noise induced resonances : coherence resonance ( cr ) and stochastic resonance ( sr ) in a threshold or excitable systems , have been studied both numerically and experimentally in many physical , chemical , biological and electronics systems  .the regularity of the dynamical behavior of an excitable system emerges by virtue of the interplay between the autonomous nonlinear dynamics and the optimum superimposed noise that is termed as cr . whereas , in sr , the response of the system to a weak periodic input signal is amplified or optimized by the presence of a particular level of noise .recently , another type of cr has been observed in plasma and electronic systems , where the coherency remains almost constant even at higher noise level  .this cr phenomenon of constant coherency may be termed as constant coherence resonance ( ccr ) .the main difference between cr and ccr is that , in cr autonomous dynamics is destroyed at higher noise level , whereas in ccr , system dynamics remains unaffected from the destructive effect of noise .one of the experimentally observed feature of the ccr is that the frequency of coherent oscillations increases or decreases with increase in noise level depending on autonomous dynamics of the system .so far there is no theoretical explanation of the ccr and frequency shift .now the qualitative features of an excitable system can be obtained using two nonlinear autonomous differential equations which have been used for modeling the noise induced resonances  . in the existing literatures ,the cr phenomenon has been modeled just by adding noise to any one of the above equations , where maximum coherency has appeared at optimum noise level  .but in this approach , one can not explain the appearance of ccr .even explanation of sr is not very clear , though there are several experiments on this phenomenon  .the main problem of adding noise directly to the equations is that the noise not only perturbs the system control parameter , but also destroys the actual dynamics .the effect of adding noise directly to the equations has been discussed in the context of the fitzhugh - nagumo model in section [ section : result ] . in this paper ,our goal is to present a general model which explains cr , sr and ccr .the model has been developed based on the excitation mechanism of the system using fitzhugh - nagumo model , a well known paradigm for modeling noise induced resonances .rest of the paper has been organized as follows : we have discussed the excitation mechanism of an excitable system and effect of noise in section [ section : excitable ] . based on the physical arguments of section [ section : excitable ] ,we have formulated the model for the noise invoked resonances in section [ section : model ] .the results and discussion of the simulation using fitzhugh - nagumo has been presented in section [ section : result ] . finally a conclusion has been drawn in section [ section : conclusion ] .the basic characteristic of an excitable system is that it shows a fixed point or coherent limit cycle oscillations depending on the value of the control parameter ( cp ) of the dynamics .the point where the system changes from oscillatory to fixed point behavior is called threshold or bifurcation point .now if the system shows fixed point behavior for the value of cp below the threshold , then the dynamics will be limit cycle oscillations on the other side of the threshold or vice versa , depending upon the system properties .the frequency of the autonomous oscillations in the excited state changes with control parameter .for example , frequency of the oscillations in fitzhugh - nagumo model increases when one increases the control parameter from the threshold value .the increase in frequency of the coherent oscillations has also been observed in real experiments  .this has important effect on the increase in the frequency of the coherent oscillations in ccr with increase in noise level , that has been discussed at the last of next paragraph .response of an excitable system at fixed point state to an external perturbation applied on the cp depends on the perturbation amplitude .when the amplitude is small such that the threshold is not crossed , the system remains at its fixed point state , and when it is large enough to cross the threshold , the system returns to its fixed point deterministically , i.e. , once the threshold is crossed , the system becomes almost independent of the perturbation and comes to its fixed point state traversing one limit cycle  .  for stochastic perturbation ( i.e. , noise ) , limit cycles will appear randomly due to random crossing of the threshold . the time between appearance of the consecutive limit cyclescan be splitted into two characteristic times : an activation time ( ) through which system resides at the fixed point state between two consecutive limit cycles , plus a refractory time ( ) which is taken to traverse one limit cycle .activation time ( ) will decrease gradually with noise , as the threshold will be crossed more frequently at higher noise amplitude .when the noise amplitude is fairly large and its changing frequency is higher than refractory frequency ( which is generally the case in experiments ) , then the threshold will be crossed several times during one limit cycle that makes and the inter - peak distances are practically determined by the time period ( ) of the limit cycle . at this stage ,the dynamics is almost similar to the coherent oscillations at the excited state and is termed as coherence resonance ( cr ) . from this stage ,increase in the noise may lead to two possible effects : firstly , it destroys the structures of the limit cycles or the actual dynamics of the system , that leads to decrease in the coherency , which has been observed in many experiments  .secondly , it perturbs the cp across the threshold so frequently that the system remains at its excited state and produces almost coherent oscillations , i.e. , constant coherence resonance ( ccr ) .ccr is not always realizable in the real experiments , as at higher noise level it is difficult to keep the noise away from its destructive effect .ccr has been observed in some recent experiments  .another interesting experimental observation regarding the ccr is that the frequency of the noise invoked oscillations increases with increase in noise level in those excitable systems whose frequency of the autonomous limit cycle oscillations increases with increase in cp from the threshold  .now we can summarize the effects of noise on excitable system : ( a ) noise helps the system to cross the threshold by perturbing the cp ; ( b ) it may destroy the system dynamics at higher noise level ; ( c ) if the destructive effect of noise is blocked , the system may remain at excited state and hence may show ccr ; ( d ) in case of ccr noise is also responsible for the effective increase in cp above the threshold .all these facts have been used to formulate a general model which has been described in the next section .the dynamical state of an excitable system can be represented by a two dimensional state vector and the equation of motion in general form can be described by two first order differential equations , whose fixed point is given by  , where , is suitable nonlinear function and is the control parameter . let be the bifurcation point ( or threshold ) of the system at which it goes from the fixed point to the oscillatory state .now , the value of is so chosen that the system remains in the excitable fixed point region .then the dynamical behaviour of the above system with gaussian noise ( ) , can be written as   where , represents the amplitude of the noise of standard deviation d. here the simulation has not been carried out just adding random noise at each step of integration as was done in the previous works . the integration has been carried out based on the physics of an excitable system discussed in the previous section . whenever cp crosses the threshold due to noise perturbation , system traverses one limit cycle .now once the threshold is crossed , the noise remains ineffective in the dynamics until the system comes to its fixed point state , even if the cp crosses the threshold several times during this refractory phase , i.e. , at the time of traversing one limit cycle  .this indicates that the noise need not be added at every step of integration during the simulation .hence cp should be kept constant up to one limit cycle during the integration .this is also clear from the points ( a ) and ( c ) discussed in the previous section [ section [ section : excitable ] ] .the possible value of the cp for each limit cycle integration can be guessed from the point ( d ) [ section [ section : excitable ] ] .if the noise would only lift the system at its excited state , then the frequency of the noise invoked dynamics would not change with increase in the noise level .but from the experimental results , it is clear that the frequency of the oscillations increases with increase in the noise level .so only possible reason for increase in the frequency is due to the increase in the effective value of cp .hence , one can guess that the effective value of cp at the start of the limit cycle can be taken to be its value at fixed point state plus the noise amplitude at the time of excitation ( which remains constant during one limit cycle ) .now for eqns .