So welcome to course on advanced geotechnical engineering module 7 lecture number 4. This is lecture number 4 on geotechnical physical modeling. So in the previous lecture we have discussed about variation of G level or gravity level with the horizontal distance as well as with the depth. In this particular slide the merit of large centrifuge is shown. So here if you look into it there is a radius which is 1 meter from the center of the shaft to certain point within the model. Similarly here this radius is about 4 meter. So if you look into it there is variation with the horizontal distance but is limited. But in the case when you take a radius of 1 meter here it is 45 gravities and here it is 47.4 gravities. So there is a variation of G level with the horizontal distance. Similarly here it is 45 gravities at the bottom it is 55 gravities. So because here it is RT omega square and here it is RT plus 0.2 omega square but whereas if you have a large beam centrifuge the variation G level is exist variation of the G level is prevalent but however the variation is negligible. So this actually shows the merit of large beam centrifuge and here in this slide the whatever we have discussed about in the normal prototype when you have got a soil strata which is having certain thickness and because of the self weight the vertical stress is sigma v and the horizontal stress if it is normally consolidated soil the distribution is shown here. But if the model if the same prototype soil is brought and model is constructed with a reduced thickness and subjected to radial acceleration field then what we have discussed is that the variation of the vertical stress with the depth is not linear is non-linear. Then we said that this is going to cause certain degree of under stress in the upper half portion or upper portion and war stress in the bottom portion. So this variation is arisen because of the variation of the G level with the depth. So you can see that in this particular slide and this particular diagram where actually this particular line shows the distribution of vertical stress as per the full scale conditions is concerned but if you look into this there is a non-linear variation of the vertical stress in the model due to radial acceleration field. And before going further and we also have discussed about two possibilities of increasing the gamma. One is that variation of gamma by increasing the seepage forces that we said that hydraulic gradient simulator method and in the centrifuge model test we said that this can be possible by subjecting a model to high gravities this is done by rotating about a vertical axis in a horizontal plane. So if you compare the basic principle one is that in centrifuge model test the centripetal forces acts towards the center and in case of so there is in centrifuge model test centripetal forces act on the model and hydraulic gradient test seepage forces act and in the porous material soil the self weight stress is increased and in the hydraulic gradient also we have the self weight stress increase and solid structure member it can be a pile member or it can be footing member it can be a retaining wall it can be any member which is being tested. The self weight stresses also increased in case of centrifuge model test but whereas in case of hydraulic gradient test and which is not done that with that self weight stresses are not affected they remain same. And the application in the sense that they are more general problems most complicated problems the construction process can be simulated or some climatic events can be simulated like rainfall or subjected to earthquake or combination of the different loading forces can be simulated in centrifuge model test which we are going to discuss in some selected applications. And as far as hydraulic gradient test is concerned we said that this mostly used for you know granular soils or sandy soils and problems with the level ground and the self weight of the structure member is not important and wherever the self weight of the structure is not important and where you have got you know problems with the level ground like you know let us say that footing resting on horizontal surface and or problems with like uplift behavior of piles subjected to you know uplift behavior of piles embedded in sandy soils. Similarly when you have got some problems like footing subjected to eccentric loads on a level ground these things can be studied by using the hydraulic gradient similitude method. So this particular slide gives comparison of hydraulic gradient similitude method and centrifuge model test. And we said that as far as the centrifuge model test are concerned two categories of test do exist one is called variable gravity level the other one is called constant gravity level mostly the constant gravity level is adopted and in the previous discussion we have discussed that when the you know gravity reaches to a constant level that means that a particular desired gravity then when the gravity reaches to a certain level then what we can do is that you know the angular velocity is maintained constant. So the tangential component of the acceleration will be 0 and secondly if the particularly the model being tested has to be safe up to that particular gravity level is being tested then it can be subjected to some sort of loading or some sort of construction process or some sort of effect due to some forces like wave forces or some earthquake forces etc. Whereas in case if you are interested in studying some collapse behavior of a certain structure like you know some collapse behavior of a some vertical cut or some slope which is being studied wherein it can be studied with variable gravity level. So here the gravity level is increased gradually at a certain speed and the speed at which this is settled for example if you are the increase it has to be you know keep in mind with the desired conditions like undrained conditions and if you are required to have then the waiting times also have to be small. So here this is you know a certain gravity level and then this is another gravity level and then up to the collapse or failure whichever actually occurred earlier. So this is called variable gravity level but each and every gravity level you know the model actually behaves like a prototype that means that if you are actually having a sandy model sandy soil model in dry sand model the generation of pore water pressures will not be there if you are having a dry sandy soil. So up to some extent like a reinforced slope constructed with a sandy soil where if you are interested in collapse behavior of a reinforced slope constructed with sandy soil they can be studied but each and every gravity level when they are subjected to they are equivalent to equivalent prototype with respect to that particular gravity level. So this is the two categories of tests one is called variable gravity level and constant gravity level but nowadays with the development of different actuator systems and you know rainfall triggering events or earthquake triggering events the constant gravity tests are you know adopted widely. So the possibility of the geotechnical centrifuge studies include we can actually model the prototype that is that if you are actually interested in understanding a particular prototype structure which has been constructed whether you know it is safe or not or what will be its performance after 5 years or 10 years of its life can be you know studied by using you know this centrifuge base of physical modeling technique but however the size of the model depends upon the size of the equipment which is being used for the purpose. So if the equipment is reasonably large and then large prototypes can be modeled very well by simulating the majority of the you know characteristics of the prototype in the model. The another possibility of geotechnical centrifuge studies is that investigation of new phenomenon that means that a new technique or a or can be you know studied by using this centrifuge base of physical modeling wherein you know this helps to draw guidelines and then to also to assess its performance over a period of time that is before failure and at failure and also lead to you know the improvement of the or you know the phenomenon being investigated. So this is you know very interesting where a new phenomenon can be studied very efficiently by using centrifuge base of physical modeling. There another possibility is that parametric studies so if you have if we have studied for certain you know dimensions then we wanted to know what will happen to the next level see the parametric study or parametric variation is very efficient and can be studied you know efficiently in geotechnical centrifuge modeling. Then as said earlier the numerical models are required to be validated mostly they are validated by using full scale physical models in the case of you know non feasibility of constructing full scale physical models let us say in certain areas like nuclear installations or where there is no access for constructing or feasibility to construct these full scale structures then you know the numerical models are required to be validated. So one of the avenues is that to you to perform series of centrifuge base physical model test and corroborate or see how the results numerical model results are actually comparable with the centrifuge model test results. So that will actually help to validate numerical models the another important application is that use of geotechnical centrifuge for geotechnical engineering instruction that is something like called centrifuge aided learning in geotechnical engineering centrifuge aided learning in geotechnical engineering. So after having discussed the possibility of the geotechnical centrifuge studies and we said that by subjecting a model to about a vertical axis you know in a rotational plane there is a possibility of this radial acceleration field effects are there. So these effects actually what we said is that you know effect the vertical stress and then consequently the effect also the horizontal stress distribution. So the X gravity is uniform for the practical range of soil depths encountered in engineering like up to 200 meters or 300 depths of you know what we encounter for soil investigations in civil engineering the X gravity is uniform for the practical range of soil depths encountered in civil engineering. So while using a centrifuge to generate high acceleration field required for physical modeling there is a variation of acceleration through the model. So that is what actually we have been discussing while using a centrifuge to generate high gravities because we wanted to have identical stresses as that in the prototype there is a variation of acceleration through the model. So what we said is that this is attributed to the inertial acceleration field given by R omega square this is the inertial acceleration field this is due to the inertial acceleration field given by R omega square where R is the radius from the center to a certain point within the model. So this apparent problem turns out to be minor if you are able to take care of the you know the hardware that is the radius of the equipment at which the gravity scale factor N is determined and this also this operating radius that is called an effective radius we are able to select properly then we can say that that particular point the stresses in model and prototype are identical. So this apparent problem turns out to be minor if care is taken to select the radius at which the gravity scale factor N is determined. So for that what we need to do is that we have to find out you know how the variation of the gravity level is there with the depth and where it is subjected to with the vertical stress distribution where it is subjected to under stress and more stress and then comparing the errors then we can actually lead to you know the point at which we can actually you know maintain this gravity level can be a certain. So the selection