the intriguing idea of one - way or cluster state quantum computing was initially developed by briegel and raussendorf .they showed that a two - dimensional array of qubits , entangled in a particular way ( through conditional phase gates ) , combined with single qubit operations , feed forward and measurements are sufficient for universal quantum computation .all the required interactions are already contained inside the system , and the computation proceeds through a series of local measurements ( with classical feed forward ) , efficiently simulating quantum circuits . in effect , the logical gates are prepared off - line and imprinted onto the qubits as they are transmitted through the cluster .  this approach was quickly applied to linear optics quantum computing and was experimentally demonstrated on the scale of several qubits ( see the review for a full set of references ) .this scenario contains a significant scaling problem in practice , due to the probabilistic nature of the logical gates .however , the cluster state method enables this problem to be pushed into the off - line preparation of the cluster .many different schemes have been proposed to efficiently generate the photonic cluster states , because of the simplicity of the interactions and the appealing coherence time of the photons .photon loss can be treated efficiently through ` indirect measurements ' and a more elaborate preparation of the cluster but at a significant cost in terms of the qubit dephasing .there remains an issue concerning storage though .initially , each photon will be flying down an optical fibre ( or two ) , meaning there is a need for an adaptive quantum memory .reliable and efficient single photon sources and detectors are a further issue for single photon approaches .  in the past few years , in an attempt to overcome the scaling properties of linear optics quantum computing , new measurement - based non - linear methods for optical quantum computing have been proposed .one approach is to effectively cascade non - demolition measurements , to enable a parity check gate on two qubits . a pair of photonic qubits prepared in a superposition of polarizationswould each in turn interact with the same coherent laser probe beam .the interaction consists of a cross - kerr non - linearity , which affects the phase of the probe beam , dependent on the state of the individual photons .the weakness of the non - linearity is compensated for by the size of . after interacting with a pair of qubits , the probe beam undergoes a homodyne measurement .the outcome of this measurement gives the parity of the two qubits , thus projecting them into a known entangled state heralded by the probe measurement result . by combining two of these parity gates and an ancillary photon ,a near deterministic c - not gate can be constructed , similar in its form to the pittman et al .c - not gate in pure linear optics .however , the difference in the weak non - linearity approach is that it is not destructive and not limited by the beam - splitters optimal success probability of .  already at this point, one can notice the usefulness of this parity gate for photonic cluster state approaches .a near deterministic entangler is all that is required to grow cluster states efficiently , be it in browne and rudolph s or in yoran and reznik s model .the required logical gates can be obtained through entanglement and local operations alone . however , as pointed out before , choosing photons as a support for one - way quantum computing may not be the best option .solid state or matter systems may be more compact and easy to deal with in this application and constituted the initial proposed system , when cluster states were first developed . in many of the solid state qubit systems proposed to date ,the multi - qubit gates arise from direct interactions between the qubits . adding extra qubits to a computationtherefore leads to changes in the required control fields and to the hamiltonian of the whole system . as a consequence ,the required setup becomes increasingly more complicated as the number of qubits in the computational system increases .a further issue is that in order for some solid state qubits to interact directly , they may need to be in such close proximity that application of individual control fields and measurements becomes infeasible .  to overcome these problems ,the concept of distributed quantum computing has arisen , in which interactions between the qubits used in a computation is mediated by a third party .this third party can be any measurable quantum physical system capable of interacting with each qubit .many proposals make use of single photons to effectively mediate interactions between matter qubits . having interacted with matter qubits , the photonsthen interact with each other in a linear optical setup before being measured , thus projecting the matter qubits into the required state without them having interacted directly .it has been shown that entanglement and logical operations can be generated in this way . however , once again there are probabilistic limits in these approaches due to the fact that simple linear optics is inherently non - deterministic .the next step was to use these probabilistic entangling schemes to prepare cluster states of matter qubits .barrett and kok first looked at this problem , proposing the use of a double - heralding probabilistic entangling procedure in order to build cluster states .the mediators are single photons , mixed on a beam splitter before being measured .the individual matter qubits comprise of two low - lying ( qubit basis ) states and a single excited state with a selection rule linking it to just one of the qubit states . applying a pulseleaves one of the low lying states unchanged , while making the other move up to the excited state .decay leads to the emission of a single photon for this amplitude .so if , after applying the pulse to both qubits , a single photon is detected after the beam splitter , the qubits are projected into the singlet state . the double - heralding procedure is used to remove mixture , generated if non - photon - number - resolving detection is used .this method has been further developed in a second paper , using a repeat until success method proposed by lim et al . , where implementation of a conditional phase gate is proposed , using a mutually unbiased basis .this enables some saving of qubit resources during the generation of the graph or cluster states .however , a further very interesting aspect of this proposal is that there are now three possible outcomes to the measurement .along with the usual success and failure outcomes , there is an insurance outcome , in which the qubits are left in a known product state , up to local operations .this means that , following the insurance outcome , a new attempt to implement the gate is possible .the corresponding scaling properties of the average number of required entangling operations follow from the various outcome probabilities for the entangling operation .this entangling operation requires a rather elaborate measurement scheme , which may be tough to implement experimentally .furthermore , as the scheme involves the detection of two photons , the success probability has a quadratic dependence on the detector efficiency .therefore on top of the inherently probabilistic aspect of linear optics , the detector efficiencies dramatically affect the scaling of the resources ( even for the highest reported efficiencies ) .it should be noted that the scheme is robust against photon loss due to the fact that this is a heralded source of error , so the fidelity of successful operation does nt suffer .nevertheless , the reduction in success probability of the gate requires a significant increase in resource overhead , which in turn increases the weight of unheralded errors in the cluster state itself .so single photon measurement has its limits in realizing entangling operations on matter qubits .however , homodyne measurements on coherent light fields can be made much more efficient than photon detection . in this paperwe will show how this and other factors make continuous variables a very powerful tool for growing cluster states .recently , the weak non - linearity concepts have been applied to the area of matter qubits and the distribution of entanglement .there are quite a number of well studied systems where one has a natural interaction between a matter qubit and the electromagnetic field .these include atoms ( real and artifical ) in cavity qed ( both at the optical and telecom wavelengths ) , nv - centers in diamond , quantum dots with a single excess electron , trapped ions and squids to name only a few .all these systems are likely to be suitable candidates for what we describe below but to illustrate the details a little more clearly let us consider a lambda based cqed system .one could use caesium atoms or an nv - diamond centre embedded in the cavity .both operate in the optical frequency range and are in consequence well matched to efficient homodyne measurements .the interaction between the coherent field mode and our matter qubit can generally be described by the jaynes - cummings interaction and in the dispersive limit ( large detunings ) one obtains an effective interaction hamiltonian of the form : where ( ) refers to the annihilation ( creation ) operator of an electromagnetic field mode in a cavity and the matter qubit is described using the conventional pauli operators , with the computational basis being given by the eigenstates of , with and .the atom - light coupling strength is determined via the parameter , where is the vacuum rabi splitting for the dipole transition and is the detuning between the dipole transition and the cavity field .the interaction applied for a time generates a conditional phase - rotation ( with ) on the field mode dependent upon the state of the matter qubit .we call this a _ conditional rotation _ and it is very similar to the cross - kerr interaction between photons . this time dependent interaction requires a pulsed probe.  