[ eqn:2 ] , whenever , the condition is satisfied , the system will be lifted to its excited state and will remain constant during each limit cycle .during this time the system does not feel the effect of noise , i.e. , it remains independent of noise perturbation .when it comes to its rest state , it feels again the noise perturbation and whenever , the condition is satisfied , the system traverse one limit cycle with . by imposing this condition , noise has been blocked from its destructive effect on the system . in some cases like fitzhugh - nagumo model ,the system shows oscillatory behaviour when . in this casethe condition to be satisfied by noise is which will be satisfied by the negative amplitudes of the noise .but the main mechanism is same for both the cases .there may be several choice of [ eqn .[ eqn:2 ] ] so that it shows an excitable dynamics  . in the present paper ,the famous two dimensional fitz hugh - nagumo model for excitable system has been studied , whose equations of motion are    ) from fft ( solid line ) and from inverse of the refractory time ( dashed line).,width=321 ]  where , and the control parameter ` ' govern the dynamics . for and ,the system [ eqn [ eqn : fitz ] ] shows fixed point and limit cycle oscillations respectively .therefore , is the threshold .hence , the dynamics is fixed point and oscillatory above and below this point respectively .frequency of the limit cycle oscillations increases between and then decreases between . in fig[ fig : freq_param ] , the solid and dashed lines show increase in the frequency obtained using fft and of the oscillations for parameter respectively and both estimations are identical .it also shows that the frequency of the autonomous dynamics increases with increase in cp below the threshold . in the next subsectionwe will see that the increase in frequency of the coherent oscillation in ccr follow the trend due to noise .to study the noise invoked dynamics , ` ' is set to so that eqn [ eqn : fitz ] shows fixed point behavior and is perturbed by noise ( ) of standard deviation d. whenever , is satisfied , the system will traverse one limit cycle before coming to its fixed point state .the refractory time will depend upon the parameter value as shown in fig [ fig : freq_param ] . for low level of noise, the limit cycle appears sparsely and increases with increase in noise level .the regularity of the appearance of these oscillations is determined by coherence parameter , where is the standard deviations of the interpeak distances ( ) of the limit cycles  .least indicates high degree of coherency in the system output .   of the limit cycles when the destructive effect of noise is blocked .the solid line shows the nv computed after interfering the output with applied noise.,width=321 ]  fig [ fig : nv ] shows vs d plot for two different cases . the dashed line [ fig [ eqn : fitz ] ] shows the , when the simulation was carried out blocking the destructive effect of noise using conditions given in section [ section : model ] [ eqn .[ eqn:2 ] ] .it shows that initially , decreases rapidly with noise lavel and after reaching the minima it remains constant .as low value of indicates coherent oscillations , the system remains in its coherent state even at higher noise level .this phenomenon may be termed as constant coherence resonance ( ccr ) as coherence remains unaltered at higher noise level .though ccr has been rarely observed in experiments , it is one of the important noise induced phenomena as both cr and sr can be obtained from the basic mechanism of ccr .when the output of ccr was directly interfered with noise , the conventional cr has been obtained in the sense that it has unique minimum for optimum noise level .nv decreases for small level of noise and then increases with increase in noise level as shown by solid line in fig [ fig : nv ] .minimum of the curve ( solid line ) represents occurrence of maximum coherence for optimum noise level . at higher level of noise nv increases due to the destructive effect of noise . foreven larger noise the output is totally governed by noise and the jumps across the threshold are so frequent that the nv decreases again which has also been found in noise induced resonances in delayed feedback system  .    in order to study the frequency change of the autonomous dynamics in ccr with noise level , we have estimated frequency using different techniques . the frequency obtained from the peak to peak distances , fast fourier transform ( fft ) and refractory time with noise has been shown in fig [ fig : freq ] by using dotted , solid and dashed lines respectively .though at small noise level , frequency obtained using above three techniques shows some difference , at higher noise level they are almost identical . from figs [ fig : freq_param ] and [ fig : freq ] , it is clear that the trend in increase in frequency due to noise and due to increase in the cp are almost identical .this may be due to increase in effective value of cp by noise lavel .  the experimental results consistent with the above numerical results have been observed in the unijunction transistor relaxation oscillator ( ujt - ro ) and plasma  . in ujt - ro, ccr appeared due to the blocking of noise from its destructive effect by ujt itself .detail of the experiment will be found in ref  . whereas , in plasma , sheath was found to be a natural blocker of noise  . in these experiments ,the frequency of the oscillations was also observed to increase with the noise . as the frequency of the autonomous dynamics in these cases increased with increase in cp, one can assume that the effective increase in cp was due to noise and this is consistent with numerical simulation .earlier , in order to show cr in fitzhugh - nagumo model , noise was added to the second equation [ of the eqns [ eqn : fitz ] ] which contains the control parameter and role of noise was interpreted as an irregular modulation of the bifurcation parameter that switches the limit cycle on and off . but adding noise to the first equation also produces cr [ eqns [ eqn : fitz ] ] , where switching mechanism can not be explained  . in our opinion , adding noise to any of the two equations basically destroys the actual dynamics and hence one does not get ccr . using our approachwe have got rid of this problem .   in order to get stochastic resonance ( sr ) in the fitz hugh - nagumo model, the cp has been perturbed by both a subthreshold periodic signal and noise .the main mechanism of the sr in an excitable system is that when a subthreshold signal is applied along with noise , system crosses the threshold at a place of occurrence of the peak of the subthreshold signal more frequently and hence system traverses one limit cycles at this place , i.e. , the system output mimics the subthreshold signal  . here, the important point is that the structure of the limit cycles is determined by the system dynamics and the time intervals between them are determined by interpeak distances of the applied subthreshold signal . to study sr , ` ' has been set to so that it shows fixed point behavior .a subthreshold pulse of the frequency , duration and amplitude ( a ) 0.1 , 0.05 and -0.21 respectively , is applied along with noise of standard deviation d. whenever , is achieved , the system will traverse one limit cycle before coming to its fixed point state .the simulation is identical as for the ccr as described above except that the cp is perturbed with a subthreshold signal and noise . for low level of noise , correspondence between output and the subthreshold signalis very little , and for optimum noise level the correspondence is excellent and at higher level of noise the system response is dominated by noise .the sr has been quantified by two stochastic parameters absolute mean difference ( amd )  and correlation coefficient(c )  .amd is defined as  ] , where , x and y are the subthreshold and output signal respectively . for sr ,amd and c should have minimum and maximum respectively .fig [ fig : corr](a ) and ( b ) shows c and amd for an applied subthreshold signal of frequency 0.1 .minimum in amd with noise and maximum in c is a typical signature of sr .the minimum corresponds to optimum noise level at which output mimics the subthreshold signal in the maximal manner .the experimental results consistent with the above numerical analysis have been found in the plasma and electro chemical system and details will be found in ref .here , we have presented an alternative modeling techniques to explain a new noise induced phenomenon ccr in an excitable system .this also explains other two noise induced resonance phenomena cr and sr .the excitation mechanism which was not clear in the earlier approach is now very much clear .though sr and cr have been observed in many excitable system , there are a very few experiments on ccr .possible reason for not getting ccr in the earlier experiments was that the system configurations were unable to block the destructive effect of noise .we hope that the experimental observation of ccr is possible in the earlier experiments , if the noise is allowed such that it only perturbs control parameter by choosing suitable parameters and system configurations .i would like to acknowledge support of amit apte at tifr and some useful discussions with a.n .sekar iyengar .i acknowledge the use of peak detection software written by eli billauer ( ) .i also aknowledge the help of a. p. choudhury to prepare the manuscript .

