed by conditional preparation with probability p 1 = |η 1 |/γ 
Here γ = |η 0 | + |η 1 | is the sampling overhead, which characterizes the magnitude of the sign problem from negative η i   For quasiprobability distributions, i e  with γ > 1, every denoiser sample has an extra sign sgn(η) = N G g=1 sgn(η g ), 2  The normalized distance between the denoised Trotter supercircuit D C and the noiseless Trotter supercircuit C (top panels), at evolution times t = 0 5, 1,    , 5, and the twopoint z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t (bottom panels), for the infinite temperature initial state 
We consider denoisers with depths M = 1, 2, 4, 6, 8 and second-order Trotter circuits with depths Mtrot = 16, 32, 64  In the top panels we use a Heisenberg chain with L = 8, and in the bottom panels with L = 14, both with periodic boundary conditions  All gates are affected by two-qubit depolarizing noise with p = 0 01



ground_truth: Symptoms of vitamin K deficiency include anemia, bruising, nosebleeds, bleeding of the gums, and heavy menstrual bleeding in women.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Can the denoiser be applied to circuits with non-Clifford noise?

Read the following text and answer briefly.

Context 3: 
The non-denoised results are labelled with M = 0, and the noiseless values with p = 0  where sgn(η g ) is the sign of the sampled coefficient of the gth channel  γ = 1 means that all signs are positive  Observables Ô p=0 for the noiseless circuit are then approximated by resampling the observables from the denoiser ensemble
where γ = N G g=1 γ g is the overall sampling overhead, with γ g the overhead of the gth gate  Clearly, a large γ implies a large variance of Ô p=0 for a given number of samples, with accurate estimation requiring the cancellation of large signed terms  The number of samples required to resolve this cancellation of signs is bounded by Hoeffding's inequality, which states that a sufficient number of samples to estimate Ô p=0 with error δ at probability 1 − ω is bounded by (2γ 2 /δ 2 ) ln(2/ω)  
Since γ scales exponentially in γ g , it is clear that a denoiser with large M and γ 1 will require many samples  We observed that decompositions with γ > 1 are crucial for an accurate denoiser  Restricting to γ = 1 leads to large infidelity and no improvement upon increasing the number of terms in or the depth M of the denoiser 
Simply put, probabilistic error cancellation of gate noise introduces a sign problem and it is crucial to find optimal parameterizations (1) which minimize γ to make the approach scalable  This issue arises in all high performance error mitigation schemes , because the inverse of a physical noise channel is unphysical and cannot be represented as a positive sum over CPTP maps 
This is clearly visible in the spectra of the denoiser, which lies outside the unit circle (cf  Fig  )  This makes the tunability of the number of gates in each denoiser sample a crucial ingredient, which allows control over the sign problem, because we can freely choose the η i in   For the parametrization (1) of denoiser channels, we try to find a set of parameters for error mitigation by minimizing the normalized Frobenius distance between the noiseless and denoised supercircuits
which bounds the distance of output density matrices and becomes zero for perfect denoising  We carry out the minimization of on a classical processor, using gradient descent with the differential programming algorithm from   Instead of explicitly calculating the accumulated global noise channel and subsequently inverting it, we approximate the noiseless supercircuit C with the denoised supercircuit D C, effectively yielding a circuit representation D of the inverse noise channel 
Results  -To benchmark the denoiser we apply it to the second-order Trotter circuits of the spin-1/2 Heisenberg chain with periodic boundary conditions (PBC) where is the Pauli algebra acting on the local Hilbert space of site i  A second-order Trotter circuit for evolution time t with depth M trot consists of M trot − 1 half brickwall layers with time step t/M trot and two layers with half time step  
We consider circuits that are affected by uniform depolarizing noise with probability p for simplicity, but our approach can be used for any non-Clifford noise  The two-qubit noise channel is which acts on neighboring qubits i and i + 1 and is applied to each Trotter and denoiser gate, and p = 0 01 unless stated otherwise 
We study circuits with depths M trot = 16, 32, 64 for evolution times t = 0 5, 1,    , 5, and denoisers D with depths M = 1, 2, 4, 6, 8  In the top panels of Fig  we show (4) for a chain of size L = 8 as a function of time t  Here it can be seen that even for M trot = 32 a denoiser with M = 1 already improves by roughly an order of magnitude at all considered t 
Depending on M trot and t, further increasing M lowers , with the biggest improvements occurring for high precision Trotter circuits with large depth M trot = 64 and short time t = 0 5, where the Trotter gates are closer to the identity than in the other cases  At the other extreme, for M trot = 16 the improvements are relatively small upon increasing M > 2  In all cases the denoiser works better at early times than at late times, again indicating that it is easier to denoise Trotter gates that are relatively close to the identity 
To probe the accuracy of the denoiser on quantities that do not enter the optimization, as a first test we consider the two-point correlator between spins at different times where we have chosen the infinite temperature initial state, and C(t) is the Trotter supercircuit for time t  In the bottom panels of Fig  we show C zz i=L/2,j=L/2 (t) for the supercircuits from the upper panels, now for a L = 14 chain 