of the effective radius for a you know beams and refuse like you know consider a model rotating about vertical axis which is shown here and the model is constructed with a material having row density the mass density row and this is the center of the shaft the model is rotating about a vertical axis and a capital V is the velocity in this direction and this is small element d z selected at a depth z from the top surface and RT is the radius from the center of the shaft to top surface of the model. So if you look into this what we can do is that this is the you know the stress in the this is the stress ordinate and this is the depth here in this direction. So here this is the you know the prototype stress distribution and this is the radial the vertical stress distribution in the radial acceleration field in the centrifuge model you can see that this portion there is an under stress and this portion there is an over stress. So this could be understood because of the increasing gravity level at this depth there is an over stress is possible so in order to determine this what we do is that again by following the same principle d sigma v in the small stress which we can actually find out by writing like d sigma v is equal to rho into z into d sigma v can be found out and from there by integrating from 0 to H m we can actually get rho omega square H m into RT plus H m by 2. So what we have is that this one and now here if the same point H m if you are able to find out you know what is the stress in the prototype so at this level we can actually find out it is nothing but rho into g that is the normal gravity the H m is correspondingly in the prototype is n times H m so rho g into n H m. So at the so the area of the stress diagram in the prototype is nothing but half base that is rho g n H m into H m so half rho g n H m into H m that is this is the area of the stress diagram. Now this area of the stress diagram in the model can be found out by integrating from 0 to H m. So the area of the stress diagram can be obtained like this wherein this is by writing am is equal to 0 to H m rho omega square z into RT plus z by 2 into dz.  So by integrating this what we get is that and then applying the limits rho the rho omega square is equal to RT H m square by 2 plus H m cube by 6. Now by equating the two areas of stress diagrams in centrifuge model and the prototype and by using n g is equal to r u omega square this r e is called the effective radius which is measured from the center of the shaft to a point within the model where we are actually going to find out the equivalency of the stresses are there. Now what we have done is that by equating the areas of the stress diagrams in centrifuge model in area of the stress diagram in the prototype then what we get is that r e is equal to RT plus H m by 3 by simplifying and substituting for n g is equal to r u omega square we get r e is equal to RT plus H m by 3. Then for finding out the location where maximum under stress occurs what we do is that now assume that it occurs at a distance x from the surface so that means that we are now assuming at a distance x below the surface the notation is different here it is just to express in terms of function x it is written in terms of vertical distance x here. Now d sigma which is nothing but rho n g x that is nothing but the x which is nothing but the ordinate of at a particular depth x below the surface of the model that x is multiplied with n to get the prototype stress equivalent stress and g level is g and so rho g into n x minus rho omega square into x into RT plus x by 2. So this is nothing but the vertical stress in the model at a depth x in the model. Now that is nothing but the d sigma the difference between prototype stress and vertical that is that which we are trying to find out the difference between a stress here and a stress here this is the stress in the prototype this is the stress in the centrifuge model. Now by taking differentiation of this by taking this as function x rho n g x minus rho omega square x into RT plus x by 2 by taking function f dash x we can write that it is rho n g minus rho omega square RT minus rho omega square that is 2 x by 2 which reduce to rho omega square into x. So for d sigma to be maximum f dash x has to be maximum by applying the rules of maxima and minima we can say that for d sigma to be maximum f dash x is equal to 0. Then with re is equal to RT plus h m by 3 and by simplifying we get this x is equal to h m by 3 that means that and again if you look into it with x is equal to 2 h m by 3 when you substitute x is equal to 2 h m by 3 in this particular expression what you get is that d sigma is equal to 0 this implies that this term and this term will be equal and then it will be equal to 0 that means that implies that at that particular point x is equal to 2 h m by 3 sigma v in model is equal to sigma v in prototype that is where the point where the centrifuge stress the distribution of a vertical stress in the centrifuge model crosses the you know the vertical stress distribution in the prototype. So when you look back into the figure so this is that point from the surface it is actually said as 2 h by 3 2 h m by 3 and this is h m and this is actually at 50 percent of the height of this point at which this is crossing where the maximum under stress is occurring that is where actually what we have located is x is equal to h m by 3 then now we can actually find out the resulting errors after having known the location of the you know where under stress is maximum and where the sigma v in model in prototype are equal then what we can do is that after having so at x is equal to h m by 3 that is the where the maximum under stress error is possible that is defined as d sigma by sigma v prototype. So by writing the expression with x is equal to h m by 3 we can write rho h m by 3 into n into g minus rho omega square h m by 3 into RT plus h m by 6 divided by rho n g into h m by 3. So by simplifying you know this particular expression which is written and substituting for n g is equal to r u omega square we get h m by 6 re. So the percentage under stress error resulting is h m by 6 re and similarly at x is equal to h m at x is equal to h m the war stress error d sigma by sigma v p is equal to rho omega square h m into RT plus h m by 2 minus rho n g h m divided by rho n g h m which also we will get it as h m by 6 re. So as we have equated the you know areas of the we have determined the effective areas by equating the areas in the centrifuge area of this stress diagrams in centrifuge model as well as in the prototype we get the percentage under stress and percentage over stress as h m by 6 re. So this implies that using this rule we can actually find out there is exact correspondence in stress in the model and prototype at two thirds of the model depth and effective centrifuge radius should be measured from the central axis to the one third depth of the model from the top surface. So generally there are they do a number of interpretation they do exist so this actually says that re is equal to RT plus one third of the h m that is 0.33 times h m but there are also some other you know investigators have presented this re is equal to RT plus somewhere like 0.4 times h m measured from the top surface. So using this rule there is exact correspondence in stresses between model and prototype at two thirds model depth and the effective centrifuge radius should be measured from the central axis to the one third depth of the model. So and then another thing is that maximum error in the model is given by Ru is equal to R0 that is h m by 6 re. So maximum error due to either under stress or over stress is given by h m by 6 re. So for most of the geotechnical centrifuges h m by re if it is less than 0.2 h m by re less than 0.2 and therefore the maximum error in the stress profile is minor that means that if the h m by re is less than 0.2 and then the maximum error in the stress profile is minor and generally less than 3% of the prototype stress. So for most geotechnical centrifuges h m by re if it is maintained as 0.2 then what you can see is that 0.2 by 6 that is into 100 about 3.33% the stress is the variation the stress is minor. If suppose h m by re let us say is equal to 0.5 or something whether the stress will be high and is greater than 3%. So if you look into the centrifuges like about 1.5 meter radius onwards what we can see is that we will be able to ensure that except for certain models the h m by re can be maintained as less than 0.2. So this actually ensures that the variation the vertical stress is maintained within the limits and this apparent error due to radial acceleration field on the vertical stress can be minimized. So in the example problem whatever we have discussed just now an example problem is given here in a 1 by 50th centrifuge model same material is used as that of in the prototype that is that the row in model prototype identical and model of the height h m is subjected to rotation with a constant angular velocity omega about a central axis. If RT is equal to 4 meter obtain the variation of vertical stress with depth at every 50 mm in the centrifuge model and compare vertical stress in model and prototype at 50 mm, 300 mm, 150 mm, 300 mm and 450 mm depth respectively. So here what we need to do is that the RT is given that is 4 meters then height of the model is about 450 mm. So what we can do is that by the height of the model which is about 450 mm and that by knowing that what we can do is that re is equal to RT is equal to 4 plus h m by 3 0.45 by 3 so re is equal to 0.15 so plus 4 then RT is equal to 4.15 meters. And by knowing n that is equal to 50 so 50 into 9.81 is equal to 4.5 omega square you can actually find out omega you will get it in radians per second and if you would like to convert this omega there from radians per second to rpm it should be multiplied by 60 by 2 pi and for converting from rpm to radians per second multiply by 2 pi by 60. So by knowing omega now by writing the depths and for writing the stresses in model and prototype at different depths we can find out what is the variation of vertical stress in the model as well as in the prototype. So if you look into it by at the depth exactly 2 h m by 3 that is 2 into 450 divided by 3 that is around 300 mm you will see that the stresses in model and prototype are identical that is where the d sigma will be 0 and at a depth of 1 that is a depth around 450 mm you will see that the under stress error is maximum and at depth h m that is at the base of the model that is h m is equal to 450 mm you will see that the vertical stress in the model is more than the prototype stress. So we can also find out by knowing 450 that is 0.45 that is h m divided by 6 into 4.15 you can also calculate what is the percentage error due to variation the vertical stress due to radial acceleration field. So this is how the solution for this problem runs and then for calculating the stresses the appropriate units by need to be used for by while substituting rho is equal to 2000 kg per meter cube. So verification of the static equilibrium of a centrifuge model so here what we have said is that we said that now the vertical stresses variations they do exist but let us see that we have a prototype and if you are having a certain element under equilibrium so we need to satisfy or check whether the static equilibrium equations are satisfied or not. Then we can say that the model material response will be equivalent to as that in the prototype. So consider an embankment which is shown here and this dimension is length L and these are the x axis and y axis and it is actually has seepage as well as it is subjected to some body forces due to seepage as well as the sulphate forces and consider an element within the model at a certain depth as shown here. So assume that the same model is reduced by 1 by n times by maintaining this length as L by n and maintaining this gravity as ng that means that we and this all