using this interaction , a two - qubit gate has been proposed based on controlled bus rotation and subsequent measurement .the coherent state interacts with both qubits so an initial state of the system : evolves to at this stage we can choose from different types of measurements on the probe beam .the first and simplest option we have is to perform a homodyne measurement of some field quadrature which for a sufficiently strong local oscillator ( compared to the signal strength ) implements a projective measurement on the probe state .the key advantages with homodyne measurement , at least in the optical regime are that it is highly efficient ( 99% plus ) and is a standard tool of continuous variable experimentalists .the easiest to perform would be that of the momentum ( ) quadrature .in that case the measurement probability distribution has three peaks with the overlap error between them given by . as long as overlap error is small ( ) and the peaks are well separated . nowa measurement of the central peak will project the two matter qubits into the entangled state .this occurs with a probability of 1/2 .detecting either of the other two side peaks will project the qubits to the known product states or .it is worth explicitly mentioning here that we have already reached the limits of conventional linear optics implementations .when realistic detector efficiencies ( ) are taken into account , the initial probability of seen in photonic cluster states decreases proportional to or depending on the specific implementation and so the probability of the operation succeeding is now significantly less than 1/2 .however homodyne measurement are highly efficient and so our success probability will remain very close to 1/2 .it is of course possible to exceed this 1/2 by changing the nature of our measurement . in principlewe could achieve a near deterministic gate if we measured the the position ( ) instead of the momentum ( ) quadrature .for the position quadrature there are two peaks in the measurement result probability distribution , corresponding to the even and the odd entangled states of the qubits .the overlap between the peaks in this case is given by which is an issue because in order to separate the peaks well enough we would require .this is much more difficult to achieve than required for the momentum quadrature measurement .another strategy for the probe measurement would be to apply an unconditional displacement on the probe beam followed by a photon number measurement .after the displacement the combined state of the matter qubits and probe beam is now a photon number measurement of the bus mode will then either pick out the vacuum state , or project onto two amplitudes without distinguishing between them .for an ideal projection onto the number basis , the state of the two qubits becomes : with an equal probability of as long as the coherent amplitudes do not contribute significantly to the vacuum .the overlap of these coherent states with the vacuum leads to an error probability of which can be made quite small with a suitable choice of and  .for example with small we can choose which leads to an error probability as low as . consequently we can obtain a near - deterministic gate if we can implement a photon number measurement .however as observed in the linear optics schemes this currently constitutes a significant technological challenge . in the futureif this issue is solved we will have a near deterministic entangling gate to build cluster states . without the photon number resolving detector , but assuming that the vacuum can be distinguished from some photons our gate does work in a heralded fashion but with a success probability of 1/2 .hence for cluster state generation the simplest option so far would then be an efficient momentum ( ) quadrature homodyne measurement , giving us a success probability close to .we will note here that this is the most accessible and robust weak - nonlinear scheme so far proposed , using a single interaction per qubit .would it be possible to further improve the success probability all the while maintaining a highly efficient measurement ?within the same framework of conditional rotations , one can envisage three qubits interacting with a single probe beam .if we limit ourselves again to efficient quadrature measurements ( which scale as ) , we could consider the generation of three qubit states .ghz states are for instance one particularly useful state .one way of projecting the qubits onto ghz - type states is to vary the strength of the interactions between the qubits and the probe beam .let us represent a rotation of the coherent probe beam by .now no combination will lead to the required ghz end states in the case that we implement a quadrature measurement .however having one of the qubits interact twice as much with the probe beam will yield the correct paths in phase space .consider the sequence which we depict in figure [ fig1 ] .the peak centered on the origin will then correspond to the ghz state ( after being detected ) .this will happen with a probability of 1/4 .next the two peaks having been rotated through will correspond to the qubit states and respectively . now in both of these possible outcomes we obtain the same bell state on qubits 1 and 2 , disentangled with qubit 3 .so overall we obtain a ghz state with probability of 1/4 and a bell state with probability of 1/2 ( on two qubits of our choice ) , heralded by the probe beam quadrature measurement outcome .the other two outcomes project the qubits to two different known product states or . consequently ,if all we want to do is entangle a pair of qubits , we can now do this with probability of 3/4 .it may seem like increasing the number of qubits taking part will further raise the success probability .this claim turns out to be valid if we allow for more and more interactions as we add extra qubits .considering for example the 4 qubit case .the optimal combination then becomes .we now have 16 possible paths in phase space with 9 different end states .all of these apart from two , under detection , will project the qubits to bell states and ghz states .focusing solely on qubits 1 and 2 ( these can be any two qubits which we choose to have interact only once with the probe beam ) , they will be entangled with probability . following this method for larger numbers of qubits , the success probability in entangling a specific pair of qubits ( here 1 and 2 ) scales as .we do nt necessarily have to view these extra qubits as ancillas .they can become ( if they are nt already ) useful elements ( ` dangling bonds ' ) for future operations when we consider the generation of 2-dimensional cluster states .however there are drawbacks to using this generalization .the setup and measurement process will become increasingly complicated .the probe beam will have to travel and interact a lot more , rapidly accentuating the errors that we could have had initially .another essential point to note , is that the gate operation time will grow exponentially with the number of qubits we are willing to use .if we only have access to a fixed interaction strength , the gate operation time will double every time we add an extra qubit .so depending on the situation we are in , a compromise will have to be made between the time we are willing to take and the success probability we want to achieve .the 3-qubit gate minimizes the ratio of operation time over probability and we shall use this 3/4 probability in the remainder of the paper .we now consider how this entangling scheme may be used for generating cluster states of matter qubits .the usual cluster state is a rectangular 2-dimensional lattice of qubits .the qubits are entangled in a particular way , through conditional phase gates , with some of their nearest neighbours , thus creating ` bonds ' .each 1-dimensional chain represents the life line of a single qubit to be processed .these chains form a full 2-dimensional lattice structure by having bonds between them .the cluster state is defined as the eigenstate of the set of operators , where represents a particular lattice site and all its nearest neighbours .building chains is a possible basis for generating cluster states .if the chains can be efficiently generated , then simple schemes can be developed to combine them to form a 2-dimensional cluster , required for quantum computing . given a parity check operation ,the simplest growing technique involves taking one qubit ( prepared in a superposition state ) at a time and linking it on the end of the chain .once this is done , a hadamard transform is performed on this new end qubit , before the next one is added . in case of failure ,the initial end qubit is left in an unknown state .thus it needs to be measured adaptive feed forward on its nearest neighbour then enables recovery of the cluster state .so the chain shrinks by one qubit in this case .this constitutes the basic sequential approach to building chains . clearly for success probabilities smaller than 1/2, the chain will shrink on average ; for a success probability of exactly 1/2 , it will remain the same length .we can immediately appreciate benefits from the relatively high probabilities achieved in our two entangling procedures .the first two - qubit procedure already constitutes the limit of simple linear optics approaches .the second one , involving a 3 qubit interaction , can already be used in a sequential fashion , ensuring fast average growth and thus limited resource consumption .  , removing all the operators anti - commuting with it . herewe apply a hadamard transform on qubit 4 after the operation , thus producing a dangling bond . ]    in the case of lower probabilities , small chains are to be built inefficiently before joining them to the main chain .the process of linking chains with an entangling operation is described in figure [ fig2 ] , using the stabilizer notation .we can see that even though we are not obliged to measure out one qubit , the actual length of the resulting chain is the sum of the two initial ones minus one qubit .a convenient way of representing this operation with states is used in . and in figure [ fig3 ], we can see how the three qubit gate allows us to directly join three chains into a ` t ' piece , as well as joining two chains together . in case of failure, the end qubits need to be measured out and each chain shrinks by one element. then the procedure is repeated .supposing we start off with two chains of equal length ( following previous analysis , is defined as the number of qubits constituting the linear cluster ) , then the average size of the resulting chain is : this approximation is nt meaningful for small chains .here is the success probability of the entangling operation .we can immediately identify a critical length , above which there is growth : this critical length varies between different entangling operations . if an actual logical gate can be immediately implemented , then for example . or if this logical gate requires the qubits from the cluster to interact directly ( non - distributive approach ) then  .so our minimal size chain , ensuring growth is and ( the next integer greater than ).  following strategies previously developped , these minimal chains are grown in a divide and conquer fashion ( in parallel , without recycling ) starting from individual qubits before being merged together .this yields scaling relations for the average time taken  ] to grow a chain of length : &= & t\sum_{i=1}^{\log_2(l_0 - 1)+1}(1/p)^i+(t / p)\log_2\left(\frac{l - l_c}{l_0-l_c}\right ) \\ n[l]&=&\frac{\left(1/2\sum_{i=1}^{\log_2(l_0 - 1)+1}(2/p)^i+1/p\right)(l - l_c)}{l_0-l_c}-\frac{1}{p }  % \label{16c}\end{aligned}\ ] ] where denotes the time taken per entangling operation .  for our first entangling procedure , and thus . growingthis 4-chain will require 14 entangling operations on average leading to =16l-50 ] and for a failure probability of 0.4 it becomes \simeq16.6l-47.7 ] .this is a vast improvement over previous proposals .we shall extend this scaling comparison to the generation of 2-dimensional cluster states . using the redundant encoding method described in , we can give the average number of qubits consumed in the creation of a vertical link .each trial to establish this link costs two qubits ( per chain ) , because we first need to create dangling bonds .if we succeed in linking the two dangling bonds , we need to measure the first dangling qubit in order to establish the c - phase gate then measure the next one , to have a direct link between the two chains . the fact that we can only implement a simple entangling operation andnot the logical gate means we lose an additional qubit , which may be used later for additional vertical links or error correction .but if we concentrate on the task of making a single vertical link , the number of qubits consumed is : we can see that this converges to 4 as p tends to unity ( this corresponds to the qubit cost of a single trial ) .then the average number of entangling operations required to make the vertical bond is given by : + 1/p \ ; .\ ] ] using the linear optics scheme proposed in , for failure probabilities of 0.6 , 0.4 and 0.2 respectively , , and .the latter failure probability would however be very difficult to implement physically . with our two and three qubit entangling procedures we obtain and 46.7 respectively .we can see that the efficiency of these schemes generalizes to the creation of 2-dimensional cluster states in a straightforward manner .our gates can also be used to build cluster states in a ` tree ' like fashion , as proposed by bodiya and duan in .the method relies on the observation that ghz - type states are locally equivalent to star shaped cluster states .a parity check is all that is needed here .we now turn back to the time scaling . solving for we end up with : =14 + 2\log_2(l-3 ) \ ; .\ ] ] of course this is only valid for .the above result is obtained with a corresponding to the average time needed to build a 5-qubit chain without recycling ( this is due to the form of the sum ) .it is pretty clear that if we allow for entangling operations to be made in parallel , alongside additional resources , this can be minimized .for we have , meaning we only keep the first term in the sum for .this results in a general closed expression for : =(t / p)\left(1+\log_2\left(\frac{l - l_c}{l_0-l_c}\right)\right ) \ ; .\ ] ] we can compare this with the time taken by a sequential adding and building , as we now have access to probabilities higher than .adding one qubit at a time , via an entangling procedure , gives the recursion relation for the length , leading to the number of rounds .so for our 3 qubit gate , on average the chain grows by one unit every two trials .the time being sequential too , , the general form for becomes : =(l-1)t / p \ ; .\ ] ] thus time now scales linearly with the length of the chain in contrast with the logarithmic dependence we had above .for the two - qubit entangling gate , we essentially stand at the same point as the photonic cluster state approaches .optimizing the resources boils down to finding the optimal strategies in combining elements of cluster states .though this is a classical analysis , relying on probabilistic gates , it is a very complex task .obtaining bounds or comparing different strategies requires computing assistance . in their recent paper , kieling , gross , and eisert these issues in significant detail .they analyse essential methods and derive bounds for the globally optimal strategy , based on an entangling operation working with probability 1/2 .  for higher probabilitieshowever , the critical length insuring average growth simply does nt exist anymore and additional truly scalable approaches are at hand. we shall go over the obvious ones . from previous works on generating cluster states , we know that the simplest way to grow short chains with probabilistic gates is through a ` divide and conquer ' approach. it also turns out to be much quicker than a sequential adding , as we allow for many gates to operate in parallel .as described earlier on , this technique attempts to link up chains of equal length on each round , and discards the chains which failed to do so .this approach can be extended to growing large chains in the aim of saving time . in this contextwe can work out some important _ average _ quantities , starting off with the time taken : =1+\log_2(l-1)=k \ ; .\ ] ] here represents the number of rounds and can easily be worked out , as we saw above , from the given chain length .thus we will only use in the following expressions .next we can give the number of chains , at a particular round ( for ) , having started off with qubits : =n(p/2)^k  \ ; .\ ] ] then the number of remaining qubits on that round is given by : =c[k]\times l = n(p/2)^k(2^{k-1}+1 ) \ ; .