Here we see that at M trot = 16 we can retrieve the noiseless values already with M = 1, but that increasing M trot makes this more difficult  At M trot = 64 we see larger deviations, and improvement upon increasing M is less stable, but nonetheless we are able to mitigate errors to a large extent  As a further test, we compute the out-of-time-ordered correlator (OTOC) ]
In Fig  we show the results for i = L/2, for a Trotter circuit with depth M trot = 32 and a denoiser with depth M = 2  Here we see that a denoiser with M M trot is able to recover the light-cone of correlations, which are otherwise buried by the noise  In the Supplementary Material we consider how the denoiser performs at different noise levels p, and how the denoised supercircuits perform under stacking 
There we also calculate domain wall magnetization dynamics, and show the distribution of the optimized denoiser parameters and the sampling overhead associated to the denoiser as a whole  In Fig  we show the eigenvalues of the noisy supercircuits for a noisy second-order Trotter supercircuit with M trot = 16 at t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised supercircuit (right) 
The eigenvalues λ of a unitary supercircuit lie on the unit circle, and in the presence of dissipation they are pushed to the center  We see that the spectrum of the denoiser lies outside the unit circle, making it an unphysical channel which cures the effect of the noise on the circuit, such that the spectrum of the denoised circuit is pushed back to the unit circle 
The noiseless eigenvalues are shown as blue bars, making it clear that the denoiser is able to recover the noiseless eigenvalues from the noisy circuit  In the Supplementary Material we show the spectra for a p = 0 036 denoiser, where we observe a clustering of eigenvalues reminiscent of Refs    There we also investigate the channel entropy of the various supercircuits  
Conclusion  -We have introduced a probabilistic error cancellation scheme, where a classically determined denoiser mitigates the accumulated noise of a (generally non-Clifford) local noise channel  The required number of mitigation gates, i e  the dimensionality of the corresponding quasiprobability distribution, is tunable and the parameterization of the corresponding channels provides control over the sign problem that is inherent to probabilistic error cancellation 
We have shown that a denoiser with one layer can already significantly mitigate errors for second-order Trotter circuits with up to 64 layers  This effectiveness of low-depth compressed circuits for denoising, in contrast with the noiseless time evolution operator compression from , can be understood from the non-unitarity of the denoiser channels 
In particu-lar, measurements can have non-local effects, since the measurement of a single qubit can reduce some highly entangled state (e g  a GHZ state) to a product state, whereas in unitary circuits the spreading of correlations forms a light-cone  To optimize a denoiser with convenience at L > 8, the optimization can be formulated in terms of matrix product operators or channels , which is convenient because the circuit calculations leading to the normalized distance and its gradient are easily formulated in terms of tensor contractions and singular value decompositions  
This provides one route to a practical denoiser, which is relevant because the targeted noiseless circuit and the accompanying noisy variant in (4) need to be simulated classically, confining the optimization procedure to limited system sizes with an exact treatment or limited entanglement with tensor networks 
Nonetheless, we can use e g  matrix product operators to calculate (4) for some relatively small t, such that the noiseless and denoised supercircuits in (4) have relatively small entanglement, and then stack the final denoised supercircuit on a quantum processor to generate classically intractable states 
Analogously, we can optimize the channels exactly at some classically tractable size and then execute them on a quantum processor with larger size  Both approaches are limited by the light-cone of many-body correlations, as visualized in Fig  , because finite-size effects appear when the light-cone width becomes comparable with system size 
1  The normalized distance (left) and z spin correlator C zz i=L/2,j=L/2 (right), for a second-order Trotter supercircuit of depth Mtrot = 16 for time t = 1, affected by various twoqubit depolarizing errors p  We compare the values obtained with and without a denoiser, i e  M > 0 and M = 0, to the noiseless values (p = 0) 
The denoiser is affected by the same noise as the Trotter circuit  We consider denoisers with depths M = 1, 2, 4, 6, 8, and we use a L = 8 Heisenberg chain with PBC for the normalized distance, while for the correlator we use L = 14  * david luitz@uni-bonn de to observe that even for larger noise strength p, the local observable C zz improves significantly even with denoisers of depth M = 1 
For large noise strengths, we generally see that the optimization of the denoiser becomes difficult, leading to nonmonotonic behavior as a function of p, presumably because we do not find the global optimum of the denoiser  It is interesting to analyze the spectra of the supercircuits considered in this work 
As mentioned in the main text, the spectrum of the ideal, unitary supercircuit C lies on the unit circle  The comparison to this case is therefore instructive  In the main text, we showed an example of the spectra in Fig  for moderate noise strength  Here, we show additional data for stronger noise p = 0 036 in Fig  for a denoiser with M = 4 layers, optimized to mitigate errors for a second-order Trotter supercircuit with M trot = 16 layers at time t = 1 