dimensions has been reduced and then here this element is actually having sizes of dx dy dz. Similarly here with the same density both models are constructed with same mass density and here the volume of the element which is considered is dx dy dz by n cube dx dy dz and here it is by reducing the 1 by n times that is dx dy dz by n cube. So this model represents the representative conditions of the prototype. Now what we do is that considering two dimensional equilibrium let us see what will happen when you wanted to derive the static equilibrium of a centrifuge model. So consider infinitesimally small element having dimensions dx dy dz in prototype that is what we have been discussing that representing homogeneous isotropic elastic continuum and where x and y which are actually shown here they are the unit weights due to body forces in x and y direction and x could be due to seepage and y could be due to sulphate of the forces. So then we have the assume that is a two dimensional element when it is subjected to the horizontal stresses and vertical stresses. So here this is the rate of change of the stress because of that there is an increase in stress. So which is nothing but sigma xx plus dou sigma xx by dou x into dx and this stress direction is sigma xs we have considered the elements in tension. Similarly the stresses in y direction here it is sigma yy plus dou sigma y by dou y into dy this is sigma y. So this x the body force acting in x direction this y body force due to sulphate acting in the y direction and this is the sign convention x is positive in this direction y is positive in this direction. Now we have shear stresses the conjugate shear stresses here dou yx plus dou yx by dou x into dx and here dou yx and here it is dou xy plus dou xy dou dou xy by dou y into dy by dou xy that is here. So when we consider now the equilibrium of you know the force equilibrium fx fy and mx and my when you take moments about the center so you will get that dou xy is equal to dou yx and by using sigma fx is equal to 0 sigma fy is equal to 0 what we get is that by for equilibrium what we for the equilibrium of the element is that we have to ensure because it is a two dimensional sigma fx is equal to 0 sigma fy is equal to 0 and sigma mx is equal to my is equal to 0. Now taking moments of the center of the element we get dou xy is equal to dou yx. Now with sigma fx is equal to 0 what we can do is that sigma fx is equal to 0. So here sigma fx is nothing but sigma xx plus tau xx tau sigma xx by tau x into dx this is you know acting on the area which is nothing but dy dz. Similarly here sigma xx stress acting on the area dy dz so the net force is nothing but dou sigma xx by dou x into dx dy dz. So if you multiply if you subtract this one similarly when you take the forces in the shear stresses in this direction and body forces in this direction and simplifying the after simplification what we get is that dx dy dz into dou sigma xx by dou x plus dou tau xy by dou y plus x is equal to 0. So this particular you know expression here we write it as dou sigma xx by dou x plus dou tau xy by dou y plus x is equal to 0 because as dx dy dz volume of the element cannot be equal to 0. So this is equilibrium equation in the x direction where it is read as dou sigma xx by dou x plus dou tau xy by dou y plus x is equal to 0. Similarly when you look into sigma fy is equal to 0 in the vertical in the y direction and simplifying the different forces what we write. So what we get is that dou sigma yy by dou y plus dou tau yx by dou x plus y is equal to 0. So this is again as dx dy dz cannot be equal to 0 what we write is that we can write this equation equilibrium equation in the so equilibrium equations for a model for the prototype is nothing but the two equations one is here in the y direction which is actually shown. So in the case of ng model considering that the small element having volume dx dy dz by n cube it yields now what we get is that dou sigma xx by dou x by n plus dou xy by dou y by n plus nx is equal to 0 and then in the y axis dou sigma yy by dou y by n plus dou tau yx by dou x by n plus ny is equal to 0. So this x and y are nothing but the body forces are equal unit weights in x direction and y direction that is the reason why they get multiplied by n. Now with that what it actually if you look into this previous expression by simplifying this the n will get cancelled n cannot be equal to 0. So dou sigma x by dou x plus dou tau xy by dou y plus x is equal to 0 dou sigma y by dou y plus dou tau yx by dou x plus ny plus y is equal to 0. So these equations whatever we have got for a prototype are analogous to the one what we have got in the centrifuge model. So this indicates that the centrifuge based physical model satisfies the static equilibrium conditions what they are actually existing in the equivalent prototype. So what we have actually reduced from this discussion that for convenience we actually have taken a two dimensional case and where we have written the forces in the x and y direction only and we have taken sigma fx is equal to 0 sigma fi is equal to 0 and by simplifying what we have reduced is that we reduced the equilibrium equations in the x and y direction and afterwards we converted them to ng model and considering a small element having volume dx dy dz by n cube it yields the equations which are actually identical to those in the prototype. So we can say that the static equilibrium conditions of the centrifuge model are satisfied. So whatever the change of boundary values is considered the solution of differential equations will not distinguish between model and prototype provided that the materials and system at corresponding points in model product can be made behave identically. So what we have noted is that whatever the change of boundary values is considered the solution of differential equations will not distinguish between model and prototype provided that the materials and systems at corresponding points in model prototype can be made behave identically. Now the limitations of you know after having discussed the like how the vertical stress can be minimized due to radial acceleration field and also how we can actually can satisfy the static equilibrium equations of a prototype in a centrifuge model. Now let us look into limitations like any modeling technique the centrifuge based on physical modeling technique also suffers from some limitations like non-homogeneity and inestropy of soil profiles is difficult to model and limitations of some modeling tools the sizes and all those things required to be considered and variation of g level with the horizontal distance and depth of the model. So this we said that water whether it is a beam centrifuge or drum centrifuge the variation of the g level with the depth cannot be eliminated but however in case of a drum centrifuge we can actually say that the variation of g level with the horizontal distance especially for small centrifuges can be eliminated because with curvature of the model but similarly even in beam centrifuges also for slope model some for example some investigators like Kumura et al in 1991 they have given the curvature for the top surface of the slopes to suit these curvature of the centrifuge and this is basically done to eliminate the variation of the g level with the horizontal distance. And then the boundary effects the models particularly especially the sandy soils they exhibit you know very severe boundary effects then nothing but the friction due to soil and the containers of the inner sides of the containers of the walls of the container. So this need to be eliminated by a special techniques particularly like applying a white petroleum grease or a lubricating agent without affecting the transparency of the front view transparent glass wall and by applying a sheets in a special way we can eliminate this friction angle up to reduce this up to you know in a big way. So the boundary effects are important need to be controlled especially for sandy soil models in case of we are having a high plastic soils there is you know this need to be arrested by applying the by lubricating the inner sides of the containers so that the boundary effects are minimized due to adhesion in case of place in case of sandy soils due to friction between the wall and the sand grains or sand grains. Then another thing is that these models they subject to scale effects particularly one of the predominance effects which we actually have discussed is that because of the our inability to not able to model the you know the grain size distribution what we said is that the scale effects are evident that these scale effects are can be minimized and also need to be found out for how they are effect in the model behavior. So this we have the one of the possible scale effects in the centrifuge based physical model means due to the grain size effects or particle size effects and which we will be discussing and how this can be solved are attended by using the same centrifuge based physical modeling technique that is nothing but what we call modeling of models. Then when we are actually reducing you know times scale factors for the time we will find that you know each and every phenomenon when it changes for example if it is for a see phase force or if it is for something like for a dynamic time or if it is for a some you know for viscous force you actually have we have different timings different scale factors for the time. So this inconsistency of the scale factors for the time is also said as you know one of the limitations sometimes you know when we have got scale factors for the time for some dynamic cases as well as for the you know some diffusion cases when they are different and if the phenomenon occurs simultaneously then we actually have to resort to some sort of you know an alternative so that the scaling considerations are satisfied. Then one of the major effects particularly the effect which is called the Coriolis effect which is actually called caused due to movement of you know particle within the model. If this suppose if that velocity with which the particle is actually moving is say V and when you are actually comparing with the velocity with which the model is rotating then we have to see that how this Coriolis effect can be you know considered. So let us look into this particular Coriolis effect in depth and what is the theory behind this Coriolis effect. So what we said is that the Coriolis effect which is nothing but it is the effect caused by the radial acceleration this is again the effect caused by the radial acceleration field when there is a movement of the model in the plane of rotation. So this movement of the model can actually occur because of certain like some certain construction process like assume that we are having a soft soil bed and with certain stresses tree and there we wanted to construct an embankment because in the construction of embankment on the soft soil happens in different stages. So let us say that we are actually simulating that so that for that what we do is that we use the sand hopper technique so inflate sand hopper is used. So when the particle is released in the when the particles are released sand particles released to construct sand embankments during flight there is a possibility that these particles are subjected to so called Coriolis acceleration or Coriolis effect. So the construction of embankments of soft soil during centrifuge test and triggering of rainfall during centrifuge test for example if you wanted to study the effect of rainfall on geotechnical structures when you are actually modeling this the rain droplets when they are actually simulated in the form of a mist or in the models and these droplets they are subjected to when they are released in the high acceleration field they can be subjected to this Coriolis acceleration that is it is basically the Coriolis effect is the effect caused by the radial acceleration field when there is a movement of the model