\ ] ] following this we can work out the number of wasted qubits =n - q[k] ] to unity . or alternatively , we can use the ratio =g[k]/c[k] ] .one could envisage in this case a form of partial recycling , to save on the qubit resources whilst still retaining the time speed up .then we would allow for two or three trials before discarding the chains ( the initial scheme set no limit on the number of trials ) .however the protocol now becomes more elaborate unless we are willing to wait between each round ( of discarding ) because some chains will link on the first trial while others will link on the second ( supposing we allow two trials ) .so it seems like savings in time could be made if we are able to manage and organize chains of different lengths .the linear time scaling for the sequential method is due to the fact that operations can not be undertaken in parallel during its growth . if we did nt have access to simultaneous entangling operations , the time scaling for the divide and conquer methods would be equivalent to $ ] which is sub - exponential .one needs to keep in mind that by adopting a sequential method , the whole procedure is simplified considerably and would be more accessible to physical implementations .divide and conquer methods require a lot of work in parallel and should in practice involve the moving about and reordering of qubits or even small cluster states . unless the actual edges linking up the vertices in the graph states can be displaced via entanglement swapping strategies, we will most probably have to physically move some vertices in order to implement additional entangling operations .adding qubits sequentially should solve some architectural problems which may arise .for example , the qubits could be perfectly static and the measuring system ( including the ancillary qubit which can be reused ) would travel along the chain ` zipping ' it up . of course the measuring system would go back and forth , with a frequency related to the success probability of a single operation .but essentially the qubits constituting the chain would nt have to move .this seems significantly more practical than moving the qubits and chains around or having to change the measuring setup every time so as to implement the operation between qubits in various places .however many of these problems may be solved by more advanced protocols making use of percolation phenomena as developed in .all this brings us to view the cluster state as having active regions in which it is being built or measured in the computation ( both can be undertaken simultaneously ) and regions in which the qubits are simply waiting .now this waiting can be minimized in the building itself , through the appropriate protocols , and in the measurement process .that is , the cluster can be built only a few layers in advance , so that the qubits have less waiting to do , between the building and the actual measurement . in any case, there will be some waiting. therefore the lowest decoherence support would be preferred , but it may not be the easiest to manipulate .thus we may envisage having two different physical realizations constituting the cluster state .for example , we could use single electron spins initially in building the cluster . once the links are made between one site and its nearest neighbours ,the qubit could be switched into a nuclear spin state which has a significantly longer coherence time , via a swap operation or some other coherent write and read actions .most of the waiting would be done in the long - lived state , before being swapped again for the readout .this follows the same scenario as using a second physical system to mediate the interaction and make the measurements , in distributed quantum computing . in the present proposal ,we use a continuous variable bus and homodyne measurements to generate the links .this physical system shows itself to be very efficient in this application .then , for example , electron spins or superconducting charge qubits could then be the matter realization interacting with the bus and serving for the final readout .these systems provide the useful static aspect required , they interact well with the mediating bus and ensure good single qubit measurements .finally a low decoherence realization such as nuclear spin could be envisaged , mainly as a storage medium .the swapping or write and read procedure should have a high fidelity for this storage to be beneficial . on the whole , we see that optimization will depend directly on the physical realization(s ) we have chosen to work with . for systems with long dephasing times we would give priority to sequential adding approaches , as we have some freedom in the time scaling andthus we can make significant savings in resources .but for realizations with short dephasing times , we would probably want to divide the task up and run operations in parallel , in order to accelerate the fabrication of the cluster state , at the expense of extra resources .looking at our entangling gates , we have seen that if we utilize four non - linear interactions and three qubits the success probability is dramatically increased . within the framework of four non - linear interactions ,another option presents itself to us . defining the conditional displacement operator by , one can simulate a conditional - phase gate between qubits 1 and 2 with the following interaction sequence :    by setting .the resulting operation is then locally equivalent to that of the conventional conditional phase .thus at the end of the sequence the coherent state is disentangled from both qubits , removing the need for a measurement .these conditional displacements can be simulated through conditional rotations and _ unconditional _ displacements , as is clear from the self - inverse quality of pauli operators and the general property of rotations :    where can be expanded in a power series .this can be realized with the following sequence     with real .the conditional phase observed arises from the different areas the probe traces out in phase space , picking up a geometrical phase .more in the spirit of the initial proposal of wang and zanardi this can easily be extended to the simulation of many - body interactions .the interactions required to build a cluster state are pairwise thus conditional displacements are sufficient .  by having the probe interact with more qubits and adapting the direction of the displacements , we can pick out the qubits we want to couple through the geometrical phase . in that way one could start from a general sequence of the form :   . \label{31}\ ] ]  but here we are simulating interactions between all qubits and from this sequence one can not directly generate linear or grid - like cluster states .that is we need to adjust the so as to choose which qubits we want to interact .we can use such a sequence to directly generate useful graph states such as star shaped graphs ( locally equivalent to ghz states ) .for example , taking real and the displacement from qubit 1 orthogonal to all the others we have   .\label{32}\ ] ]  clearly if we set we will obtain a star shaped graph of our qubits , centered on qubit 1 ( figure [ fig6](a ) ) .we note that from this condition on and equation ( [ 30 ] ) , the scaling and magnitude requirements for here are comparable to those of the measurement - induced entangling schemes . to generate a linear clusterwe need to switch to another interaction sequence .we need to disentangle the probe with the qubits as the sequence evolves so as not to create extra links .coming back to our conditional phase operation we notice that after the third interaction the probe becomes disentangled from qubit 1 . furthermore by setting the entirety of the geometric phase is already acquired ( by the corresponding two - qubit state ) at this point .along with this observation and a correct ordering of the displacements we can propose a sequence of the form   \label{33}\ ] ] again setting all the couplings are locally equivalent to conditional phase gates , yielding a linear cluster state ( see figure [ fig6](b ) ) .we can view the probe as creating the links as it travels along the chain .the main advantage with these generalizations is the reduced number of interactions per qubit .if we were to use the simple conditional phase sequence then the number of interactions per qubit would be where is the degree of the qubit in the graph state .in other words , to build a qubit star shaped graph , the center qubit would have to interact times at most , or with local operations to swap the center qubit as the star is being generated we could bring this down to four interactions per qubit . as we see from our generalized sequence each qubit would only need to interact twice with the probe mode .now if we think of a grid - like structure , the qubits inside the graph will have , thus using would mean we require 8 interactions for each of these qubits .but here again if we switch to the linear cluster sequence each one of these qubits would interact 4 times only , twice in each of the two chains that go through it .  .the first and second interaction , for each qubit , are in opposite directions in phase space .in ( a ) we have the sequence leading to a star shape while in ( b ) the one leading to a linear graph , both on 5 qubits . for larger numbers of qubits we have the same patterns . ]these multi - qubit approaches could be envisaged in different contexts .for example we may view it as an expensive resource ( be it in time or work ) alongside a cheaper one such as probabilistic two qubit parity gates .many schemes make use of a basic building block such as star shaped graphs . for instancethis approach could be used to directly generate the building blocks needed in the percolation techniques derived in , then easier measurement based gates would take on from there , following the same methods . when we begin to think of loss in the probe mode however, this star shaped graph sequence is a lot less robust than the linear graph sequence .the reason being that halfway through the interactions the probe holds information about all qubits , meaning that correlated errors will be quite significant .whereas in the linear graph sequence , the probe holds information about at most 2 elements because it is constantly disentangling from the previous ones . meaning that the correlated errors will be limited to pairs of qubits . in that sense the simplest and apparently most robust procedure for building a grid like structure is to generate chains that would then overlap at the intersections . in any casewe can compensate for the loss in the probe and make sure it disentangles from the qubits .this will leave us with weighted graphs and some dephasing on the qubits .we can then resort to purification protocols such as those proposed in .the important aspect of this measurement free approach is that it is significantly quicker .also it does precisely simulate the two qubit gates required for generating the cluster state , unlike the measurement based gates which are simple parity checks .this means that no feed forward or local operations are required . butthis comes to a price .the constraints for the strength of the interactions are greater , that is they are now fixed , in contrast with our parity gates for which there simply was a lower bound needed for the distinguishability of the measurement outcomes .  