Abstract: recently , it is observed [ md . nurujjaman et al , phy . rev . e * 80 * , 015201 ( r ) ( 2009 ) ] that in an excitable system , one can maintain noise induced coherency in the coherence resonance by blocking the destructive effect of the noise on the system at higher noise level . this phenomenon of constant coherence resonance ( ccr ) can not be explained by the existing way of simulation of the model equations of an excitable system with added noise . in this paper , we have proposed a general model which explains the noise induced resonance phenomenon ccr as well as coherence resonance ( cr ) and stochastic resonance ( sr ) . the simulation has been carried out considering the basic mechanism of noise induced resonance phenomena : noise only perturbs the system control parameter to excite coherent oscillations , taking proper precautions so that the destructive effect of noise does not affect the system . in this approach , the cr has been obtained from the interference between the system output and noise , and the sr has been obtained by adding noise and a subthreshold signal . this also explains the observation of the frequency shift of coherent oscillations in the ccr with noise level .


the use of non - polynomial basis functions in the context of finite element methods dates back at least to the early 1980s . in recent years, there has been much interest in using non - polynomial basis functions to discretize wave propagation problems in the frequency domain .a prominent example is the use of plane wave bases to solve the helmholtz equation at high frequencies .the motivation behind this approach is to reduce the number of degrees of freedom per wavelength required to obtain accurate results .thus obtained trefftz methods have proved very successful in practice , hence it is a natural question to ask whether they can be extended and whether they can be equally successful in the time - domain .the most natural way of including space - time trefftz basis functions is within the confines of a space - time discontinuous galerkin ( dg ) method . in this work ,we construct and analyse a new space - time interior penalty dg method for the second order wave equation , that can utilize trefftz polynomials as local basis functions .the method discretizes the wave equation in primal form and is defined using space - time slabs to ensure solvability on each time - step , as well as to aid the presentation and the analysis. however , with minor modifications , completely unstructured space - time meshes are , in principle , possible in the proposed space - time dg framework .this results in a stable , dissipative scheme for general polynomial bases . for trefftz basis , we prove quasi - optimality in the dg energy norm , which we show to be an upper bound for a standard space - time energy norm .numerical results in the dg norm show that the theoretical estimates of the convergence order are optimal .furthermore , the numerical results show that higher order schemes have excellent energy conservation properties and work well for systems with energy at high frequencies .comparison with standard polynomial bases show that the same convergence order and approximation properties is obtained with considerably fewer degrees of freedom . furthermore , the implementation is less expensive due to integration being restricted to the space - time skeleton .space - time variational methods for wave - type problems have appeared in the late 1980 s with the works of hughes and hulbert and johnson .first numerical experiments with trefftz space - time dg methods were performed in .currently there is significant activity on the topic . in particular in trefftz space - time local dg method for the maxwell equations , written as a first order system , resulting in a two - field formulation , has been analysed .the dg method designed and analyzed in the present work involves interior - penalty - type numerical fluxes in space , resulting in one - field approximation in space , as well as a one - field dg method in time , with very similar computational stencil widths compared to the local dg method introduced in .typically , finite element methods for linear ( and some spatially nonlinear ) wave problems are based on a continuous or discontinuous finite element discretisation of the spatial variables complemented with standard time - stepping schemes , the most popular of which are explicit , such as the leap - frog scheme , due to the acceptable cfl restrictions .though even the low order methods can be conservative , for acceptable accuracy when energies at high frequencies are excited , higher order methods are essential .compared to these methods , introduction of higher order approximations is much more straightforward in the context of space - time dg methods .moreover , space - time dg methods , such as the one presented below , do _ not _ require any cfl - type restrictions , owing to their implicit time - stepping interpretation . if a general space - time mesh can be used , a judicious choice of the mesh can result in a quasi - explicit method where only small local systems need to be solved .the remainder of this work is structured as follows . in the next section ,we introduce the model problem . in sections [ sec :space ] and [ sec : method ] we construct the space - time interior penalty method and we show its stability .we proceed in section [ sec : tspace ] to analyse polynomial trefftz spaces and prove quasi - optimality . finally in section [ apriori ] ,we prove convergence rates for the -dimensional method in space , ; moreover , we also provide -version a priori bounds for .a series of numerical experiments in section [ sec : numerics ] , illustrates the theoretical findings and highlights the good performance of the proposed method in practice .we consider the wave equation , \\u & = 0 & & \text{on } \partial\omega \times [ 0,t],\\  u(x,0 ) = u_0(x ) , \ ;  \dot u(x,0 ) & = v_0(x ) , & & \text{in } \omega , \end{aligned}\ ] ] where is a bounded lipschitz domain in , its boundary and a piecewise constant function .if and are two subsets of with the boundary separating them and with in and in , then if we denote by and we further have the transmission conditions where is the exterior normal to ( or ) .we denote by , , the standard lebesgue spaces , , , with corresponding norms ; the norm of will be denoted by .further , denotes the standard -inner product on ; when the arguments are vectors of -functions , the -inner product is modified in the standard fashion .we denote by the standard hilbertian sobolev space of index of real - valued functions defined on ; in particular signifies the space of functions in whose traces onto the boundary vanish . for ,we denote the standard bochner spaces by , with being a banach space with norm .finally , we denote by the space of continuous functions \to x ] , will be denoted by ) \;:\ ; u|_{\omega \times i_n } \in { s_{n}^{h , p } } , \ ;n  = 0,1,\dots , n-1\}.