within the plane of rotation. Then another thing is that studies on the geotechnical structures subjected to earthquakes for example when the model is subjected to earthquake perturbance due to by with the help of some actuator which can actually induce this phenomenon then what it does is that this is subjected to the particles velocities or such that they can actually get experienced with this Coriolis effect. So for some models which is actually required like construction of embankments on soft soils or let us say when we are studying rainfall effects on some geotechnical structures or geotechnical structures subjected to earthquake models it is mandatory to check whether the model is actually free from Coriolis effect or not. So in order to understand the theory of the Coriolis effect consider a model rotating about a vertical axis in a horizontal plane what we are seeing is the plan view and here this is the center of the shaft and x and y are the global coordinates at the center of the shaft and the model is rotating in this direction and this is the velocity v is equal to r omega where let us say that this is the radius which is actually up to the element a element which is considered here this is an embankment having a physical dimension length from the this axis and constructed with a material having rho. So what we need to see is that when this is subjected to the so called perturbance what will happen to the acceleration components in this. So what we need to do is that we have to first express this in the terms of polar coordinates and try to get the local accelerations at the center of the shaft and then by using the method of transformation it transform the local acceleration to the global accelerations to the local accelerations within the model that is the local axis within the model they are x dash and y dash so now what we do is that the coordinates of x and y can be expressed in radial polar coordinates as x is equal to r cos theta where r is the radius from the center of the shaft the center distance where x and y origin is there to up to the element which is actually considered in the previous slide. So y is equal to r sin theta so we have x is equal to r cos theta and y is equal to r sin theta so local velocities can be obtained by differentiating this dx by dt is equal to dr by dt cos theta minus r d theta by d sin theta similarly dy by dt is equal to dr by dt sin theta plus r d theta by dt cos theta. So let this be regenerate as one so these are the local velocities then by differentiating once again what we get is the local accelerations but these are with reference to the axis the global axis at the center of the shaft. So the local accelerations are given as d square x by dt square is equal to d square r by dt square cos theta minus 2 dr by dt into d theta by dt sin theta minus r d square theta by dt square sin theta minus r d theta by dt whole square cos theta.  This is let us regenerate as number 2. Now what we do by applying the transformation of from the global axis to local axis by applying the method of transformation what we can write is that the basic equation is that x dash so here you can see that one axis is getting transformed from the center of the shaft to the axis within the model which is actually rotating. So x dash is along x axis of the model that is horizontal axis of the model y dash along the vertical axis of the model. So we can write x dash is equal to C minus x sin theta plus y cos theta c is a constant y dash is equal to constant minus x cos theta minus y sin theta. So in order to get these things what we do is that to get the local accelerations with reference to axis within the model first you do the differentiation get the dx dash and dt and then after further doing successive differentiation to this what we get is the d square x dash by dt square. So the d square x dash by dt square term works out to be minus d square x by dt square into sin theta plus d square y by dt square cos theta. Similarly in the y dash direction d square y dash by dt square is equal to minus d square x by dt square cos theta minus d square y by dt square sin theta. So by we know that d square x by dt square d square y by dt square with reference to global coordinates at the center of the shaft similarly we also know here and here by substituting and simplifying what we get is that the following terms which is nothing but by simplifying the substituting for d square x by dt square and d square y by dt square what we get is the d square x dash by dt square is equal to 2 dr by dt d theta by dt plus r d square theta by dt square. So if you look into it this particular element when it is actually it has two components of acceleration one is this 2 dr by dt d theta by dt plus r d square theta by dt square and similarly we have d square by dash by dt square is equal to minus d square r by dt square plus r d theta by dt whole square. So if you look this particular element which is considered in the centrifuge model what we can write is that this particular term in the horizontal direction that is 2 dr by dt d theta by dt plus r d square theta by dt square that is this particular term. So this particular term is called as the Coriolis acceleration term. So when the model when the element is subjected to certain amount of acceleration so what we are actually acceleration field what we see is that there is some lateral acceleration which is actually acting on the element which is actually called as Coriolis term is also called as the Coriolis acceleration term which is nothing but 2 v omega ac a suffix c is indicated as Coriolis acceleration which is nothing but 2 v omega plus this term is due to inertial effect in horizontal direction because we actually assume that this model is subjected to some subparted bonds with due to some shaking let us say in that case this is actually due to inertial effect in horizontal direction. When you look into the elements here the when the vertical component d square r by dt square is due to is the inertial effect in vertical direction then r t theta r d theta by dt which is whole square which is nothing but the n g term that is nothing but r omega square term that acceleration acting towards the element acceleration acting towards the center. So here in this particular derivation by taking from global accelerations from global coordinates to the local coordinates and the local accelerations have actually expressed as we have that they have got two components in x direction two components in y direction and in the x direction what we have observed that the term which is called as 2 v omega that is the local acceleration term which is called the Coriolis acceleration term. Then we have the Coriolis effect which is actually defined as ratio of Coriolis acceleration to the radial acceleration that is ac by ar so which is nothing but we can write now ac is nothing but 2 v omega by ar let us assume that up to this ar which is nothing but like we have discussed the r can be effective radius so it is re omega square. So by simplifying using v omega is equal to v by r or v by re we can write that Coriolis effect is nothing but 2 v by v so here what we are doing is that we are actually comparing two velocities one is small v is nothing but the model velocity and capital V is nothing but the small v is nothing but the velocity of the particles within the model it can be due to seepage velocity or it can be due to some wall or wall movement the soil particles are moving or it can be due to the particle velocity which are generated because of the some seismic perturbance. And then capital V is nothing but the model velocity the velocity with which the model is rotating. So if you further you know analyze this one it is actually generally said that if the you know the percentage error due to Coriolis effect ac by ar if it is say less than 10% if that actually happens that means that if it is less than 10% then what it actually means is that the Coriolis effect can be neglected. So if the ratio of v that means that for ac by ar less than to be equal to 10% what we need to do is that the v has to be equal to 0.05 v if this is less than or equal to 0.05 v we can say that the Coriolis effect can be you know negligible that means that the model is not subjected to Coriolis effect. So there are also you know many events the tests which can actually take place let us say that we are actually investigating the effect of you know igniting a blast load on the you know performance of some structures buried structures. So when the explosive is say subject is ignited remotely then what you can see that the ejecta which is actually thrown with very very high velocities that means that this particles soil particles are thrown with very very high velocities. So in order to calculate this radius of the curvature subtended by these particles Fokhrowsky and Fedorov in 1968 they have given an empirical expression they saying that Coriolis acceleration is given by v square by RC v is the velocity with which the particle is actually is being throwing out and the RC is nothing but the radius of the curvature. So they try to compare the radius of the curvature subtended by the particles as well as with the radius of curvature which is with which the model is being rotating from which what they have reduced is that RC is equal to v square by AC and AC by substituting we get 2 v omega so we can that RC is equal to v by 2 omega so using omega is equal to v by r re what we can get is that v by 2 into v by re by taking re this side we can write RC by r is equal to v by 2 v. So it actually appears is that if RC the radius of the curvature subtended by the particles is say much much much higher than the radius of the curvature of the centrifuge then the effective radius then v is supposed to be 2 v that means that the particle will be thrown with a very high velocity and it is subjected to and then it will hit the container boundary and then falls. So in general Coriolis effect cannot be neglected for any object moving within the velocity range that means that if the velocity is within 0.05 v to 2 v then we can say that Coriolis effect cannot be neglected if the velocity is actually more than 2 v then we can say that the Coriolis effect is nonexistent that means that the model will not be affected by the Coriolis effect. So for example if you are having a model with v is equal to 30 m per second Coriolis effect can occur if the model velocity occur between so as per our this thing 0.05 v 0.05 times 30 is 1.5 m per second and this is 2 v that is so if the velocity of the moving object within the model is within 1.5 m per second to 60 m per second then we can say that the model is subjected to Coriolis effect otherwise the model is actually free from Coriolis effects. So in this particular lecture what we try to understand is that how the static equilibrium of centrifuge model can be established and also we also try to look into some aspects of how to calculate the effective radius and then we discussed about the limitations of centrifuge based on physical modeling as we have said in this particular slide where we have got the non-homogeneity and isotope of soil profile but with the advancement of the techniques this is also being you know worked out now with the modern physical modeling in geotechnical engineering and some limitations of modeling tools and boundary effects and scale effects and the inconsistence of scale factors which we are going to discuss in detail and then we actually had a discussion on the Coriolis effect.