now given these near deterministic operations ,the number of interactions required to build a certain cluster state becomes fixed .the question of time then simply reduces to the number of gates we can implement in parallel . looking at the process in a dynamical way , we can see now that the size of the cluster state at a certain time during the computation is significantly reduced .this ` buffer ' region of the cluster state may still be a couple of layers , but the off - line part of the cluster , which is nt attached yet , can be made very small .previously , the size of the buffer that is yet to be linked up was dictated by the success probabilities of the entangling operations .the bigger this off - line prepared buffer is , the more time it takes to build it and the more time it takes to attach it . in other wordsthe more errors it contains . now depending on the amount of near deterministic gates we can implement in parallel, this off - line buffer only needs to consist in a couple of layers , freshly built , purified and attached . as a matter of fact we may not even need this off - line aspect anymore .the individual qubits could be added directly to the existing cluster as it is being measured .this represents huge savings in the number of qubits we are dealing with and minimizes the error they may have picked up , as they spend a minimal time inside a cluster state .the issues raised at the end of the last section are still of concern here . therealways will be _ some _ waiting , between the building and the readout , so change in support during that time , from electron spin to nuclear spin for example , in order to minimize the dephasing , is still an important idea .in this paper we have considered the usefulness of weak non - linearities in the building of matter qubit cluster states enabling us to work in the success probability regime of .we first developed a 2-qubit parity check , based on a single non - linearity per qubit and a simple measurement of the probe bus .at this point we already noticed the advantage of using continuous variables to mediate an interaction between the qubits and to provide us with an efficient measurement system .then we extended the setup to a 3-qubit entangling operation , increasing the probability of entangling a pair of qubits to 3/4 .we saw how this scheme could be generalized to using more qubits , rapidly increasing the success probability .the 3-qubit interaction already provides new possibilities for the schemes used in building cluster states .after what we considered the vital issues of scaling , by going through previous results and adapting them to our own gates .the results themselves were already significant improvements to previously proposed schemes .the time scaling was particularly emphasized and discussed .this lead us to notice that there will always be a compromise to be made , between the time and the number of entangling operations required .we also observed that within this framework we have access to measurement free approaches which can be generalized to more qubits .they make use of the geometrical phase acquired by the probe in phase space , simulating conditional phase gates between the qubits in the cluster .the constraints in implementation are higher , but many scaling issues are immediately resolved .finally , we remark that the non - 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Abstract: we propose a scalable approach to building cluster states of matter qubits using coherent states of light . recent work on the subject relies on the use of single photonic qubits in the measurement process . these schemes can be made robust to detector loss , spontaneous emission and cavity mismatching but as a consequence the overhead costs grow rapidly , in particular when considering single photon loss . in contrast , our approach uses continuous variables and highly efficient homodyne measurements . we present a two - qubit scheme , with a simple bucket measurement system yielding an entangling operation with success probability 1/2 . then we extend this to a three - qubit interaction , increasing this probability to 3/4 . we discuss the important issues of the overhead cost and the time scaling . this leads to a no - measurement " approach to building cluster states , making use of geometric phases in phase space .


a geometry lies in the foundation of physics , and a true conception of geometry is very important for the consequent development of physics .it is common practice to think that all problems in foundations of geometry have been solved many years ago .it is valid , but this concerns the geometry considered to be a logical construction .physicists are interested in the geometry considered as a science on mutual disposition of geometrical objects in the space or in the space - time .the two aspects of geometry are quite different , and one can speak about two different geometries , using for them two different terms .geometry as a logical construction is a homogeneous geometry , where all points have the same properties .well known mathematician felix klein believed that only the homogeneous geometry deserves to be called a geometry .it is his opinion that the riemannian geometry ( in general , inhomogeneous geometry ) should be qualified as a riemannian geography , or a riemannian topography . in other words, felix klein considered a geometry mainly as a logical construction .we shall refer to such a geometry as the mathematical geometry .the geometry considered to be a science on mutual disposition of geometric objects will be referred to as a physical geometry , because the physicists are interested mainly in this aspect of a geometry .the physical geometries are inhomogeneous , in general , although they may be homogeneous also . on the one hand ,the proper euclidean geometry is a physical geometry . on the other hand ,it is a logical construction , because it is homogeneous and can be constructed of simple elements ( points , straights , planes , etc . ) .all elements of the euclidean geometry have similar properties , which are described by axioms .similarity of geometrical elements allows one to construct the mathematical ( homogeneous ) geometry by means of logical reasonings .the proper euclidean geometry was constructed many years ago by euclid .consistency of this construction was investigated and proved in .such a construction is very complicated even in the case of the proper euclidean geometry , because simple geometrical objects are used for construction of more complicated ones , and one can not construct a complicated geometrical object without construction of more simple constituents of this object .  note that constructing his geometry , euclid did not use coordinates for labeling of the space points .his description of the homogeneous geometry was coordinateless .it means that the coordinates are not a necessary attribute of the geometry .coordinate system is a method of the geometry description , which may or may not be used .application of coordinates and of other means of description poses the problem of separation of the geometry properties from the properties of the means of the description .usually the separation of the geometry properties from the coordinate system properties is carried out as follows .the geometry is described in all possible coordinate systems .transformations from one coordinate system to the another one form a group of transformation .invariants of this transformation group are the same in all coordinate system , and hence , they describe properties of the geometry in question .  