[ [ section ] ]  the biham - middleton - levine ( bml ) traffic models was first introduced in published 1992 .the model involves two types of cars : `` red '' and `` blue '' .initially the cars are placed in random with a given density on the torus .the system dynamics are as follows : at each turn , first all the red cars try to move simultaneously a single step to the right in the torus .afterwards all blue cars try to move a single step upwards .a car succeeds in moving as long as the relevant space above / beside it ( according to whether it is blue / red ) is vacant .[ [ section-1 ] ]  the basic properties of this model are described in and some recent more subtle observations due to larger simulations are described in . the main and most interesting property of the system originally observed in simulations is a _ phase transition _ : for some critical density one observes , that while filling the torus randomly with cars in density the system self organizes such that after some time all cars flow freely with no car ever running into another car ( see figure [ cars1 ] ) , by slightly changing the density to some not only does system not reach a free flow , but it will eventually get stuck in a configuration in which no car can ever move ( see figure [ cars2 ] ) .[ [ section-2 ] ]  very little of the above behaviour is rigorously proven for the bml model .the main rigorous result is that of angel , holroyd and martin , showing that for some fixed density , very close to 1 , the probability of the system getting stuck tends to 1 as . in can find a study of the very low density regime ( when ) .[ [ section-3 ] ]  first , we introduce a slight variant of the original bml model by allowing a car ( say , red ) to move not only if there is a vacant place right next to it but also if there is a red car next to it that moves .thus , for sequence of red cars placed in a row with a single vacant place to its right - all cars will move together ( as oppose to only the rightmost car in the sequence for the original bml model ) . not only does this new variant exhibits the same phenomena of self - organization and phase transition , they even seem to appear more quickly ( i.e. it takes less time for the system to reach a stable state ) . actually the demonstrated simulations ( figures [ cars1 ] , [ cars2 ] ) were performed using the variant model .note that the results of appear to apply equally well to the variant model .  :the initial configuration in the left organizes after 520 steps to the `` diagonal '' configuration on the right which flows freely ]  : the initial configuration on the left gets to the stuck configuration on the right after 400 steps , in which no car can ever move ]  [ [ section-4 ] ]  in the following , we will analyze a simplified version of the bml model : bml on a single junction .meaning , we place red cars in some density on a single row of the torus , and blue cars are placed in density on a single column. we will show that for all the system reaches _ _ optimal speed __ , depending only on . for we will show the system reaches speed 1 , while for the speed can not be 1 , but the system will reach the same speed , regardless of the initial configuration .moreover , at the system s behaviour undergoes a phase transition : we will prove that while for the stable configuration will have linearly many sequences of cars , for we will have only different sequences after some time .we will also examine what happens at a small window around .[ [ section-5 ] ]  note that in the variant bml model ( and unlike the original bml model ) car sequences are never split .therefore , the simplified version of the variant bml model can be viewed as some kind of 1-dimensional coalescent process .[ [ section-6 ] ]  much of the proofs below rely on the fact that we model bml on a symmetric torus .indeed , the time - normalization of section [ normalization ] would take an entirely different form if the height and width were not equal .we suspect that the model would exhibit similar properties if the height and width had a large common denominator , e.g. if they had a fixed proportion .more importantly , we believe that for height and width with low common denominator ( e.g. relatively prime ) , we would see a clearly different behaviour . as a simple example , note that in the relatively prime case , a speed of precisely 1 can not be attained , no matter how low is , in contrast with corollary [ speed1cor ] .this dependence on the arithmetic properties of the dimensions is also apparent in .