After my talk Yeah, I was the same yesterday I got back to my hotel at three and I just slept Yeah, hey, I'd like to introduce Josh Josh holds a PhD in quantum information from the University of Western Australia and is now working for the quantum full stack startup Xanadu in Toronto where he's part of the quantum machine learning software team And quantum open-source project that he's actively working on includes strawberry fields and penny lane and today He's going to talk about quantum machine learning with penny lane. So please welcome Josh Hi, everyone. So as Mark mentioned, I'm a my name is Josh I'm a quantum software developer at Xanadu and Xanadu were a full-stack quantum confusing company So we're working on both the continuous variable hardware as well as open-source quantum software development so as mentioned one of our software platforms of strawberry fields, this is for Integration and submitting jobs to our quantum hardware I encourage you to check it out strawberry fields to AI for the purposes of this talk I'll be focusing on one of our other libraries penny lane, which is a open-source quantum machine learning framework So for this talk, so I gave this a similar talk last year introducing penny lane this year I'll be taking going back a bit Discussing where the ideas for penny lane came from and also moving forward a year So discussing what's changed in penny lane over the last year since I last talked at plus them so starting off Penny Lane was conceived about July 2018 we were noticing that a Quantum machine learning had become quite a buzzword the number of papers in quantum machine learning the archive had started to increase significantly and as we were in this noisy era of quantum computation where we have these small-scale devices People were wondering what could we do to combine machine learning with quantum computing and what could we get out of it? So we sat down and we asked ourselves. Why was deep learning so successful over the last Decade or so and it basically came down to three main points Hardware advancements, so we'd seen GPUs to become immensely important in machine learning the ability to scale up these Thousands of small computations that people have been using in graphics for decades for machine learning caused a seismic shift in what we could do and This is in addition to tensor processing units. So TPUs as well have So revolutionized what we can do in machine learning In addition the abundance of workhorse algorithm, so things like back propagation gradient descent We were able to suddenly train massive neural networks using Exacted differentiation techniques that were much more stable than what we were doing previously and Finally, and this is probably a really large one is specialized user-friendly software So things like TensorFlow, PyTorch, Jax These machine learning frameworks that people most of you in the room probably familiar with these have done a huge amount to increase the accessibility of machine learning Yeah, so this is kind of just Leaning more into that so you can see that the number of projects using TensorFlow have increased almost exponentially over the last 10 years or so and As the industry is matured these software frameworks are becoming more accessible more easy to use and there's more choice as well So TensorFlow, PyTorch, Jax the question is Can we train quantum circuits? so It was a question people had and people start playing around with in the quantum computing field It turns out quantum computing is also differentiable So you have a small quantum circuit which is controlled by quantum gates These quantum gates take classical parameters You perform a quantum computation on the hardware and then you perform a quantum measurement at the end and from these quantum measurements You can compute measurement statistics So you can measure certain quantum observables and take expectation values or variances and these things are deterministic If you put in the same parameters to this quantum circuit you get the same output So it turns out that these quantum circuits are differentiable The outputs depend smoothly on gate parameters Can we practically train quantum circuits though? It turns out we can and Quantum machine learning methods called variational quantum algorithms have become one of the main near-term quantum algorithms for near-term devices So this happened around 2011-2012-2013 Ideas start forming in the quantum community things like the variational quantum eigensolver Which was a really neat idea at the time saying that these quant variational quantum devices They give us access to this enormous state space that we can't normally access with classical techniques Can we use can we train the classical parameters of these gates? to generate a state that outputs a measurement statistic that we are looking for that we want to find So this is kind of this variational quantum algorithms described on the right-hand side here So we have a quantum sub routine. This is a quantum black box We just have quantum gates on our quantum hardware We prepare a quantum state and we have some circuit parameters that drive these quantum gates We run the quantum circuit we get a measurement output and then we perform some classical processing and classical optimization so we use techniques such as optimization and Gradient free optimization to update these parameters via some classical optimization loop So this was first these two algorithms ventured up here VQE and QAOA These were sort of the first two major variational quantum algorithms, but since then the fields really grown and There'd be natural extensions to other circumstances that we've seen from standard machine learning so things like Quantum classifies can we distinguish different quantum states by using the source of variational quantum algorithm? Nowadays there are many many proposals just because of how well the scales to noisy devices We don't have fault tolerance yet in quantum computing But these algorithms seem to work so far very very well on the noisy devices that we have So what we want to do is we want to be able to train Quantum variational quantum algorithms, but we want to do them using some of the workhorse algorithms They've been developed over the last 10 20 30 years of machine learning So one technique we have of doing this. It's called the parameter shift method so Initially we were using gradient free methods to train quantum circuits This shifted a bit and we started using something called the parameter shift method So this is an analytic method of computing Gradients coming out of quantum devices So the main insight here is we're able to use the same quantum hardware to evaluate its own gradients And this is a significant insight Because if we weren't able to do this We'd have to classically simulate the circuit to get the gradients and that's intractable is the size of the quantum devices We have increases You'd no longer be able to classically compute the gradient to perform this classical optimization loop So you're constrained to gradient free optimization techniques. So this is just a Very short diagram that sort of demonstrates this parameter shift method So on the left we have our variational quantum circuit It's an arbitrary circuit We have a couple of gates running on the quantum device and we have our input gate parameters and our output measurement statistics All the parameters shift rule says is that if we Take the exact same quantum circuit that we want to train We shift the parameters forward by some value s backward by some value s and take the difference The quantum device will actually output the gradient Of the quantum expectation value at that parameter value And the crucial thing here is that this shift is macroscopic So It's not finite differences. We're not doing numerical differentiation. We're doing analytic differentiation And we're able to get the output of the gradient with noisy devices even when Even outside of regimes or we might be drowned by noise So the quantum variational algorithms we've had so far have been quite simple in scope They've been mostly maybe one or a couple of quantum devices And we're getting the output of that quantum device and we've got that classical optimization loop directly training this quantum circuit So Back in july 2018 or so we were thinking whatever we want to do more than that Whatever we want to do something like this So we have an arbitrary computational graph On this diagram here the green uh boxes the green nodes. Those are quantum circuits The yellow nodes are just arbitrary classical processing And we want it to be as arbitrary as we want Why can't we construct something like this say we have a machine learning model Involving lots of classical layers. We want to replace a couple of them with quantum layers We want to do something like that in open source quantum software and we want to see what happens So that's kind of where we were. We knew what we wanted to do. We wanted an open source framework That could interface with the accessible machine learning framework such as pi torch tends to flow jacks We wanted it to drive real quantum hardware devices using these techniques for analytic gradient computation on real devices And we wanted it to be accessible intuitive And allow you to perform arbitrary computations mixing classical and quantum however you want So we knew what we wanted to do. We weren't really sure what it should look like So I for this presentation actually scrolled Up quite high into the deep dark depths of my slack history To see what we were discussing when it came to the UI of penny lane So I've got a couple of the messages here before I actually show you what the UI looks like This is the very first one the coach would look as much as like numpy as possible This was a very popular statement in our slack channel when we were discussing the architecture of penny lane So there was lots of agreement there We kind of got a bit distracted though and we started looking at context managers We were like maybe we want to train things using context managers in python We spent I'm a bit surprised looking at slack history, but we spent about two to three days trying to make this work Before realizing it's absolutely atrocious idea. People don't want to train things using context managers But hey a quantum node is kind of like a function. You have input classical parameters you have output Classical data that's come from taking measurement statistics of your quantum device You want to define it once and then you want to evaluate it multiple times To be able to train it. So we started thinking hey Quantum devices in this framework should just be functions that you call And compose with your classical functions So that's kind of where we ended up. So that's our prototype on the left of what this quantum machine learning framework should look like And that's what we ended up with so In penny lane all quantum circuits are functions within these functions. You define your quantum operations The function state classical parameters and they output Measurement statistics So this is just a quick overview of what A quantum machine learning task looks like in penny lane the main Things we want to do with penny lane. I've mentioned them slightly before but just to reiterate It's open sourced and developed openly So not only is the code available on github But we also develop it openly on github So I think that's quite important and that anyone who wants to contribute can Look at our github issues Look at our pull requests and see the roadmap for where we want to take penny lane And this goes back all the way to the very beginning as well If you go back through the history of the github repo you'll see all those ideas In terms of the ui and what we want penny lane to do as pull requests and issues So we want penny lane to be functional Penny lane is functional. It has a python interface. It's designed to integrate Seamlessly with the machine learning frameworks that we have right now. So numpy pie torch and TensorFlow You can train the quantum computer the same way as you can within your own network And crucially it's designed to scale as quantum computers grow in power So in penny lane All quantum functions Are actually usable on the quantum hardware we have available. So we have quantum hardware available over the cloud from regetti from ibm There are plugins available to connect penny lane to that And by