The eigenvalues λ of the noisy supercircuit C are clustered close to zero, far away from the unit circle (except for λ = 1), showing that the circuit is strongly affected by the noise  To mitigate the impact of the noise, the denoiser consequently has to renormalize the spectrum strongly  If it accurately represents the inverse of the global noise channel, its spectrum has to lie far outside the unit circle, which is the case 
Interestingly, we observe a clustering of eigenvalues which is reminiscent to the spectra found in   By comparison to these works, we suspect that this is due to the local nature of the denoiser, and warrants further investigation  The right panel of Fig  shows the result of the denoiser, pushing the eigenvalues back to the unit circle, nearly with the exact same distribution along the circle as the noiseless eigenvalues (blue bars) 
Due to the strong noise, this is not achieved perfectly, and it is clear that this cannot work in principle if the global noise channel has a zero eigenvalue  The complexity of an operator can be quantified by its operator entanglement entropy   Here we calculate the half-chain channel entanglement entropy S of the noiseless C, noisy C, denoiser D, and denoised D C supercircuits 
We define S as the entanglement entropy of the state that is related to a supercircuit C via the Choi-Jamio lkowski isomorphism, i e  ψ C = χ C /N , where the process matrix χ ab,cd C = C ac,bd is simply a reshaped supercircuit and N ensures normalization  Then we have S = −Tr [ψ C ln ψ C ]  This entropy measure is a particular instance of the "exchange entropy", which characterizes the information exchange between a quantum system and its environment  
In Fig  we plot the various S for a second-order Trotter circuit with M trot = 16 at t = 2, for a denoiser with M = 4, both affected by two-qubit depolarizing noise with p ∈ [10 −3 , 10 −1 ]  The Trotter circuit is for a Heisenberg model with L = 6 and PBC  We see that at large p, the noise destroys entanglement in the noisy supercircuit, and that the denoiser S increases to correct for this, such that the denoised supercircuit recovers the noiseless S 
Here we investigate how denoised supercircuits perform upon repeated application  We optimize the denoiser for a Trotter supercircuit for a fixed evolution time t  Then, to reach later times, we stack the denoised supercircuit n times to approximate the evolution up to time nt: In Fig  we stack a denoised t = 1 supercircuit up to n = 20 times and calculate the correlation function, defined in the main text, for the middle site 
We consider Trotter depths M trot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8, for a L = 14 Heisenberg chain with p = 0 01 depolarizing two-qubit noise  The noisy results correspond to M = 0 and the noiseless results to p = 0  In Fig  we calculate the OTOC, defined in the main text, with stacked time evolution for a denoised t = 2 supercircuit with M trot = 32 and M = 2, stacked up to ten times 
We see that the stacked supercircuit performs very well, and the additional precision obtained by using deep denoisers (M = 8) pays off for long evolution times, where we see convergence to the exact result (black dashed lines in Fig  ) as a function of M   FIG    The two-point z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t, for the infinite temperature initial state, for denoised second-order Trotter supercircuits that are optimized at evolution time t = 1 and then stacked up to twenty times 
We use Trotter depths Mtrot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8  The calculations were performed for a periodic Heisenberg model with L = 14 and PBC, affected by two-qubit depolarizing noise with strength p = 0 01, which also affects the denoiser  The non-denoised results are labelled with M = 0, and the noiseless results with p = 0 
The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively  The costliest and most noise-susceptible operation is the two-qubit ZZ rotation with angle α, which is the foundation of the unitary piece in our channel parameterization, defined in the main text 
For completeness, we here present the α angles of the optimized denoisers  The results are shown in Fig  , which contains histograms for the channel count N G versus α  The histograms are stacked, with the lightest color corresponding to the angles of the denoiser at t = 0 5 and the darkest at t = 5  The top four panels are for a denoiser with M = 2 and the bottom four with M = 8 
We consider M trot = 8, 16, 32, 64  We see that in both cases the distribution widens upon increasing M trot , indicating that the unitary channels start deviating more from the identity  Moreover, while the M = 2 denoisers in all cases except M trot = 64 have ZZ contributions close to the identity, this is clearly not the case for M = 8 
For simplicity, we did not focus on obtaining denoisers with the smallest sampling overhead γ, which is required to minimize the sign problem and hence ease the sampling of mitigated quantities  Instead, we let the optimization freely choose the η i in the denoiser parameterization, as defined in the main text 
In Fig  we show the sampling overhead of the denoisers from Fig  of the main text  We see that for M = 1 and M = 2 the sampling overhead is relatively small and uniform across the different t, whereas for M > 2 the optimization sometimes yields a denoiser with large γ and other times with small γ  This could be related to the difference in α distributions from Fig   
The large fluctuations of γ appears to stem from the difficulty in finding optimal deep denoisers, and our optimization procedure likely only finds a local minimum in these cases  Here C(t) is the Trotter supercircuit for time t  In Fig  we show Z dw for the circuits from Fig 



ground_truth: Yes, the denoiser works for non-Clifford local noise channels.