\ ] ]  we require some notation .the _ skeleton _ of the mesh is defined by and the interior skeleton by .moreover , we define the union of two skeletons of two subsequent meshes by ; see figure [ figure : mesh ] .the numerical method presented below naturally admits variable spatial meshes from one space - time slab to the next .this flexibility will be crucial for future developments in the context of space - time adaptive dg methods , aiming to use local spatio - temporal resolution to resolve sharp moving fronts .if the spatial mesh remains fixed , all the formulas hold with for all .let and be two spatial elements sharing a face , with respective outward normal vectors and on . for and , let and be the traces on with limits taken from .we define the respective jumps and averages across each face by }|_e & = u^+{\mathbf{n}}^++u^-{\mathbf{n}}^- , & { \left [ \mathbf{v } \right]}|_e & = \mathbf{v}^+\cdot{\mathbf{n}}^++\mathbf{v}^-\cdot{\mathbf{n}}^- ; \end{aligned}\ ] ] if , we set and }|_e = u^+{\mathbf{n}}^+ ] , defined by if for and ; when , we set to be the average . finally we assume that there exists such that where is the radius of the inscribed circle of .  for simplicity of the presentation _ only _ of the a priori error bounds below , we shall later make a shape - regularity assumption on the space - time mesh ( cf . , assumption [ as : st_reg ] ) .we stress , however , that the stability results presented below do not depend on assumption [ as : st_reg ] and , therefore , the numerical method proposed below is unconditionally stable for any choice of spatial and temporal meshsizes . indeed ,an important advantage in using such space - time methods is that they do _ not _ require any cfl - type restrictions .finally , the broken spatial gradient will be denoted by , given by for all and a ; collectively , we shall denote the broken gradient by defined as , for , i.e. , is allowed to be discontinuous both in space and in time .to derive the weak form suitable for dg discretisation we will follow an energy argument .we start by assuming that is a smooth enough solution of and let .the standard symmetric interior penalty dg weak formulation on the time - slab when tested with is given by })_{\gamma_n \times i_n}\\  & - ( { \left [ u \right]},{\left\ { a \nabla \dot v \right\}})_{\gamma_n \times i_n}+ ( \sigma_0 { \left [ u \right ] } , { \left [ \dot v \right ] } ) _ { \gamma_n \times i_n } = 0 , \end{split}\ ] ] where for a positive constant to be made precise later .this motivates the following definition of _ discrete energy _ at time , for : }\|^2_{\gamma_n}-{\left({\left\ { a \nabla v(t ) \right\ } } , { \left [  v(t ) \right]}\right)_{\gamma_n}}.\ ] ] using the classical inverse inequality , for all , ( see , e.g. , , ) we see that hence , if the penalisation parameter is chosen large enough : in particular suffices we have that }\|^2_{\gamma_n } + \tfrac{1}{{{4}}}\|\sqrt{a}{\widetilde{\nabla}}v(t)\|^2_{\omega } ,  \end{split}\ ] ] ensuring the non - negativity of the energy for functions in : choosing as test function in and summing over , we obtain }\right)_{\gamma_n } } + \tfrac12 { \|{\sqrt{\sigma_0 } { \left [ u \right]}}\|_{\gamma_n}}^2 \big ) dt \\ & = e_h(t_n^-,u)-e_h(t_0^+,u)-\sum_{n = 1}^{n-1}{\lsem e_h(t_n , u ) \rsem}. \end{split}\ ] ] in order to allow for discontinuities in time , the formulation needs to be modified ( in a consistent fashion ) to control the terms that have no sign . to this end, we shall use the elementary algebraic identity for some scalar quantities which may , in general , be discontinuous across .the idea here is to add terms that change the jump of a product to the product of jumps in the above energy identity , without compromising its consistency .for instance , to in we add the extra term . for ,this does not change the consistency of with respect to in weak form , as the smoothness assumptions on the initial data ensure that the exact solution satisfies ; l^2(\omega)) ] and } \rsem},{\left\ { a\nabla  v(t_n^+ ) \right\}}\right)_{{\hat{\gamma}_n}}} ] and .the use of in the last terms also merits a brief explanation : since we assume that both the solution and its spatial flux are continuous within a space - time element , it follows that }\right)_{\gamma_n \times i_n } } =  { \left({\left\ { a \nabla u \right\}},{\left [ \dot v \right]}\right)_{{\hat{\gamma}_n}\times i_n}}\ \text { and }\  { \left({\left [ u \right]},{\left\ { a \nabla \dot v \right\}}\right)_{\gamma_n \times i_n } } = { \left({\left [ u \right]},{\left\ { a \nabla \dot v \right\}}\right)_{{\hat{\gamma}_n}\times i_n}}.\ ] ] for consistency , the terms }\right)_{\gamma_0}}- { \left ( { \left [ u_0 \right]},{\left\ { a\nabla  v(t_0^+ ) \right\}}\right)_{\gamma_0}} ] in the energy identity ; these do not have a sign but can be bounded by the other terms in the energy using .in view of the above considerations , we can now state the space - time weak formulation of our method : }\right)_{\gamma_n \times i_n } } - { \left({\lsem { \left\ { a{\widetilde{\nabla}}u(t_n ) \right\ } } \rsem},{\left [  v(t_n^+ ) \right]}\right)_{{\hat{\gamma}_n } } }  \\ & - { \left({\left [ u \right]},{\left\ { a \nabla \dot v \right\}}\right)_{\gamma_n \times i_n } }  - { \left({\lsem { \left [ u(t_n ) \right ] } \rsem},{\left\ { a\nabla  v(t_n^+ ) \right\}}\right)_{{\hat{\gamma}_n } } }  \\ & + { \left(\sigma_0  { \left [ u \right]},{\left [ \dot v \right]}\right)_{\gamma_n \times i_n } } + { \left(\sigma_0 { \lsem  { \left [ u(t_n ) \right ] } \rsem } , { \left [  v(t_n^+ ) \right]}\right)_{{\hat{\gamma}_n } } }  \\ & +  { \left(\sigma_1 { \left [ u \right]},{\left [ v \right]}\right)_{\gamma_n \times i_n } } + { \left(\sigma_2 { \left [ a\nabla u \right]},{\left [ a\nabla v \right]}\right)_{\gamma_n \times i_n } }  = { b^{\text{init}}}(v ) ,  \end{aligned}\ ] ] where is given by }\right)_{\gamma_0}}\\ & - { \left ( { \left [ u_0 \right]},{\left\ { a\nabla  v(t_0^+ ) \right\}}\right)_{\gamma_0 } } + { \left(\sigma_0  { \left [ u_0 \right]},{\left [  v(t_0^+ ) \right]}\right)_{\gamma_0}}.  \end{aligned}\ ] ] note that the last two terms in the definition of are zero if the initial data is continuous in space . the terms with positive penalty parameters and in the weak formulation do not affect the consistency of the weak formulation ; the need for their inclusion in the method will become apparent in the convergence analysis .   although the choice of , , and will only become evident in the analysis , for easy referral we state the choice of the stabilisation parameters , , and here also : is chosen as , ( see also , ) and we set   thus , we arrive at a space - time discrete method , which can be thought of in two ways : as a method for obtaining a discrete solution on a fixed space - time domain  ] is then defined as ) \;:\ ; u|_{\omega \times i_n } \in { s_{n,\text{trefftz}}^{h , p } } , \ ;n = 0,1\dots , n-1\right\ } \subseteq { v^{h , p}}.\ ] ]  polynomial plane waves are examples of functions in this space   [ prop : trefftz_poly ] the local dimension of the trefftz space in is given by   the proof for is clear . for in is shown that the dimension of the space of trefftz , homogeneous polynomials of degree is , hence the total dimension is given by the case is proved similarly by noticing that the dimension of the space of trefftz , homogeneous polynomials of degree is .  