just changing a couple of lines you can switch What devices you're running your quantum functions on? So the main abstraction layer in penny lane is I've discussed this previously by a quantum node so the quantum node is what encapsulates your quantum computation and That's the diagram there on the left. So you have your quantum circuit input parameters input state preparation Output expectation values or variances or samples or probabilities. These are all things you can train on analytically in penny lane so the Goal is that a computation is end-to-end differentiable We want you to be able to use a workhorse algorithms like gradient descent across your quantum classical hybrid model So all quantum nodes in penny lane will always be Differentiable end-to-end you can combine it with your other models in Classical machine learning and the entire computational graph will be end-to-end differentiable So how do we do this? So with penny lane we support three main interfaces. We support numpy py torch and tensorflow And the way this works is we simply we accept the tensor objects from these three frameworks So numpy if you're using autograd and array box the other two a torch tensor or tensorflow tensor When it comes time to evaluate the quantum portion of your computation What we do is we just unwrap and extract the numerical value We evaluate it we use our formulas and internal architecture to evaluate the gradient Converse it back to a tensor object and as well pass on the gradient information to the machine learning framework So this is how we're able to make the entire thing end-to-end differentiable Even though we're using a mixture of quantum and classical hardware So you can almost think of the Quantum hardware as accelerators in a sense similar to gpus and tpus Penny lane is framework and hardware agnostic So I mentioned previously that one of our framework hard one of our hardware frameworks is strawberry field This is what we use to connect to our photonic hardware that we're developing in our photonic labs In addition We want to be able to mix and match so as we're in the noisy era there are advantages to our photonic technology There are also disadvantages to our photonic technology This is similar to superconducting cubits. This is similar to d waves quantum annealer All these noisy Devices have their pros and cons. They have applications. They're best suited for and applications that may be a better in a different architecture So we want to be able to combine these different frameworks these different quantum devices Um, so you can take any of these there are plugins available for kiskit forest Cirque and strawberry fields and combine them with either tensor flow pure non-py for scikit-learn or pytorch The cool thing is you can even In one computation have devices from different providers. So You can have a for instance This circuit down here in the bottom left. That's a photonic circuit. It's going into a classical processing node doing the cosine and square root And that's going up to a qubit circuit So you can take a provider from regetti or take a device from regetti or ibm combine this with a photonic device Train over the entire architecture So for those familiar with machine learning, but maybe not quantum machine learning Here's a very quick example of what you can do. So we have a quantum circuit. We're rotating Uh We have one qubit in this quantum circuit We're rotating the qubit around the x-axis Controlled by a classical parameter doing the same thing around the y-axis controlled by another classical parameter And just measuring where this uh The state the qubit ends up in measuring the z the poly z expectation value So this is pretty simple in penny lane. You just uh Create your device so the device is just an abstraction for the qpu that you want to run and train your computation on in this case i'm using a qpu device from regetti first You define your quantum function. So this uh, this is a q-node. It corresponds exactly to that quantum circuit in the previous slide I've got my two rotations. I'm returning expectation value of the poly z measurement And the last part is just pure classical optimization exactly like you'd see in the pi torch or a tensor flow Uh machine learning example in this case. I'm just using pi torch to um optimize So i'm defining my cost function which takes In the uh the q-node and i'm just optimizing that to reduce the expectation value of that q-node So the key thing I guess is that The penny lane q-node is It makes pi torch tensor uh quantum aware in this example You can take a q-node And use it within pi torsion from pi torches point of view or tensor flows point of view This function is a classical function that's end to end differentiable So these examples are available on our website as a tutorials Just on the right hand side is the animation of this Short script actually running on a qpu. So we're training the qubit to To enter the state that's defined by the cross on the block sphere and you can see the qubit being trained and approaching The required state So over the last year we've worked hard on making penny lane more accessible So we want it to be a high level abstraction. A lot of the other frameworks are They're object oriented. They're low level and they're very hardware focused. So you can access things that Maybe specific to a particular hardware and things that we want to do sometimes so sometimes We need to think about the underlying hardware that we're running our quantum programming on Other times we don't we want to think in a high level abstraction We want to maybe do something assuming that The hardware will do the best it's can based on the underlying framework So that's kind of where we're positioning penny lane and sources the num pi interface Wrapping will pack underneath So a couple of things we've added to the library of the last year to make it more accessible to provide an Easier way of constructing your quantum machine learning models Built a library of circuit ansatz from the qml literature So it's kind of an unanswered question about what quantum circuits ansatz are easiest to train and best for particular machine learning models and quantum algorithms Papers are continuously coming out with new ansatz that provide benefits for different different use cases So these are easily accessible in penny lane. We have a qml.templates library and you can just include these in your quantum node We heard feedback from a lot of users that said that they had Codes already using kiss kit or already using pi call that they wanted to Um interface with their machine learning models, but They didn't want to have to rewriting it using penny lanes interface So we added the ability to import from these quantum libraries So you can import your code from kiss kits from chasm from pi quill from quill Um, it runs the same circuit. It gets converted into a q node and it just becomes end-to-end differentiable So you can take your existing code And then interface it with pi torch or TensorFlow straight away So automatically adds analysis differentiation to existing pi quill and kiss kit frameworks Cool thing is you can take these programs loaded from regessie forest convert it into q nodes and then deploy it on other quantum hardware if you'd like We're also working on adding optimization that's it may be better suited for a quantum landscape so things like The quantum natural gradient So taking into account what the cost landscape of your variational circuit looks like In order to ensure better convergence. So this is internal research that we've been working on at zanidu. Um, this is just a Outline of what this algorithm looks like. It's a kind of a quantum equivalent to the natural gradient if you've heard of that Which uh takes into account classical euclidean space when you're doing gradient descent Quantum natural gradient takes into account the quantum geometry when doing gradient descent And in general can lead to much better convergence with variational algorithms So this is just a plot comparing for a couple of circuits Uh gradient descent some blue versus Adam and green and then quantum natural gradient descent in uh reds and black for different approximations Um, we've also heard feedback that people want their simulations to be faster So with pinning lane, we took the approach that it was hardware first If you defined a q-node it should always be executable on hardware But there are shortcuts we can take with classical simulators One of those is if we write the simulator using tensorflow or pie torch, for example We can do classical back propagation techniques that scale better insertion regimes So penny lane now includes a tensor network simulator Uh built using google's tensor network software and tensorflow Uh, so this uses classical back propagation inside the q-node To compute the gradient which if you have a lot of parameters can be faster. So enables routines where maybe you're Training your quantum circuit on a simulator you get your Trains parameters and then you deploy using test data on the quantum device We also have a new quantum chemistry plugin. So Right in the beginning of the talk I mentioned that one of the First variational quantum algorithms to really create this qml second 2.0 wave Was the vqe algorithm the variational quantum eigen solver So this algorithm allows us to use quantum computers to Probe the electronic molecular Energies of various molecules So penny lane now interfaces with another open source quantum library open fermion by the quantum team at google So using open fermion you can Converse at any electronic molecular structure To a quantum observe a set of quantum observables and these are now importable into penny lane for training on quantum hardware So the other thing we really want to do is we I mentioned at the beginning We want to make it accessible and part of that is the ui and what frameworks we interface with Another part of it is having good documentation stable api and Collection of tutorials for people to explore how to use penny lane So if you go to penny lane dot ai slash qml, please check it out Where we have a constantly growing collection of quantum machine learning tutorials Using penny lane across a mixture of different hardware devices hardware frameworks and also machine learning frameworks And these are things from just looking at variational quantum eigen solver looking at quantum state preparation To things such as quantum transfer learning doubly stochastic gradient descent in quantum systems Quantum structure learning etc So please have a look this not only is penny lane completely open source and developed in the open But the penny lane dot ami dot ai slash qml website is open sourcing on github So any of these true tutorials that are available there can be downloaded Can be edited with pull requests and we also accept pull requests for new tutorials We also have a twitter account penny lane ai so please follow and we'll We try every week or every two weeks to tweet out new qml results We see as well as new tutorials that are available using penny lane So we wanted to see where we could go with penny lane once we had this base ui the space abstraction We noticed that The core abstraction penny lane is this q node concept So a quantum device corresponds to a quantum function in python that is an end-to-end differentiable and can be used with any framework But we started to notice that it wasn't enough for us as internal researchers at zandu We needed The ability to process these q nodes At a higher level of traction we found that we would have groups of q nodes And we wanted to work with these groups of q nodes without thinking about the low level quantum circuit within each q node And this is something that happened when we were thinking about vqe Essentially the vqe module was two things. It was a mapping operation We were taking a quantum circuit And mapping it across different return values different expectation values. We wanted to measure