at this pointwe are to make a very important remark ._ any geometry is a totality of all geometric objects _ _ _ and of all relations _ _ _ _ between them__. any geometric object is a subset of points of some point set , where the geometry is given . in the riemannian geometry ( and in other inhomogeneous geometries )the set is supposed to be a -dimensional manifold , whose points are labelled by coordinates .this labelling ( arithmetization of space ) is considered to be a necessary attribute of the riemannian geometry .most geometers believe that the riemannian geometry ( and physical geometry , in general ) can not be constructed without introduction of the manifold . in other words, they believe that the manifold is an attribute of the riemannian geometry ( and of any continuous geometry , in general ) . this belief is founded on the fact , that the riemannian geometry is always constructed on some manifold .but this belief is a delusion .the fact , that we always construct the physical geometry on some manifold , does not mean that the physical geometry can not be constructed without a reference to a manifold , or to a coordinate system .of course , some labelling of the spatial points ( coordinate system ) is convenient , but this labelling has no relations to the construction of the geometry , and the physical geometry should be constructed without a reference to coordinate system .application of the coordinate system imposes constraints on properties of the constructed physical geometry .for instance , if we use a continuous coordinate system ( manifold ) we can construct only continuous physical geometry . to construct a discrete physical geometry ,the geometry construction is not to contain a reference to the coordinate system .  herewe present the method of the physical geometry construction , which does not contain a reference to the coordinate system and other means of description .it contains a reference only to the distance function , which is a real characteristic of physical geometry .  if a geometry is inhomogeneous , and the straights located in different places have different properties , it is impossible to describe properties of straights by means of axioms , because there are no such axioms for the whole geometry .mutual disposition of points in a physical ( inhomogeneous ) geometry , which is given on the set of points , is described by the distance function the distance function is the main characteristic of the physical geometry .besides , the distance function is an _ unique characteristic _ of any physical geometry . _ the distance function _ _ _ determines completely the physical geometry ._ _ this statement is very important for construction of a physical geometry .it will be proved below . any physical geometry is constructed from the proper euclidean geometry by means of a deformation , i.e. by a replacement of the euclidean distance function by the distance function of the geometry in question .for instance , constructing the riemannian geometry , we replace the euclidean infinitesimal distance by the riemannian one .there is no method of the inhomogeneous physical geometry construction other , than the deformation of the euclidean geometry ( or some other homogeneous geometry ) which is constructed as a mathematical geometry on the basis of its axiomatics and logic .unfortunately , conventional method of the riemannian geometry construction contains a reference to the coordinate system . but this reference can be eliminated , provided that we use finite distances instead of infinitesimal distances .  for description of a physical geometry one uses the world function  , which is connected with the distance function by means of the relation .the world function of the -space is defined by the relation application of the world function is more convenient in the relation that the world function is real , when the distance function is imaginary and does not satisfy definition ( [ b1.1 ] ) .it is important at the consideration of the space - time geometry as a physical geometry .in general , a physical geometry can not be constructed as a logical building , because any change of the world function should be accompanied by a change of axiomatics .this is practically aerial , because the set of possible physical geometries is continual .does the world function contain full information which is necessary for construction of the physical geometry ? it is a very important question .for instance , can one derive the space dimension from the world function in the case of euclidean geometry ?slightly below we shall answer this question in the affirmative .now we formulate the method of the physical geometry construction .let us imagine that the proper euclidean geometry can be described completely in terms and only in terms of the euclidean world function .such a description is called -immanent .it means that any geometrical object and any relation between geometrical objects in can be described in terms of in the form and . to obtain corresponding geometrical object and corresponding relation between the geometrical objects in other physical geometry ,it is sufficient to replace the euclidean world function by the world function of the physical geometry in description of and . e shows that the geometric object is constructed on the basis of the euclidean axiomatics .thus , one can obtain another physical geometry from the euclidean geometry by a simple replacement of by .for such a construction one needs no axiomatics and no reasonings .one needs no means of descriptions ( topological structures , continuity , coordinate system , manifold , dimension , etc . ) .in fact , one uses implicitly the axiomatics of the euclidean geometry , which is deformed by the replacement .this replacement may be interpreted as a deformation of the euclidean space .absence of a reference to the means of description is an advantage of the considered method of the geometry construction .besides , there is no necessity to construct the whole geometry . we can construct and investigate only that part of the geometry which we are interested in .any physical geometry may be constructed as a result of a deformation of the euclidean geometry .the geometric object is described by means of the skeleton - envelope method .it means that any geometric object is considered to be a set of intersections and joins of elementary geometric objects ( ego ) .the finite set of parameters of the envelope function is the skeleton of elementary geometric object ( ego ) .the set of points forming ego is called the envelope of its skeleton . for continuous physical geometrythe envelope is usually a continual set of points .the envelope function , determining ego is a function of the running point and of parameters .the envelope function is supposed to be an algebraic function of arguments , .each of arguments is a -function of two arguments , either belonging to skeleton , or coinciding with the running point .thus , any elementary geometric object is determined by its skeleton and its envelope function .for instance , the sphere with the center at the point is determined by the relation where is a point belonging to the sphere .the elementary object is determined in all physical geometries at once .in particular , it is determined in the proper euclidean geometry , where we can obtain its meaning .we interpret the elementary geometrical object , using our knowledge of the proper euclidean geometry .thus , the proper euclidean geometry is used as a sample geometry for interpretation of any physical geometry .we do not try to repeat subscriptions of euclid at construction of the geometry .we take the geometrical objects and relations between them , prepared in the framework of the euclidean geometry and describe them in terms of the world function .thereafter we deform them , replacing the euclidean world function by the world function of the geometry in question . in practice the construction of the elementary geometry objectis reduced to the representation of the corresponding euclidean geometrical object in the -immanent form , i.e. in terms of the euclidean world function .the last problem is the problem of the proper euclidean geometry .the problem of representation of the geometrical object ( or relation between objects ) in the -immanent form is a real problem of the physical geometry construction .it is very important , that such a construction does not use coordinates and other methods of description , because the application of the means of description imposes constraints on the constructed geometry .any means of description is a structure given on the basic euclidean geometry with the world function .replacement is sufficient for construction of unique physical geometry .if we use an additional structure for construction of physical geometry , we obtain , in general , other geometry , which coincide with not for all , but only for some of world functions .thus , a use of additional means of description restricts the list of possible physical geometries .for instance , if we use the coordinate description at construction of the physical geometry , the obtained geometry appears to be continuous , because description by means of the coordinates is effective only for continuous geometries , where the number of coordinates coincides with the geometry dimension .  constructing geometry by vtans of a deformation we use essentially the fact that the proper euclidean geometry is a mathematical geometry , which has been constucted on the basis of euclidean axiomatics and logical reasonings .we shall refer to the described method of the physical geometry construction as the deformation principle and interpret the deformation in the broad sense of the word .in particular , a deformation of the euclidean space may transform an euclidean surface into a point , and an euclidean point into a surface .such a deformation may remove some points of the euclidean space , violating its continuity , or decreasing its dimension .such a deformation may add supplemental points to the euclidean space , increasing its dimension .in other words , the deformation principle is a very general method of the physical geometry construction .the deformation principle as a method of the physical geometry construction contains two essential stages :  \(i ) representation of geometrical objects and relations of the euclidean geometry in the -immanent form , i.e. in terms and only in terms of the world function .