[ [ section-7 ] ]  we start with the exact definition of our simplified model . on a cross shape , containing a single horizontal segment and a single vertical segment , both of identical length , red cars are placed in exactly randomly ( and uniformly ) chosen locations along the row , and blue cars are similarly placed in locations along the column . will be called the density of the configuration . for simplicity and consistency with the general bml model, we refer to the cars placed on the horizontal segment as red and those on the vertical segment as blue . for simplicitywe may assume that the _ junction _, i.e. the single location in the intersection of the segments , is left unoccupied .the segments are cyclic - e.g. a red car located at the rightmost position ( ) that moves one step to the right re - emerges at the leftmost position ( 0 ) .  [ [ section-8 ] ]  at each turn , all the red cars move one step to the right , except that the red car that is just left of the junction will not move if a blue car is in the junction ( i.e. blocking it ) , in which case also all the red cars immediately following it will stay still . afterwards , the blue cars move similarly , with the red and blue roles switched . as in the original bml , we look at the asymptotic speed of the system , i.e. the ( asymptotic ) average number of steps a car moves in one turn . it is easily seen that this speed is the same for all blue cars and all red cars , since the number of steps any two cars of the same color have taken can not differ by more than .it is somewhat surprising , perhaps , that these two speed must also be the same .for instance , there is no configuration for which a blue car completes 2 cycles for every 1 a red car completes .[ [ section-9 ] ]  though less natural , it will be sometimes useful to consider the equivalent description , in which two rows of cars - red and blue - are placed one beneath the other , with a `` special place '' - the junction - where at most one car can be present at any time . in every stepfirst the red line shifts one to the right ( except cars immediately to the left of the junction , if it contains a blue car ) and then the blue line does the same .furthermore , instead of having the cars move to the right , we can have the junction move to the left , and when a blue car is in the junction , the ( possibly empty ) sequence of red cars immediately to the left of the junction moves to the left , and vice verse .figure [ timeimg ] illustrates the correspondence between these models .[ [ section-10 ] ]  from the discussion above we get the following equivalent system , which we will call the time - normalized junction :    1 .  fix and fix some initial configuration representing the red and blue cars respectively , i.e. iff there s a red car at the -th place .we require that , and at place ( ) there is at most one car in both rows. 2 .  in each turn : * if place contains a blue car , and place contains a red car ( if ) , push this car one step to the left . by pushing the car we mean also moving all red cars that immediately followed it one step to the left , i.e. set , for  ] ) . * set   note that indeed the dynamics above guarantee that after a turn place - the new junction - contains at most one car . generally , as long as cars flow freely , the time - normalized system configuration does not change ( except for the location of the junction ) .cars in the time - normalized system configuration actually move only when some cars are waiting for the junction to clear ( in the non - time - normalized system ) .[ [ section-11 ] ]  our analysis of the junction will argue that for any , regardless of the initial configuration , the system will reach some optimal speed , depending only on .first let us state what is this optimal speed :  for a junction with density , the maximal speed for any configuration is .[ maxspeedthm ]  [ [ proof ] ] proof + + + + +  obviously , the speed can not exceed 1 .observe the system when it reaches its stable state , and denote the speed by .thus , at time , a car has advanced steps ( on average ) which means that it has passed the junction times . as only one carcan pass the junction at any time , the total number of cars passing the junction at time is bounded by . therefore , we have which implies .   [ [ section-12 ] ]  we will now show that the system necessarily reaches these speeds from any starting configuration .[ [ section-13 ] ]  we begin by proving that for the junction will eventually reach speed very close to 1 . a system in a stable stateis said to be _ free - flowing _ if no car is ever waiting to enter the junction .a junction is free - flowing iff the time - normalized junction satisfies :  \(1 ) for all there is only one car in place in both rows .\(2 ) for all if place contains a blue car , place   does not contain a red car .[ violate1 ]  [ [ proof-1 ] ] proof + + + + +  obviously , this is just a reformulation of free - flowing .  [ [ section-14 ] ]  we will now turn to show that for , following the system dynamics of the time - normalized junction will necessarily bring us to a free flowing state , or at worst an `` almost '' free flowing state , meaning a state for which the system speed will be arbitrarily close to 1 for large enough .[ [ section-15 ] ]  for this let us consider some configuration and look at the set of `` violations '' to lemma [ violate1 ] , i.e. places that either contain both a blue and a red car , or that contain a blue car and a red car is in the place one to the left . as the following lemma will show , the size of the set of violations is very closely related to the system speed , and posses a very important property : it is non - increasing in time .[ [ section-16 ] ]  more formally , for a configuration we define two disjoint sets for the two types of violations :  also , let be the set of all violations .it will be sometimes useful to refer to a set of indicators , where each is 1 if and 0 otherwise , thus .for a junction with some initial configuration , let be the system configuration at time , and let be the set of violations for this configuration , and be the corresponding indicator vector .similarly , let denote the junction s position at time .1 .  2 .  for any ,the system speed is at least   [ violate2 ]  [ [ proof-2 ] ] proof + + + + +  property ( 1 ) follows from the system dynamics . to see this , examine the three possible cases for what happens at time :  1 .if in turn place does not contain a violation then the configurations do not change during the next turn , i.e. , hence clearly .2 .  if , then the configuration changes in two places : 1 . is changed from 1 to 0 .thus , place is no longer in , i.e. changes from 1 to 0 . 2 . is changed from 0 to 1 for .this may affect , and .however , for place , by changing from 0 to 1 no new violation can be created ( since by definition of , , so iff regardless of ) . + for other indices since do not change , so between times and , we have that changes from 1 to 0 , and at worst only one other place - changes from 0 to 1 , so .3 .  similarly ,if place then the configuration changes in two places : 1 . is changed from 1 to 0 .thus , changes from 0 to 1 is changed from 0 to 1 for , affecting , .however for place changing does not affect whether this place is a violation or not , so at worst changed from 0 to 1 .+ by the same argument we get .  therefore , .[ [ section-17 ] ]  for property ( 2 ) we note that in the time - normalized system , following a specific car in the system , its `` current speed '' is where is the number of times the car was pushed to the left during the last system turns .we note that if a car at place is pushed to the left at some time , by some violation at place , this violation can reappear only to the left of , so it can push the car again only after passes .hence any violation can push a car to the left only once in a car s cycle ( i.e. moves ) . since by ( 1 ) at any time from onwards the number of violations in the system is at most , then each car is pushed left only times in turns , so its speed from time onwards is at least as asserted .  [ [ section-18 ] ]  with lemma [ violate2 ] at hand we are now ready to prove system self organization for .we will show that for , after system turns , and hence deduce by part ( 2 ) of lemma [ violate2 ] the system reaches speed ( as ) .as the junction advances to the left it pushes some car sequences , thus affecting the configuration to its left .the next lemma will show that when , for some , the number of cars affected to the left of the junction is only a constant , independent of .  consider a junction with density .there exists some constant , independent of , for which :  from any configuration with junction at place there exist some such that after turns :  \(1 ) for , ( i.e. there are no violations there )  \(2 ) for , and ( are unchanged there ) [ ugly ]  [ [ proof-3 ] ] proof + + + + +  first let us consider . for to not satisfy the lemma conclusions , there need to be a car sequence ( either red or blue ) of length exceeding , which is pushed left by the junction as it moves .progressing through the process , if , for some , the sequence currently pushed by the junction is shorter then , then this is the we seek .therefore , for the conclusions _ not _ to hold , the length of the car sequence pushed by the junction must exceed for all .if this is the case , then leaving the junction we see alternating red and blue sequences , all of lengths exceeding and one vacant place after any blue sequence and before any red one .  however , if this is the case for all then the average number of cars per location in at time must be at least ( at least cars between vacant places ) . therefore , the total number of cars in at time is more than ( recall that ) .since there are only cars in the system , this can not hold for all up to .thus , there must be some time for which the conclusions of the lemma are satisfied.  [ [ section-19 ] ]  we are now ready to easily prove the main result for this section .a junction of size with density reaches speed of from any initial configuration .[ speed1thm ]  [ [ proof-4 ] ] proof+ + + + +  let be some initial configuration , with and let the corresponding set of violations and the matching indicators vector .by lemma [ ugly ] there exist for which for  ] , and also , as long as , for  ] and blue cars at places  ] 2 . for  ]  [ uglier1 ]  [ [ proof-6 ] ] proof + + + + +  since it is easy to lose the idea in all the notations , a visual sketch of the proof is provided in figure [ proofimg ] .[ [ section-28 ] ]  by the assumptions :  - for  ] ,   [ [ section-29 ] ]  to get ( 1 ) , we note that with the assumption , the blue sequence will be pushed to the left in the next turns , so by restriction ( b ) , for  ] and for  ] which all vanished after turns .possible violations at places  ] .since for place either or are empty by the assumption , we must therefore have for  ] will be pushed left until eventually hitting the red car already proven to be present at  ] giving ( 3 ) .   ( steps 1 - 3 ) and [ uglier2 ] ( step 4 ) ]  [ [ section-33 ] ]  the following lemma is completely analogous when reversing the roles of , and can be proven the same way .let be a stable configuration with junction at and .assume that there is a sequence of exactly consecutive red cars at places  ] , .then :  1 . for  ] 3 . for  ] and blue cars at places  ] .place is empty - i.e. 3 .starting at there is a sequence of places for which ) ] .[ ugliest ]   ( a ) and [ uglier2 ] ( b ) combined together yield lemma [ ugliest ] ]  [ [ proof-7 ] ] proof + + + + +  figure [ proof3img ] outlines the proof , without the risk of getting lost in the indices . for the full proof ,first get from lemma [ uglier1 ] that :  1 . for  ] 3 . for  ] , and for  ] , and place is empty , proving claims 1,2 in the lemma .[ [ section-36 ] ]  from ( 2),(3 ) we get for  ] , thus places  ] .[ [ section-37 ] ]  now running the system for turns , we will have the junction at place , , followed by sequences of reds and blues .applying lemma [ uglier2 ] for the system ( rotated by i.e. for ) :  1 . for = [ n-\max(s_r , s_b)-\min(s_r , s_b)-s_r'-2 , n-\max(s_r , s_b)-s_r'-2 ] = [ n - s_r - s_b - s_r'-2 , n-\max(s_r , s_b)-s_r'-2] ] 3 . for = [ n-\max(s_r , s_b)-s_r'-1 , n-\max(s_r , s_b)-\min(s_r , s_b)-1 ] = [ n-\max(s_r , s_b)-s_r'-1 , n - s_r - s_b - 1] ] : we were previously assured this is true up to place , and for places \subseteq [ n-\max(s_r , s_b)-s_r'-1 , n-\max(s_r , s_b)-s_b'-1] ] and blue cars at places  ] , and at places  ] , for which properties of theorem [ stablethm ] hold ]  [ [ proof-8 ] ] proof + + + + +  this is merely applying lemma [ ugliest ] repeatedly . applying lemma [ ugliest ] we know that there exist such that : place is empty , followed by consecutive places with only red cars and consecutive places with only blue cars left to it , thus the assertion holds for the segment  ] , and this is enough to advance the system for turns .the blue segment is pushed left places , further pushing the red sequence places to the left such that its last cars now overlap with the blue sequence .[ [ section-42 ] ]  so after turns the system evolves to a state where , , and left to there are consecutive blue cars and exactly consecutive red cars . noting that this time , once again we can deduce from lemma [ ugliest ] that : there are no additional cars in  ] for .[ [ section-43 ] ]  repeatedly applying lemma [ ugliest ] as long as , we repeatedly get that the assertion holds for some  ]     [ [ section-44 ] ]  we have worked hard for theorem [ stablethm ] , but it will soon turn to be worthwhile .let us obtain some useful corollaries .let be a stable configuration with junction at and .assume that there is a sequence of exactly consecutive red cars at places  ] .denote then :  1 .the number of blue segments in the system equals the number of red segments .system speed is at least 3 .total number of cars in the system is at most 4 .total number of cars in the system is at least 5 .total number of segments in the system is at most   [ stablecor ]  [ [ proof-9 ] ] proof+ + + + +  \(1 ) follows by the structure described in [ stablethm ] , since there is exactly one red and one blue sequence in  ] since any red sequence is immediately proceeded by a blue sequence .[ [ section-45 ] ]  for ( 2 ) we note that by [ stablethm ] the configuration contains exactly violations : the overlapping places of the segments in  ] contains a single car , and no red car can be immediately to the left of a blue car . by [ violate2 ]the system speed is hence at least .