And then a reduction operation Once we evaluated all these different quantum nodes We wanted to just take the dot product between these quantum nodes And coefficients that defined the problem we were solving So we wanted a better abstraction for working with groups of q nodes that was algorithm independent We also noticed that we were getting to the point where multiple q p's were available right now on the cloud So if you're using ibm, there are I think five or six ibm devices available Uh for the public. So if you sign up to their accounts, um, you have access to multiple devices right now um The same thing is available on riggedi forest. There are I think two chips now available on the qcs aspen 4 and aspen 7 so Maybe a year ago we were getting excited by having access to one device and that was fine now we have access to couple of devices And it can be frustrating at sometimes now that we know we can have more than one. We want a thousand so We started thinking what can we do with rather than have a single big q pu which is what a lot of hardware providers are Working really hard to get towards What can we do in the interim if we had a thousand small q p's? So we recently introduced the concept of a q node collection So a q node collection is a group of q nodes that are evaluated that can be evaluated independently Um, so they can evaluated uh at one go asynchronously Um at the same time We wanted to be in the abstraction that you don't have to think about too much And a common way for creating q node collections and various, uh quantum algorithms is you map a ansatz of template Over measurements, so we wanted to extend that as well. Not only can you map a Ansatz or a quantum template over measurements, but you can also map it over devices So I have a small example of what this looks like here So here I have a quantum circuit ansatz Um, it just does two x-axis rotations Uh, both of these parameterized and then it does a c-naught which is a quantum entangling gate Then I have a list of uh observables. I want to measure So these are just two qubit tensor observables. I want to measure poly x cross poly z And I also want to measure poly z cross poly x And then say I have access to two devices. So in this example here, I'm just using the regetti forest qvm simulator Uh with Two devices that are currently available on the simulator aspen 4 and aspen 7 So previously you would create two q nodes. You would be forced to evaluate them sequentially as you normally are in python We introduced a way to do this that makes it slightly simpler So you can treat you can use this new qml dot map operation to map your ansatz over both your observable list and your list of devices Say what you want your measurement to be And the output of that is not just a single q node, but a collection or a group of q nodes And these group of q nodes are treated as one single object in penny lane So in this example here, you have some parameters. You pass it to your q nodes Both of the quantum devices are evaluated At the same time and you get your results back as a single numpy array And the really cool thing about this is because we have access to these devices that we can access At the same time It's embarrassingly paralyzable. So we've added this asynchronous dispatch support to penny lane So this is experimental. It's only recently been added Uh, but if you call your q nodes normally in this case, it takes about five seconds If I pass the parallel equals true arguments, penny lanes dispatching both of these quantum circuits to be evaluated at the same time Uh, you get your results back in almost half the time So we added this ability to create q node collections, but we also want them to be easy to use So we introduce some high level, uh, quantum function high level processing functions They allow you to impose your q node collections With other, uh, classical processing functions from your machine learning libraries So things like q ml dot sum q ml dot dot q ml dot apply So in this very simple three line example here I'm creating a uh, so i'm using pie torch in this case creating a tensor of observables a tensor of coefficients rather And then you can see in the second line I've got my q nodes and i'm just composing it with a whole bunch of different classical processing functions uh to manipulate the output of these q nodes and With the ability to manipulate q node collections in this way, we've gone from something that might take Uh 10 or 15 lines in python to something that only becomes a single one liner in python So in this particular case, I'm applying the torch sign function to my q nodes And then performing the dot product between The sign of those q nodes and the coefficients I just defined All this happens lazily in penny lanes So the quantum evaluation only ever happens when you actually Evaluate your q node collection your cost function So the third line here when i'm calling my new cost function with the parameters That's when the dispatching happens and penny lane starts Dispatching the quantum computations to the external quantum devices and Performs the classical processing so I hope this talk is kind of given uh You guys an idea of our vision for quantum machine learning at zanidu We want uh quantum machine learning to be intuitive accessible Um be easily explored and we want anyone to have the ability to run the quantum machine learning algorithms Whether it's on simulators or actually on hardware that's available in the cloud We want quantum machine learning models to be widely widely shared and re-implemented by others So that's sort of the impetus behind the penny lane dot ai slash q ml collection Um circuits should be reusable embeddings pre and post processing should be easy We want that to be something that you don't have to think about We want the documentation to be able to To be designed so that you can find these things easily and that maybe you don't even have to look at the documentation Rather than searching throughout what methods are available uh You just find a function in penny lane that applies to that thing for you and you can do function Compositioning on your quantum uh devices Um penny lane is open source development is open source and uh, I think as uh Mark mentioned in tomas mentioned in his first talk a lot of the physicists working on quantum open software at the moment A lot of the developers are physicists. I'm a physicist a lot of my team are physicists um And sometimes It helps to get feedback and contributions from people outside of physics So something we'd love is for people in this room to have a look at our open source code Have a look at the issues that are opened and see if there's anything they feel they can contribute because maybe From someone outside of physics you guys can Add features or implement things in a way that we would never have thought of doing ourselves Uh, so thank you. Uh, I've got some links at the top. So this is our twitter link and our qml tutorials page


The Grape Nuts Flakes program starring Orson Welles who's pinch-hitting for Jack Benny with Mary Livingston, Phil Harris, Dennis Day, Rochester, and yours truly, Don Wilson. Friends, you've all heard that line about stretching a point, haven't you? Well, here's where I turn out to be the best little stretcher-bearer of a precious ration point you ever heard about. All you have to do to save a ration stamp at breakfast is to make it Grape Nuts Flakes every morning. For Grape Nuts Flakes are non-racioned, nourishing, whole-grain cereal. But that's only half the story.  It's your old friend that sweet-as-a-nut Grape Nuts flavor turned out in extra crisp, extra appetizing flake form.  And when you buy this swell and ration-stampless breakfast cereal, be sure you ask for the big 12-ounce economy-sized package of Grape Nuts Flakes. Trained to Memphis, played by the orchestra. Now, ladies and gentlemen, once again, we take you to the little drug store to the NBC building here in Hollywood, where our whole gang always gathers for a bite to eat before the show. The time, exactly 15 minutes before this broadcast. Take it away, drug store! All right, Cliff. Yes, Gilroy? How's that three-decker sandwich coming along? Fine, I'm just putting the poop back on it. Here you are. Right, Cliff, that's a beauty. Well, here comes Don Wilson and Phil Harris is with him. That old-time glamour boy. If he pulls any of his lulus today, I'll smack him right in the kisser with his lemon pie. Radcliffe, put that down. Hello, Mr. Wilson. Hi, Phil. Hello, boys. What a beat, tWitch. Well, I ain't hungry, but give me a donut. I want to practice blowing smoke rings through the hole. It's a lulu. You ask for it, brother. Radcliffe. Bullseye. The way he threw this, who hit me with that pie? Now, calm down, Phil. Here's my handkerchief. Wipe your face. As soon as I get this stuff out of my eyes, I'm gonna murder somebody. Duck, Radcliffe. He's plenty mad. Here I go, under the counter. Whoop, bop my head. That's what's wrong with him. He's always buffing his head. I wonder what happened to Barry. She promised she'd beat us here. If you're looking for Miss Livingston, she's over there in the phone booth. She got a call a few minutes ago from old man Benny in Phoenix. And she asked me to tell you to be sure and wait for her. Well, how do you feel, Jack? Well, what do you mean you still have your coat, but you've got your strength back? Oh, now you can blow a hole right through the Kleenex. When will you be back on the show? Next Sunday? Well, you don't have to hurry. Orson Wells has been very funny on the program. I said Orson has been very funny. Oh, Jack, for heaven's sake, stop crying. Of course we'll be glad to see you again. Goodbye. What's that? No, Jack, I won't. I don't care. I won't do it. I will not send your old sailor suit so you can get a cut right on the train. Goodbye. Oh, hello Don, Phil. Hello, Larry. Hiya, Mary. Phil, I'm ashamed of you. What's the idea of giving yourself a shampoo here at the counter? That ain't no shampoo. It's lemon pie. I just walked in and pulled a gag and that Radcliffe let me have it. Let you have it? Yeah, and I only said, I ain't hungry, but give me a donut. I don't want to practice blowing smoke rings through the hole. I love it, love it, love it. Radcliffe. Now cut that out. Uh, what did Jack have to say? Oh, he feels fine now. He'll be back on the program next week. Well, I'll be glad to see the old boy, but I'm sure going to miss Orson. You know, him and me was getting to be cultural buddies. What do you mean? What do you mean, Phil, cultural buddies? Well, one night we go to an opera and he explains the barber of Seville to me, and then the next night we go to a burlesque show and I wise him up on Betty Roll in that ball of fire. What? Well, we got sort of a lend-lease on brains agreement, kind of. Oh, comb the pie out of your hair. Hello, Dennis. Hello, Miss Livingston. You know what, Dennis? Jack will be back on the show next week. Jack who? Jack and the beanstalk. Who? Jack and the beanstalk. That'll be a novelty if nothing else. Oh, quiet. Well, look who's coming in. It's Miss Harrington, Orson's secretary. Hello, Miss Harrington. Hello, Mr. Wilson. Good afternoon, everybody. Hi, Emerson. Oh, young man, will you please take an order? Yes, ma'am. Mr. Wells will be here shortly for a malted milk with an egg in it, and the egg must be absolutely fresh. Yes, ma'am. All right, Cliff. Yes, Gilroy. One fresh egg. One fresh egg coming up. Get busy, Gwendolyn. Give out. The leg. Attention, everybody. Mr. Wells is approaching. Good afternoon. This is Orson Wells. Mary, what do you hear from Jack?  