\(ii ) replacement of the euclidean world function by the world function of the geometry in question .  a physical geometry , constructed by means of the only deformation principle ( i.e. without a use of other methods of the geometry construction ) is called t - geometry ( tubular geometry ) .the t - geometry is the most general kind of the physical geometry .application of the deformation principle is restricted by two constraints .\1 . describing euclidean geometric objects and euclidean relation in terms of , we are not to use special properties of euclidean world function . in particular , definitions of and are to have similar form in euclidean geometries of different dimensions .they must not depend on the dimension of the euclidean space .the deformation principle is to be applied separately from other methods of the geometry construction . in particular, one may not use topological structures in construction of a physical geometry , because for effective application of the deformation principle the obtained physical geometry must be determined only by the world function ( metric ) .the crucial point of the t - geometry construction is the description of the proper euclidean geometry in terms of the euclidean world function . we shall refer to this method of description as the -immanent description .unfortunately , it was unknown for many years , although all physicists knew that the infinitesimal interval is the unique essential characteristic of the space - time geometry , and changing this expression , we change the space - time geometry . from physical viewpointthe -immanent description is very reasonable , because it does not contain any extrinsic information .the -immanent description does not refer to the means of description ( dimension , manifold , coordinate system ) .absence of references to means of description is important in the relation , that there is no necessity to separate the information on the geometry in itself from the information on the means of description .the -immanent description contains only essential characteristic of geometry : its world function . at firstthe -immanent description was obtained in 1990 .the first question concerning the -immanent description is as follows .does the world function contain sufficient information for description of a physical geometry ?the answer is affirmative , at least , in the case of the proper euclidean geometry , and this answer is given by the prove of the following theorem .let -space be a set of points with the given world function let the vector be the ordered set of two points , , and its length is defined by the relation ._ theorem _  the -space is the -dimensional proper euclidean space , if and only if the world function satisfies the following conditions , written in terms of the world function .  i. condition of symmetry :   \ii . definition of the dimension : where is the gram s determinant the scalar product of two vectors and is defined by the relation vectors , are basic vectors of the rectilinear coordinate system with the origin at the point , and the metric tensors , ,  in are defined by the relations structure of the euclidean space : where coordinates  of the point are covariant coordinates of the vector , defined by the relation   iv : the metric tensor matrix has only positive eigenvalues   v. the continuity condition : the system of equations considered to be equations for determination of the point as a function of coordinates ,  has always one and only one solution . conditions ii v contain a reference to the dimension of the euclidean space .as far as the -immanent description of the proper euclidean geometry is possible , it is possible for any t - geometry , because any geometrical object and any relation in the physical geometry is obtained from the corresponding geometrical object and from the corresponding relation in the proper euclidean geometry by means of the replacement in description of an .for such a replacement be possible , the description of and is not to refer to special properties of , described by conditions ii v. a formal indicator of the conditions ii v application is a reference to the dimension , because any of conditions ii v contains a reference to the dimension of the proper euclidean space .if nevertheless we use one of special properties ii v of the euclidean space in the -immanent description of a geometrical object , or relation , we refer to the dimension and , ultimately , to the coordinate system , which is only a means of description .let us show this in the example of the determination of the straight in the -dimensional euclidean space .the straight in the proper euclidean space is defined by two its points and  as the set of points  where condition means that vectors and are collinear , i.e. the scalar product of these two vectors satisfies the relation where the scalar product is defined by the relation ( [ b11a ] ) .thus , the straight line is defined -immanently , i.e. in terms of the world function .we shall use two different names ( straight and tube ) for the geometric object .we shall use the term straight , when we want to stress that is a result of deformation of the euclidean straight .we shall use the term tube , when we want to stress that may be a many - dimensional surface .  in the euclidean geometry one can use another definition of collinearity .vectors and are collinear , if components of vectors and in some coordinate system are proportional .for instance , in the -dimensional euclidean space one can introduce rectangular coordinate system , choosing points and forming basic vectors , .then the collinearity condition can be written in the form of equations where is some constant. relations ( [ b17 ] ) are relations for covariant components of vectors and in the considered coordinate system with basic vectors , .let points be chosen in such a way , that . then eliminating the parameter from relations ( [ b17 ] ), we obtain independent relations , and the geometrical object defined according to ( [ b17 ] ) , depends on points . this geometrical object defined -immanently .it is a complex , consisting of the straight line and the coordinate system , represented by points . in the euclidean spacethe dependence on the choice of the coordinate system and on points determining this system , is fictitious .the geometrical object depends only on two points and coincides with the straight line .but at deformations of the euclidean space the geometrical objects and are deformed differently .the points cease to be fictitious in definition of , and geometrical objects and become to be different geometric objects , in general . but being different , in general , they may coincide in some special cases .  what of the two geometrical objects in the deformed geometry should be interpreted as the straight line , passing through the points and in the geometry ?of course , it is , because its definition does not contain a reference to a coordinate system , whereas definition of depends on the choice of the coordinate system , represented by points . in general ,definitions of geometric objects and relations between them are not to refer to the means of description .but in the given case the geometrical object is , in general , -dimensional surface , whereas is an intersection of -dimensional surfaces , i.e. is , in general , a one - dimensional curve .the one - dimensional curve corresponds better to our ideas on the straight line , than the -dimensional surface .nevertheless , in physical geometry it is , that is an analog of the euclidean straight line .it is very difficult to overcome our conventional idea that the euclidean straight line can not be deformed into many - dimensional surface , and _ this idea has been prevent for years from construction of t - geometries_. practically one uses such physical geometries , where deformation of the euclidean space transforms the euclidean straight lines into one - dimensional lines .it means that one chooses such geometries , where geometrical objects and coincide . condition ( [ b19 ] ) of coincidence of the objects and , imposed on the t - geometry , restricts list of possible t - geometries .let us consider the metric geometry , given on the set of points .the metric space is given by the metric ( distance ) . at first sight the metric space is a special case of the -space ( [ a1.3 ] ) , and the metric geometry is a special case of the t - geometry with additional constraints ( [ c2.5 ] ) , ( [ c2.6 ] ) imposed on the world function .however it is not so , because the metric geometry does not use the deformation principle .the fact , that the euclidean geometry can be described -immanently , as well as the conditions ( [ b10 ] ) - ( [ b14 ] ) , were not known until 1990 .additional ( with respect to the -space ) constraints ( [ c2.5 ] ) , ( [ c2.6 ] ) are imposed to eliminate the situation , when the straight line is not a one - dimensional line .the fact is that , in the metric geometry the shortest ( straight ) line can be constructed only in the case , when it is one - dimensional .let us consider the set of points  if the metric space coincides with the proper euclidean space , this set of points is an ellipsoid with focuses at the points and the large semiaxis .the relations , , determine respectively external points , boundary points and internal points of the ellipsoid . if , we obtain the degenerate ellipsoid , which coincides with the segment } ] between the points , , we truncate the list of possible geometries , because such an identification is possible only in some physical geometries . identifying ( [ a1.1 ] ) and ( [ a1.2 ] ) , we eliminate all discrete physical geometries and those continuous physical geometries , where the segment } ] , etc . )are defined on the -space in the same way , as they are defined -immanently in the proper euclidean space .practically one uses the deformation principle , although it is not mentioned in all definitions .  