in two dimensions , with a constant , scalar diffusion coefficient , the following is a basis for with    next we prove that the discontinuous galerkin energy norm is indeed a norm on the subspace of trefftz polynomials .this also includes piecewise linear polynomials as for .[ prop : seminorms ] with the choice of as in and , bilinear forms and give rise to two semi - norms and these are in fact norms on trefftz subspaces and .  recalling and using , we deduce that and is hence a semi - norm .suppose for .then , and have no jumps across the space skeleton and hence is a weak solution of the homogeneous wave equation on with zero initial and boundary conditions .uniqueness implies and hence that is a norm on this space .the analysis of is similar recalling , which shows that is a semi - norm if the stabilization parameter is chosen correctly . proceeding as in the first case, shows that it is in fact a norm on the trefftz spaces .[ cor : ex_un ] under the conditions of the above proposition and with initial data , , the discrete system with has a unique solution .the uniqeness of the solution to over the trefftz space follows from being a norm on this space .existence of the solution to the linear system follows from uniqueness .next , we present the convergence analysis of the trefftz based method.   we shall now establish the quasi - optimality of the proposed method .let and , then for some constant and }\|^2_{\gamma_n}+\|\sigma_0^{-1/2}{\left\ { a\nabla w(t_n^- ) \right\}}\|^2_{\gamma_n}\big)\\ & + \sum_{n = 0}^{n-1}\left({\|{\sqrt{\sigma_1}{\left [ w \right]}}\|_{\gamma_n \times i_n}}^2 + { \|{\sqrt{\sigma_2}{\left [ a\nabla w \right]}}\|_{\gamma_n \times i_n}}^2 + { \|{\sigma_2^{-1/2}{\left\ { \dot w \right\}}}\|_{\gamma_n^{\rm int } \times i_n}}^2\right.\\ & \qquad\qquad+\left.{\|{\sigma_1^{-1/2 } { \left\ { a\nabla \dot w \right\}}}\|_{\gamma_n \times i_n}}^2+{\|{\sigma_0\sigma_1^{-1/2}{\left [ \dot w \right]}}\|_{\gamma_n \times i_n}}^2\right ) .\end{split}\ ] ]  integration by parts gives },{\left\ { a \nabla v \right\}}\right)_{\gamma_n\times i_n}}-{\left({\left\ { \dot w \right\}},{\left [ a \nabla v \right]}\right)_{\gamma_n^{\text{int}}\times i_n}}\\ & + { \left(\dot w(t_{n+1}^- ) , \dot v(t_{n+1}^-)\right)_{\omega}}-{\left(\dot w(t_{n}^+ ) , \dot v(t_{n}^+)\right)_{\omega}}\\ & + { \left(a\nabla w(t_{n+1}^- ) , \nabla v(t_{n+1}^-)\right)_{\omega}}-{\left(a\nabla w(t_n^+ ) , \nabla v(t_n^+)\right)_{\omega } } , \end{split}\ ] ] since and by using the ( elementary ) identity },{\left\ { a \nabla v \right\}}\right)_{\gamma_n\times i_n}}-{\left({\left\ { \dot w \right\}},{\left [ a \nabla v \right]}\right)_{\gamma_n^{\text{int}}\times i_n}},\ ] ] in the second step .further applications of integration by parts in time yield },{\left\ { a \nabla v \right\}}\right)_{\gamma_n\times i_n } } \\= \ & { \left({\left [  w \right]},{\left\ { a \nabla \dot v \right\}}\right)_{\gamma_n\times i_n}}-{\left({\left [  w(t_{n+1}^- ) \right]},{\left\ { a \nabla  v(t_{n+1}^- ) \right\}}\right)_{\gamma_n } } + { \left({\left [  w(t_{n}^+ ) \right]},{\left\ { a \nabla  v(t_{n}^+ ) \right\}}\right)_{\gamma_n } } , \end{split}\ ] ] and }\right)_{\gamma_n\times i_n } } + { \left(\sigma_0 { \left [ w \right ] } , { \left [ \dot v \right]}\right)_{\gamma_n\times i_n}}\\ = \ & { \left({\left\ { a\nabla \dot w \right\ } } , { \left [ v \right]}\right)_{\gamma_n\times i_n } } -{\left(\sigma_0 { \left [ \dot w \right]},{\left [  v \right]}\right)_{\gamma_n\times i_n } } \\ & -{\left ( { \left\ { a\nabla w(t_{n+1}^- ) \right\ } } , { \left [  v(t_{n+1}^- ) \right]}\right)_{\gamma_n } } + { \left ( { \left\ { a\nabla w(t_{n}^+ ) \right\ } } , { \left [  v(t_{n}^+ ) \right]}\right)_{\gamma_n}}\\ & + { \left(\sigma_0{\left [ w(t_{n+1}^- ) \right ] } , { \left [  v(t_{n+1}^- ) \right]}\right)_{\gamma_n } } -{\left(\sigma_0 { \left [ w(t_{n}^+ ) \right ] } , { \left [  v(t_{n}^+ ) \right]}\right)_{\gamma_n}}. \end{split}\ ] ] substituting these into , we obtain }\right)_{\gamma_n \times i_n}}- { \left({\left\ { a\nabla w(t_{n+1}^- ) \right\}},{\left [  v(t_{n+1}^- ) \right]}\right)_{\gamma_n } } \\ & - { \left(\sigma_0  { \left [ \dot w \right]},{\left [ v \right]}\right)_{\gamma_n \times i_n}}+ { \left(\sigma_0  { \left [ w(t_{n+1}^- ) \right]},{\left [  v(t_{n+1}^- ) \right]}\right)_{\gamma_n}}\\ & - { \left ( { \left\ { \dot w \right\ } } , { \left [ a \nabla v \right]}\right)_{\gamma_n^{\rm int}\times i_n}}- { \left({\left [ w(t_{n+1}^- ) \right]},{\left\ { a\nabla  v(t_{n+1}^- ) \right\}}\right)_{\gamma_n } } \\ & + { \left(\dot w(t^-_{n+1}),\dot v(t_{n+1}^-)\right)_{\omega } } + { \left(a\nabla w(t_{n+1}^- ) , \nabla v(t_{n+1}^-)\right)_{\omega}}\\ & + { \left(\sigma_1  { \left [ w \right]},{\left [  v \right]}\right)_{\gamma_n \times i_n } } + { \left(\sigma_2  { \left [ a\nabla w \right]},{\left [ a \nabla v \right]}\right)_{\gamma_n \times i_n}}.  \end{aligned}\ ] ] therefore , upon adopting the notational convention we have }\right)_{\gamma_n\times i_n } } - { \left(\sigma_0  { \left [ \dot w \right ] } , { \left [ v \right]}\right)_{\gamma_n\times i_n } }  - { \left ( { \left\ { \dot w \right\ } } , { \left [ a\nabla v \right]}\right)_{\gamma_n^{\rm int}\times i_n } }  \\ & + { \left(\sigma_1  { \left [ w \right ] } , { \left [  v \right]}\right)_{\gamma_n \times i_n } } + { \left(\sigma_2  { \left [ a\nabla w \right ] } , { \left [ a \nabla v \right]}\right)_{\gamma_n \times i_n}}\big)\\ & - \sum_{n = 1}^{n}\big({\left ( \dot w(t_n^- ) , { \lsem \dot v(t_n ) \rsem}\right)_{\omega } } + { \left(a\nabla w(t_n^- ) , { \lsem \nabla v(t_n ) \rsem}\right)_{\omega}}\\ & - { \left({\left\ { a\nabla w(t_n^- ) \right\}},{\lsem { \left [  v(t_n ) \right ] } \rsem}\right)_{\gamma_n } }  - { \left ( { \left [ w(t_n^- ) \right]},{\lsem { \left\ { a\nabla  v(t_n ) \right\ } } \rsem}\right)_{\gamma_n}}\\ & + { \left(\sigma_0  { \left [ w(t_n^- ) \right]},{\lsem { \left [  v(t_n ) \right ] } \rsem}\right)_{\gamma_n}}\big ) . \end{split}\ ] ]  it is now clear how to estimate most of the terms to obtain the stated result using the cauchy - schwarz inequality .the first two terms on the right hand side in the above sum are estimated as follows }),\sqrt{\sigma_1}{\left [  v \right]}\right)_{\gamma_n \times i_n } } \leq { \|{\sigma_1^{-1/2}({\left\ { a\nabla\dot  w \right\}}-  \sigma_0 { \left [ \dot w \right]})}\|_{\gamma_n \times i_n } } { \|{\sqrt{\sigma_1 } { \left [ v \right]}}\|_{\gamma_n \times i_n}};\ ] ] for the third term , we have }\right)_{\gamma_n^{\rm int}\times i_n } }   & \le { \|{\sigma_2^{-1/2 } { \left\ { \dot w \right\}}}\|_{\gamma_n^{\rm int } \times i_n } }  { \|{\sqrt{\sigma_2 } { \left [ a\nabla v \right]}}\|_{\gamma_n^{\rm int } \times i_n}}. \end{split}\ ] ]  [ rem : cheaper ] note that shows that for trefftz functions the bilinear form can be evaluated without computing integrals over the volume terms .this can bring considerable savings , especially in higher spatial dimensions ; see figure [ fig : conv_time ] .it is possible to define the space - time dg method above with the classical ( discontinuous ) space of _ all _ polynomials of degree ( total degree or of degree on each variable ) .the resulting method then appears to work in practice also . some numerical experiments and comparison with the smaller polynomial trefftz spaceare given section [ sec : numerics ] .the error analysis of the method with the full polynomial space of degree is not straightforward , though .in particular , it is not immediately clear how to treat the volume terms which do not vanish in this case. this would be essential in completing an error analysis for such spaces also .