Yeah, I just had him on the phone. Jack's feeling fine now, and says he's back on the show next week. Well... Looks like you'll have to hit the road, bug. Dennis. Naturally, now that Jack feels better, he should resume his rightful position. As for myself, I shall hold my tent like an Arab and silently steal away. When you put it that way, Orson, I could sob. Thank you, Phil. Thank you. Oh, Miss Harrington, you ordered me a malted milk with an egg in it. Yes, Mr. Wells. And are you sure the egg was absolutely fresh? Yes. I was here for the unveiling. Excellent. Oh, Orson. I'm afraid you won't have time for that malted milk. We're due on the air in just a couple of minutes. Very well. And I shall have it delivered to me. Oh, boy. Yes, Mr. Wells. I want this malted milk delivered to Studio B as soon as possible. Well, bring it right over, Mr. Wells. Hey, wait a minute, Gil. We ain't no carrier pigeons. Get back under that counter. Yes, Mr. Wells. That's not an egg, Gwendolyn. That's his head. Gwendolyn, control yourself. Hurry up, Orson. We'll be late for the show. Follow me, everybody. Last one in the studio is an old tomato. I'm taking a chance on love, as who doesn't?  And, uh... Phil, that's the last time I'll be announcing your band number. Just think, after tonight's broadcast, I shall fold my tent like an air-rap, and silently steal away. Don't say that, Orson. Look, tears is tinkling down my cheek. That's trickling, Phil.  Well, you've got to admit their bloodshot. That's another story. Seriously, I'm going to miss all of you, and if I ever get my own program, I'd like this gang to sign up with me. You mean you'd like to have us work for you? Yes, of course. I don't know if I could meet the salaries that Jack's been paying you. They're really lousy. Why do you want to meet them? Very good, Mary. Incidentally, before I forget it, I'd like to have all of your phone numbers. Okay, here's mine. Here's mine, Wellsy. Here's my number, too. Thanks, fellows. How about yours, Mary? I don't know.  Oh, no, no, no, no, Mary. If I call, I assure you it'll be strictly business. Then you can't have us. Girls are certainly aggressive nowadays. You're telling me I went up to Alice Fay to preview one night and asked her for autographs. The next thing I know, I'm married to her. Found it. Women are sure hard to figure out, eh, Wellsy? Yes, Phil, but it's delightful research. And now, ladies and gentlemen, this evening, for our feature attraction, O-W-P will present... O-W-P, what's that? Orson Welles Productions, Mary. I was trying to be modest. Don't cry. Dennis. Whoops, he bumped my head. Next time, it'll be harder. As I was saying, for our feature attraction this evening, go away, Dennis. Orson Welles Productions will present a dramatic thunderbolt which we trust will thrill and delight you. Well, I hope it's something Shakespeare-ing, like the taming of the stew. The shrew. No, Phil, it isn't the taming of the shrew. Well, is it Hamlet? No, it isn't, Mary. Is it The Merchant of Venice? No, Donna. As a matter of fact, it isn't Shakespeare-ing at all. Tonight, ladies and gentlemen, we will present our own version of... Little Red Riding Hood. Isn't that delightful? Little Red Riding Hood. Isn't that kind of juvenile for you and me, Orson? Perhaps, Phil, but I think the kid is, you're listening and deserving of a playlist, especially for the heaven. Now I will play the part of Old Man Hood, an elderly farmer who lives at the edge of a forest. Mary will be my daughter, Red, and Dennis will be my son, Robin. Robin Hood. Are you giggling, kiddies? If they are, they got feathers and they're rompers. Now, let's see. Don and Phil. Oh, yes. Don will be a tree in the woods. Only God can make, and Phil Harris will be a wood chopper. Phil Harris will be a wood chopper who cuts Wilson down. I'll take it easy, Phil. You know how ticklish I am. No kidding, Donna, you really? Oh, didn't you know that, Mr. Harris? Watch this. A kudgy-kudgy-kudgy-kudgy-kudgy-kudgy. Stop it, Dennis. He'll get hysterical. No, I like it. Don! Don, we haven't time. Now, there's one other part to play. That of the old lady is to be my wife. Well, maybe I can play a double part, too. How's this, Arson? Very good, Mariah. The part is yours. Bye, Cracky. Oh, murdered. If Lomenevner listening in, this does not constitute an endorsement of rude humor. We'll do all right. And now, ladies and gentlemen... Oh, my goodness. I forgot the most important part of all the wolf. I know I will be the wolf. In that case, here's my phone number, babyface. Thanks, Mary. This play is really crazy. Mary is her mother and her daughter. Orson is her father and a wolf. And Wolfson is a tree that giggles. I worry about things like that. Now, there's no need for you to tax your mind. What are you going to sing tonight? A song. A song? That's fine. And what's the name? Dennis Day. The name of the song! I'm going to sing as time goes by, and I dedicate it to a little girl who stole my wristwatch. That's a short story. Go ahead, Dennis. Make it a lulu second. I'll take it. Hello? Hello, Mr. Wells. This is Rochester. Hello, Rochester. What do you want? I'm leaving your employer after four weeks of sheer bliss. Yes, I understand that Mr. Benny is coming back. But Rochester, isn't there any way you can stay and be my butler? I don't think so. I got a contract with Mr. Benny that takes me up to and within the pearly gauge. Well, there must be some loophole in the contract. Is your lawyer read it? Yeah, he just shook his head and said Lincoln wouldn't like this. Well, I'm sorry you're leaving, Rochester. You've been a splendid butler and very trustworthy. You know when you came to work for me, there was a brand new bottle of bourbon in my liquor cabinet. And the cork hasn't been out of that bottle. That's the toughest cork I ever broke a tooth on. Well, in spite of that, Rochester, I'm certainly sorry that you're leaving me. Goodbye. Goodbye. Oh, Mr. Wells. What is it, Rochester? Just think, tonight I shall fold my tent like an arrow and silently steal away. That's my material. Yes, I know. I'm tired of that tent. I'm going to get myself a room, sir, Mr. Wells. Goodbye, Rochester. And now, ladies and gentlemen, before Dennis sings this song, I have a message of importance for all of you. Tomorrow your B-ration stamps, that is the red ones, may be used for purchasing meats, cheese, or fats. This is your second set of red stamps, the B stamps, in ration book two. You have a whole row of them to use this week in eight, five, two, and one-point values, totaling 16 points in all for each person. Spend your ration stamps wisely and supplement them as much as you can with the plentiful, nourishing, unrationed foods. And it's very good, Don. She could hear what I could have done with it. Go ahead and sing, Dennis. This day and age we're living in gives cause for our apprehension with speed and new invention and things like third dimension yet we get a trifle weary with Mr. Einstein's theory so we must get down to earth at times relaxed, relieve the tension no matter what the progress packs of life a kiss is still a kiss a sigh is just a sigh of love and glory. It's sung by Dennis Day. And Dennis, that really was very, very good. Thanks, Mr. Wells. It's decent of you to admit it. What's so decent about that? You have a marvelous voice. There's no getting away from it. Well, if it's so marvelous, why do you want to get away from it? I don't want to get away from it.  Now, ladies and gentlemen, on second thought, Dennis, here's your phone number back. And now, ladies and gentlemen, for our feature attraction, Little Red Riding Hood. Say, Arson, I just thought of something. Who's going to be the grandmother? You know that old dame that gets set up by the wolf. Phil, Phil, you're giving away the plot. Darn it, I had opened my big mouth. However, Phil, I can't tell you who plays the part of Red Riding Hood's grandmother. That's our big surprise. So, here we go with Little Red Riding Hood. The play opens in the farmhouse of old Mr. and Mrs. Riding Hood. It is a cold Christmas Eve long, long ago. And outside, the wintry wind is howling. But inside the farmhouse, the fireplace is burning brightly. We find the elderly couple busy decorating their Christmas tree. Curtain. Christmas tree sure is a beauty. Yeah, sure. It's pretty there all lit up. Where's my jug?  Ain't that a loo-loo? I was for supper. Well, it's up to your face. Start eating. Are you through with your chores, son? Yep. Did you milk the cow? Sure did. I creamed her, too. Where's your sister? Hello, Red. Hello, Pa. Hello, Ma. Hello, Annie. Annie? That's the cat. Oh. Say, we were worried about you, Red, out there alone with all those wolves. Yeah, did any of them come close to you? How do you think I got this fur close? Hmm. Might as pretty. That must be Grandma calling. I wonder how her cold is coming along. I wonder who... Then I'll come in. Hello? Hello, Grandma. How are you feeling? I feel terrible, son. How are you? Your cold seems to be getting worse.  Yes, and I'm loaded. Now, Grandma, behave yourself. I want you to get in bed right away. Okay. I'll send Red over with some hot soup for you. Good. Have her bring some cigars, too. All right. Goodbye, Grandma. Goodbye, Pa. Now, Red, take this hot soup over to your Grandma and be careful going through the woods, Red. Okay. So long, Pa. So long, Ma. Goodbye, Annie. Joe and Pete. Joe and Pete. Annie just had kittens. And right in my hat. Get out of the woods, Red. Okay. So long, everybody. Wind howls. And listen to that snow fall. See, it's fun walking through the snow. Crunch, crunch, crunch. Gosh, it's dark. Who's that I see lurking in Yon Bush? It's me, kiddies. I'm a wolf now. So, are you a wolf? Here's my button. Hollywood Lodge Number 23. Where are you going, my little chickadee? I'm going to my Grandma's with this basket. It's got soup and crackers in it. Attention, kiddies. Little Red Riding Hood's basket also contains a package of toasty brown sweet as a nut, grape nut flakes. Well, a talking treat.  I remember, children, when you're taking a basket of food to your grandma's mouth, include a big 12-ounce economy-sized package of America's fastest-growing flake cereal. What are you howling for, Orson? I just sat on my wolf pin. Now to get back to little Red Riding Hood. Hey, Red, where are you? She ran away. Chris is oiled again. Snarl, snarl. I know what I'll do. I'll beat little Red Riding Hood back to her grandmother's house. And when I get there, I'll eat her up. Are you shivering, Shabers? This must be the place. And now to get in. It's little Red Riding Hood. Guys, you've got a cold, too. Oh, Grandma.  It'll be over in a minute. Boots and all. Here you go. I just ate Grandma, kiddies. Darn those shoes. Must be little Red Riding Hood now. I know what I'll do. I'll put on this knife and get under the covers. That'll fool her. I'll talk just like Grandma. Come in, Red. Hello, Grandma. I brought you a basket of food. Goodie, goodie. Come over here by the bed, my dear.  Oh, better to see you with my dear.  And I'm going to take my axe and chop him right into... Patriotism on the food front begins with breakfast and never skips a single meal. So why not do as our government urges? Why not start every day with the whole-grain cereals that are plentiful, nutritious, and rations-stamplest? Now, for a rousing and right kind of breakfast, just try heaping bowls of crisp, toasty brown grape nut flakes.  An every-appetizing dish of grape nut flakes and milk supplies many of the food essentials of scarce foods, including meat. Essentials such as proteins, iron, phosphorus, calcium, plus those two important B vitamins, niacin, and B1. So you'll have fewer problems getting the proper food values into the rest of the day's ration menus if you serve grape nut flakes, not only at breakfast, but as a dish for light lunches, soppers, or bedtime snacks. Eat plenty of whole-grain cereal, friends. Remember, they can help to make up for many of today's food scarcities. And delicious, thrifty, grape nut flakes is one of your very best best. And so, ladies and gentlemen, this is your obedient servant, Orson Welles. It's really been a pleasure pinching for Jack Benny. I'm sure you'll all be glad to welcome back Jack at the same time next Sunday night. Good night, folks. Good night, Al. This program is written by Bill Mara and Ed Palloy.