let us consider a simple example of the space - time geometry , described by the t - geometry on 4-dimensional manifold . the world function is described by the relation where and are some constants .the quantity is the world function in the minkowski space - time geometry . in the orthogonal rectilinear ( inertial )coordinate system the world function has the form where is the speed of the light .let us compare the broken line ( [ a1.2 ] ) in minkowski space - time geometry and in the distorted geometry .we suppose that is timelike broken line , and all links } ] , and the quantity is interpreted as its ( geometric ) mass .it follows from definition ( [ b11a ] ) and relation ( [ b3.3 ] ) , that for timelike vectors with  calculation of the shape of the segment } \left ( \sigma _ { \mathrm{d}}\right ) ] in the coordinate system , where points and have coordinates , and is a parameter along the segment } \left ( \sigma _ { \mathrm{d}}\right ) $ ] ( , ) .one can see from ( [ b3.7a ] ) that the characteristic value of the segment radius is .let the broken tube describe the world line of a free particle .it means by definition that any link is parallel to the adjacent link definition of parallelism is different in geometries and . as a result links , which are parallel in the geometry , are not parallel in and vice versa .let describe the world line of a free particle in the geometry .the angle between the adjacent links in is defined by the relation the angle , and the geometrical object is a timelike straight line on the manifold .let now describe the world line of a free particle in the geometry .the angle between the adjacent links in is defined by the relation the angle also . if we draw the broken tube on the manifold , using coordinates of basic points and measure the angle between the adjacent links in the minkowski geometry , we obtain for the angle the following relation substituting the value of , taken from ( [ b3.10 ] ) , we obtain hence , .it means , that the adjacent link is located on the cone of angle , and the whole line has a random shape , because any link wabbles with the characteristic angle .the wabble angle depends on the space - time distortion and on the particle mass .the wabble angle is small for the large mass of a particle .the random displacement of the segment end is of the order , i.e. of the same order as the segment width .it is reasonable , because these two phenomena have the common source : the space - time distortion .one should note that the space - time geometry influences the stochasticity of particle motion nonlocally in the sense , that the form of the world function ( [ b3.3 ] ) for values of is unessential for the motion stochasticity of the particle of the mass .such a situation , when the world line of a free particle is stochastic in the deterministic geometry , and this stochasticity depends on the particle mass , seems to be rather exotic and incredible . but experiments show that the motion of real particles of small mass is stochastic indeed , and this stochasticity increases , when the particle mass decreases . from physical viewpoint a theoretical foundation of the stochasticity is desirable , and some researchers invent stochastic geometries , noncommutative geometries and other exotic geometrical constructions , to obtain the quantum stochasticity . but in the riemannian space - time geometry the particle motion does not depend on the particle mass , and in the framework of the riemannian space - time geometry it is difficult to explain the quantum stochasticity by the space - time geometry properties .distorted geometry explains the stochasticity and its dependence on the particle mass freely . besides , at proper choice of the distortion the statistical description of stochastic leads to the quantum description ( schrdinger equation ) .it is sufficient to set , where is the quantum constant , is the speed of the light , and is some universal constant , connecting the geometrical mass with the usual particle mass by means of the relation . in other words , the distorted space - time geometry ( [ b3.3 ] ) is closer to the real space - time geometry , than the minkowski geometry .further development of the statistical description of geometrical stochasticity leads to a creation of the model conception of quantum phenomena ( mcqp ) , which relates to the conventional quantum theory approximately in the same way as the statistical physics relates to the axiomatic thermodynamics .mcqp is the well defined relativistic conception with effective methods of investigation , whereas the conventional quantum theory is not well defined , because it uses incorrect space - time geometry , whose incorrectness is compensated by additional hypotheses ( quantum principles ) . besides , it has problems with application of the nonrelativistic quantum mechanical technique to the description of relativistic phenomena .the geometry is a homogeneous geometry as well as the minkowski geometry , because the world function is invariant with respect to all coordinate transformations , with respect to which the world function is invariant . in this connectionthe question arises , whether one could invent some axiomatics for and derive the geometry from this axiomatics by means of proper reasonings .note that such an axiomatics is to depend on the parameter , because the world function depends on this parameter . if , this axiomatics is to coincide with the axiomatics of the minkowski geometry .if , this axiomatics can not coincide with the axiomatics of , because some axioms of are not satisfied in this case .in general , the invention of axiomatics , depending on the parameter and in the general case on the distortion function , seems to be a very difficult problem . besides, why invent the axiomatics ?we had derived the axiomatics for the proper euclidean geometry , when we constructed it before .there is no necessity to repeat this process any time , when we construct a new geometry .it is sufficient to apply the deformation principle to the constructed euclidean geometry written -immanently .application of the deformation principle to the euclidean geometry is a very simple and very general procedue , which is not restricted by continuity , convexity and other artificial constraints , generated by our preconceived approach to the physical geometry .( bias of the approach is displayed in the antecedent supposition on the one - dimensionality of any straight line in any physical geometry ) .thus , we have seen that the nondegeneracy of the physical geometry as well as non - one - dimensionality of the straight line are properties of the real physical geometries .the proper euclidean geometry is a ground for all physical geometries , and it is a degenerate geometry .nevertheless , it is beyond reason to deny an existence of nondegenerate physical geometries .   1.  the deformation principle uses results obtained at construction of the proper euclidean geometry and does not add any additional supposition on properties of geometrical objects . 2 .the deformation principle uses only the real characteristic of the physical geometry its world function and does not use any additional means of description .the deformation principle is very simple and allows one to investigate only that part of geometry which one is interested in .application of the deformation principle allows one to obtain the true space - time geometry , whose unexpected properties can not be obtained at the conventional approach to physical geometry .

Abstract: physical geometry studies mutual disposition of geometrical objects and points in space , or space - time , which is described by the distance function , or by the world function . one suggests a new general method of the physical geometry construction . the proper euclidean geometry is described in terms of its world function . any physical geometry is obtained from the euclidean geometry as a result of replacement of the euclidean world function by the world function of . this method is very simple and effective . it introduces a new geometric property : nondegeneracy of geometry . using this method , one can construct deterministic space - time geometries with primordially stochastic motion of free particles and geometrized particle mass . such a space - time geometry defined properly ( with quantum constant as an attribute of geometry ) allows one to explain quantum effects as a result of the statistical description of the stochastic particle motion ( without a use of quantum principles ) .