[ th : best_approx ] let be the discrete solution of the trefftz time - space discontinuous galerkin method and let be the exact solution .then , we have   by galerkin orthogonality for any . hence , giving   to conclude this sectionwe show that in the case of trefftz polynomials , the discrete norm can be bounded below by an -temporal norm .for simplicity of the presentation _ only _ , we shall , henceforth , make the following assumption .[ as : st_reg ] we assume that , for all , .  in the simplifying context of space - time meshes consisting of prismatic meshes without local time - stepping , assumption [ as : st_reg ] implies global quasi - uniformity .given the tensor - product / prismatic structure of the space - time elements , it is by all means possible to extend the error analysis below to space - time meshes not satisfying assumption [ as : st_reg ] .moreover , with minor modifications only , it is possible to extend the findings of this work to meshes with space - time elements with variable temporal dimension lengths ; the notational overhead for such a development is deemed excessive given the a - priori error analysis point of view of this work .  [prop : positivee ] for any it holds where and , . the same estimate holds with replaced by .consequently , under assumption [ as : st_reg ] , for all with a constant independent of the meshsize.  note that for an element with exterior normal  where we have used the discrete trace inequality .gronwall inequality now gives us as well as for all  ] and start from the basic observation that the exact solution to the wave equation on each space time element is of the form where we can define and by and where it is not difficult to see that these are well - defined , i.e. , that the right - hand sides indeed depend only on by virtue of satisfying the equations respectively .  for , we define the -projection operator , , defined by setting , for , with being the -orthogonal projection operator onto .now , upon considering the linear scalings , , such that and , , such that we define the univariate space - time elemental projection operators , , piecewise by using these , we can now define the _ trefftz projection _ of a function of the form element - wise by  , .the approximation properties of follow from the respective properties of , .space - time shape regularity implies , .we denote by the quantity with real numbers such that and being the gamma function ; we also adopt the standard convention . making use of _ stirling s formula _, , , we have , , for , with and constant depending only on .we have the following -approximation results for , .  [ approx_theorem ]let , for , and let for an open , bounded interval ; finally let be any of the , .then the following error bounds hold :    and with , . also , let , with .then , the following bound holds : with .finally , let , with .then , the following bound holds : with .the proof of ( [ ltwobounis ] ) and ( [ dxbounis ] ) for the -projection can be found , e.g. , in .the proof of can be found in , while the proof of follows along the same lines as in the proof of from .these -approximation estimates imply the following bound .[ approx_theorem_dg_norms ] let , for be the exact solution to . then, for space - time meshes satisfying assumption [ as : st_reg ] , the following error bounds hold : for and , with constant , independent of , , , and of . moreover ,if is analytic on a neighbourhood of , there exists , depending on the analyticity region of in a neighbourhood of , such that   the proof of follows by combining the -approximation bounds from with lemma [ lemma : abstract_error_bound ] .  for, we work as follows .analyticity of implies that there exists a , such that for all , for some , along with stirling s formula , we can arrive to the bound with the precise choice of remaining at our disposal .the function has a minimum at , giving .setting , now , the result follows .the bound is suboptimal in by one order .this is a standard feature of -version dg methods whose analysis requires the use of -type inverse estimates .it is possible to slightly improve on this result and obtaining only order -suboptimal bounds , using the classical -approximation results from , instead of the -projection operator as done above .these results , however , are not suitable for the proof of the exponential rate of -convergence .we present a series of numerical experiments aiming to highlight the performance of the proposed method above . in each experiment ,the spatial meshes are kept fixed and uniform time - step is used . in the one - dimensional ( ) examples the spatial mesh is a uniform set of intervals , whereas for , the spatial mesh is a quasiuniform triangulation .the resulting linear systems at each timestep are solved by standard sparse direct solvers .     where .note that the initial data is not exactly zero at the boundary , but is less than in the range of parameter that we consider .this slight incompatibility with the boundary condition does not influence in any visible way our numerical results .since the energy of the exact solution stays constant it is given for all times by where the approximation in the second step is of the order of for reasons given above and the final equality is obtained by using integration by parts to reduce it to the gaussian integral .  as the problem is in one spatial dimension ,the exact solution is not difficult to obtain .the error will be computed in the dg norm since the exact solution is smooth , note that , see ,    we first investigate the convergence order of the numerical method .though we did not analysed full polynomial spaces , we also give numerical experiments for these as it is interesting to compare the two sets of results .  in this subsection, we choose and .note that we choose such a small time interval in order to reach the asymptotic regime earlier this will especially be important for lower orders . in figure [fig : conv ] and tables [ tab : conv_order ] and [ tab : linear_order ] , the convergence curves and numerically computed convergence orders are given .these confirm the theoretical results .note that the errors obtained by the full and the trefftz spaces are very similar for the same order , but the trefftz spaces require fewer degrees of freedom and cheaper implementation ; see remark [ rem : cheaper ] and figure [ fig : conv_time ] .we have also found that higher order approximation converges without the two extra stabilisation terms , i.e. , with , but with the piecewise linear functions it stagnates .i. babuka and j. e. osborn .finite element methods for the solution of problems with rough input data . in _ singularities and constructive methods for their treatment ( oberwolfach , 1983 ) _ , volume 1121 of _ lecture notes in math ._ , pages 118 .springer , berlin , 1985 .

Abstract: a new space - time discontinuous galerkin ( dg ) method utilising special trefftz polynomial basis functions is proposed and fully analysed for the scalar wave equation in a second order formulation . the dg method considered is motivated by the class of interior penalty dg methods , as well as by the classical work of hughes and hulbert . the choice of the penalty terms included in the bilinear form is essential for both the theoretical analysis and for the practical behaviour of the method for the case of lowest order basis functions . a best approximation result is proven for this new space - time dg method with trefftz - type basis functions . rates of convergence are proved in any dimension and verified numerically in spatial dimensions and . numerical experiments highlight the effectivness of the trefftz method in problems with energy at high frequencies .


the interest in quantum computation is stimulated by the discovery of quantum algorithms which can solve problems of significance much more efficiently than their classical counterparts .quantum algorithms could be implemented using various types of qubits [ 1 ] or quantum simulators [ 2 ] . among important quantum algorithms ,there exist the deutsch algorithm [ 3 ] , the deutsch - jozsa algorithm [ 4 ] , the shor algorithm [ 5 ] , the simon algorithm [ 6 ] , the quantum fourier transform algorithm , and the grover search algorithm [ 7 ] .in addition , proposals for implementing other quantum algorithms have been also presented ( e.g. , see [ 8 ] ) .as is well known , the deutsch algorithm and the deutsch - jozsa algorithm were the first two that make use of the features of quantum mechanics for quantum computation . compared with other quantum algorithms ,these two algorithms are easy to be implemented and thus have been considered as the natural candidates for demonstrating power of quantum computation .  in this article , we restrict ourself to a refined deutsch - jozsa ( dj ) algorithm . we will propose a way to implement a three - qubit _ refined _ dj algorithm .as shown below , this proposal is based on the construction of the 35 -controlled phase gates , which employs single - qubit gates and two - qubit _ standard _ controlled - phase ( cp ) gates . to the best of our knowledge ,our work is the first to consider how to construct the 35 -controlled phase gates using these two kinds of basic gates .the construction of the 35 -controlled phase gates here is general , and can be applied to implement the three - qubit refined dj algorithm in a cavity - based or noncavity - based physical system .there are several motivations for this work :  \(i ) in ref .[ 9 ] , the authors showed that implementing a one - qubit or two - qubit _ refined _ dj algorithm does not require entanglement between the input query qubits , and thus test of the one - qubit or two - qubit _ refined _ dj algorithm is not meaningful because it can be solved in a classical way .however , they showed that entanglement exists between the input query qubits during performing a -qubit refined dj algorithm with , and thus the meaningful test of the refined dj algorithm occurs only and only if .hence , as far as the refined dj algorithm is concerned , a three - qubit refined dj algorithm ( i.e. , the case for ) is the smallest one that needs to be tested or implemented , in order to demonstrate the full power of quantum computation as applied to the deutsch problem . for a detailed discussion , see [ 9 ] .\(ii ) a second motivation of this work is as follows .a single - qubit gate can be easily realized by applying a single classical pulse .in addition , it is well known that a two - qubit cp gate ( i.e. , the key element in the construction of the 35 -controlled phase gates , see table ii below ) has been experimentally demonstrated in many physical systems such as quantum dots , trapped ions , atoms in cavity qed , and superconducting qubits / qutrits coupled to a single circuit cavity .thus , it is straightforward to implement a three - qubit refined dj algorithm by applying the present proposal .\(iii ) quantum information processing using superconducting qubits coupled to a cavity has attracted considerable interest during the past ten years .based on cavity qed , many theoretical proposals have been presented for realizing two - qubit gates [ 10 - 14 ] and multiple qubit gates [ 15,16 ] with superconducting qubits . moreover ,experimental demonstration of two - qubit gates [ 17 - 20 ] and three - qubit gates [ 21 - 23 ] with superconducting qubits in cavity qed has been also reported . however , after a deep investigation , we noted that how to implement a three - qubit ( original and refined ) dj algorithm with superconducting qubits or qutrits in cavity qed has not been reported in both theoretical and experimental aspects . as is known , the experimental realization of the three - qubit dj algorithm with a cavity - superconducting - device system is important because it would be an important step toward more complex quantum computation in circuit cavity qed .\(iv ) over the past decade , there has been much interest in quantum information processing with atoms in cavity qed .based on cavity qed technique , many theoretical methods have been proposed for implementing a two - qubit cp or controlled - not gate and multiple qubit controlled gates with atoms [ 24 - 28 ] .moreover , a two - qubit cp gate between a cavity mode and an atom has been experimentally demonstrated [ 29 ] .however , after a thorough search , we found that only several proposals [ 30 - 32 ] were proposed for implementing the original or refined -qubit dj algorithm for , using atoms in cavity qed .as discussed below , these proposals have some drawbacks . for instances , the one in [ 30 ]is incomplete , and the ones in [ 31,32 ] are difficult to implement in experiments when compared with our current proposal .the dj algorithm is aimed at distinguishing the constant function from the balanced functions on inputs .the function takes either or for the constant function , the function values are constant ( or ) for all inputs .in contrast , for the balanced function , the function values are equal to 1 for half of all the possible inputs , and for the other half . using the dj algorithm , whether the function is constant or balancedcan be determined by only one query . however, a classical algorithm would require queries to answer the same problem , which grows exponentially with input size .the _ original _ dj algorithm involves the input query qubits each initially prepared in the state and an auxiliary working qubit initially prepared in the state  .it operates by : ( i ) applying a hadamard gate on each input query qubit , which results in and ) , ( ii ) performing a -controlled - not gate and ( iii ) applying a hadamard gate again on each input query qubit followed by a measurement on each input query qubit along the single - qubit base formed by the two states here , is defined as where is the state of the input query qubits with being an -qubit binary decomposition , is the state of the working qubit with and means modulo 2 .note that the operation results in the following transformation [ 2 ] shows that there is no entanglement between the query qubits and the working qubit in the output of the .thus there is no need for any coupling between the query qubits and the working qubit during the entire process .hence , in the _ original _ dj algorithm the working qubit is completely redundant .-qubit deutsch - jozsa algorithm . here, represents a single - qubit hadamard gate on an input query qubit , which results in and ) .in addition , indicates a -controlled phase gate on the input query qubits , described by eq .( 2).,width=10 ]  the _ refined _ dj algorithm was proposed by collins _et al . _ in 2001 [ 9 ] , which is illustrated in fig . 1 .this refined dj algorithm follows a similar pattern of operations as the original dj algorithm , which is described below :  \(i ) each input query qubit is prepared in the initial state   \(ii ) perform a hadamard transformation on each qubit . as a result ,the -qubit initial state changes to the state ( denoted as )      \(iii ) apply the -controlled phase shift , described by which leads the state to the state ( denoted as ) .\(iv ) perform another hadamard transformation on each qubit . as a result ,the state becomes   \(v ) measure the final state of the qubits .if the qubits are measured in the state then is constant .however , if they are measured in other -qubit computational basis states , then is balanced .this is because the amplitude of the state is given by which is for a constant while for a balanced .one can see that when compared with the original dj algorithm , this refined dj algorithm does not need a working qubit .hence , it requires one qubit fewer than the original dj algorithm .consequently , its physical implementation requires one fewer two - state system .for both of the original and refined dj algorithms , there are a total of balanced functions , among which only balanced functions are nontrivial if the symmetry is taken into account . for the three - qubit dj algorithm , i.e. , there thus exist nontrivial balanced functions ( see table i ) .  a single qubit gate results in the transformation while as is well known , a two - qubit _ standard _ cp gate on qubits and is described as follows which implies that if and only if the control qubit ( the first qubit ) is in the state , a phase flip happens to the state of the target qubit ( the second qubit ) , but nothing happens otherwise .the construction for each of the 35 -controlled phase gates is listed in table ii .one can see from table ii that the 35 -controlled phase gates are classified into the following four types : ( i ) type 1 includes seven -controlled phase gates each constructed with single - qubit gates only ; ( ii ) type 2 contains twelve -controlled phase gates each constructed with single - qubit gates and one two - qubit cp gate ; ( iii ) type 3 has twelve -controlled phase gates each constructed by using single - qubit gates and two two - qubit cp gates ; and ( iv ) type 4 involves four -controlled phase gates each implemented with two single - qubit gates and three two - qubit cp gates .as discussed previously , a single - qubit gate here can be easily realized by applying a single classical pulse , and a two - qubit cp gate ( i.e. , the central element in our gate construction ) has been experimentally demonstrated in various physical systems .hence , the gate construction given in table ii is implementable .after a thorough search , we found that only several proposals [ 30 - 32 ] were proposed for implementing the original or refined dj algorithm for , using atoms in cavity qed . in the followingwe will briefly introduce these previous proposals and then give a comparison of them with our current proposal .\(i ) the authors in [ 30 ] proposed an _incomplete _scheme for realizing the -qubit original dj algorithm using atoms coupled to a cavity , because they only discussed how to implement balanced function , which completes the state transformation from the input state to the output state , as shown in [ 30 ] .furthermore , this state transformation can be reached in a classical way , since it can be alternatively achieved by simply performing a single - qubit gate on each of individual qubits of the register .\(ii ) in ref .[ 31 ] , a scheme was proposed for the implementation of the -qubit original dj algorithm via atoms interacting with a cavity . as discussed there , to implement each of balanced functions, this scheme requires different -qubit controlled - not gates each with qubits simultaneously controlling a target qubit . for a three - qubit dj algorithm , thus , by applying this scheme to realize each balanced function for a three - qubit dj algorithm , four different three - qubit controlled - not gates ( each with 2 qubits simultaneously controlling a target qubit ) would be required .\(iii ) ref .[ 32 ] presented a way for implementing a 3-qubit refined dj algorithm with three atoms in cavity qed .this proposal is based on four three - qubit controlled phase gates ( each with 2 qubits simultaneously controlling a target qubit ) . in addition , as discussed in [ 32 ] , it is required to send three atoms through four cavities at the same time , or simultaneously sending the three atoms through a common cavity four times .  from the description given above , one can see that :  \(i ) the present proposal is different from the previous one in [ 30 ] , since the latter is _ incomplete _ , which only considered realizing one balanced function that can be achieved in a classical way .in contrast to [ 30 ] , we have presented a _complete _ protocol for implementing all 35 -controlled phase gates for a three - qubit refined dj algorithm .\(ii ) to implement a three - qubit dj algorithm , the proposal in [ 31 ] requires using four different three - qubit gates , and the one in [ 32 ] requires sending three atoms through a cavity four times or needs the use of four cavities . in contrast , it can be seen from table ii that our proposal needs three two - qubit cp gates at most , which can be implemented using a single cavity or resonator only , as discussed below .hence , when compared with the previous proposals in [ 31,32 ] , it is simple and easy to implement a three - qubit refined dj algorithm based on cavity qed , by using the present proposal .we point out that it is not our intention to cast aspersions on existing approaches [ 30 - 32 ] to the dj algorithm implementation ; rather we simply wish to present a protocol to implement a three - qubit refined dj algorithm .  as relevant to the cavity qed - based implementation of the present proposal , some points may need to be addressed here .first , the present proposal can be implemented with three superconducting qubits / qutrits placed in a single cavity or resonator .this is because during performing each two - qubit cp gate one can have the irrelevant qubit / qutrit to be decoupled from the cavity by adjusting its level spacings ( e.g. , see , [ 15,16 ] and discussion therein ) .second , it can be realized using three atoms plus a single cavity . during performing each two - qubit cp gate on atoms, the irrelevant atom can be made to be decoupled from the cavity , for instance , by moving it out of the cavity via translating optical lattices ( e.g. , see [ 16,28 ] ) .we have proposed a way to implement the three - qubit refined deutsch - jozsa quantum algorithm .the construction of the -controlled phase gates can be applied to implement a three - qubit refined deutsch - jozsa algorithm in a cavity - based or noncavity - based physical system .this work is of interest because it provides a simple and general protocol for implementing a three - qubit refined deutsch - jozsa algorithm , which is an important step toward more complex quantum computation .c.p.y . was supported in part by the national natural science foundation of china under grant no .11074062 , the zhejiang natural science foundation under grant no .lz13a040002 , the open fund from the sklps of ecnu , and the funds from hangzhou normal university under grant no .q.p.s . was supported by zhejiang provincial natural science foundation of china ( grant no .lq12a05004 ) .j. q. you , f. nori , nature * 474 * , 589 ( 2011 ) ; physics today * 58 * ( 11 ) , 42 ( 2005 ) ; i. buluta , s. ashhab and f. nori , reports on progress in physics * 74 * , 104401 ( 2011 ) ; p.d .nation , j.r .johansson , m.p .blencowe and f. nori , rev .mod . phys . * 84 * , 1 ( 2012 ) ; z. l. xiang , s. ashhab , j.q .you and f. nori , rev .mod . phys . * 85 * , 623 ( 2013 ) .h. wang , l .- a .liu and f. nori , phys . rev .a * 82 * , 062303 ( 2010 ) ; h. wang , s. ashhab and f. nori , phys .a * 79 * , 042335 ( 2009 ) ; phys .a * 83 * , 062317 ( 2011 ) ; phys .a * 85 * , 062304 ( 2012 ) .

Abstract: we propose a way to implement a three - qubit refined deutsh - jozsa ( dj ) algorithm . the present proposal is based on the construction of the 35 -controlled phase gates , which uses single - qubit gates and two - qubit _ standard _ controlled - phase ( cp ) gates only . this proposal is implementable because a single - qubit gate can be easily realized by applying a single classical pulse and a two - qubit cp gate has been experimentally demonstrated in various physical systems . finally , it is noted that this proposal is quite general , and can be applied to implement a three - qubit refined dj algorithm in a cavity - based or noncavity - based physical system .