[ [ section-46 ] ]  by [ stablethm ] any place in  ] contain both a red car and a blue car .thus the total number of cars in the system is at most giving ( 3 ) .[ [ section-47 ] ]  on the other hand , since any sequence of a red car or a blue car is of length at least , an empty place in  ] , and cars in places $ ] ( since each place contains one car , except transitions between segments that are empty ) . thus the configuration contains at most segments .  [ [ section-55 ] ]simulation results show that these bounds are not only tight , but typical , meaning that a junction with a random initial configuration with density indeed has segments in the stable state , with the largest segment of size near .this suggests that the system undergoes a second order phase transition .[ [ section-56 ] ]  following are computer simulation results for the junction for critical and near critical , demonstrating the phase transition . the columns below consist of , the average asymptotic speed , the average number of car segments in the stable state , the average longest segment in the stable state , and the average number of segments divided by and .   [[ section-57 ] ]  for large , system reaches speed and the average number of segments is linear ( approx . ) .[ [ section-58 ] ]  [ cols="^,^,^,^,^,^",options="header " , ][ [ section-63 ] ]  the fascinating phenomena observed in the bml traffic model are still far from being completely understood . in this paperwe showed a very simplified version of this model , which , despite its relative simplicity , displayed very similar phenomena of phase transition at some critical density and of self - organization , which in our case both can be proven and well understood .we used two approaches in this paper : the first one was to use some sort of `` independence '' in the way the system evolves , as in the way we handle .we showed that the system self - organizes `` locally '' and with a bounded affect on the rest of the configuration , thus it will eventually organize globally .the second approach is the notion of the stable configuration , i.e. we characterize the combinatorial structure of any state that the system can `` preserve '' and use it to show it is optimal ( in a way saying , that as long as we are not optimal the system must continue to evolve ) .can these results be extended to handle more complicated settings than the junction ?possibly yes .for example , considering a -lanes junction ( i.e. consecutive red lines meeting consecutive blue rows ) , one can look at a time - space - normalized version of the system , as shown in figure [ klanesimg ] , with the junction now being a parallelogram traveling along the red and blue lines , and `` propagating violations '' ( which now have a more complicated structure depending on the configuration within the junction ) . stable states of this configurations seem to have the same characteristics of a single junction , with a red ( blue ) car equivalent to some red ( blue ) car in any of the red ( blue ) lines .thus a zero - effort corollary for a -lanes junction is that it reaches speed 1 for , but for nearing this bound is clearly non - significant .it is not surprising though , since combinatorics alone can not bring us too far , at least for the complete bml model even as little as cars in an size torus can be put in a stuck configuration - i.e. reach speed 0 .

Abstract: in the biham - middleton - levine traffic model cars are placed in some density on a two dimensional torus , and move according to a ( simple ) set of predefined rules . computer simulations show this system exhibits many interesting phenomena : for low densities the system self organizes such that cars flow freely while for densities higher than some critical density the system gets stuck in an endless traffic jam . however , apart from the simulation results very few properties of the system were proven rigorously to date . we introduce a simplified version of this model in which cars are placed in a single row and column ( a junction ) and show that similar phenomena of self - organization of the system and phase transition still occur .


quantum error correction is inherently a feedback process where the error syndrome of encoded qubits is measured and used to apply conditional recovery operations .most formulations of quantum error correction treat this feedback process as a sequence of discrete steps .syndrome measurements and recovery operations are performed periodically , separated by a time - interval chosen small enough to avoid excessive accumulation of errors but still comparable to the time required to implement quantum logic gates .there is , however , mounting evidence from the field of real - time quantum feedback control that continuous observation processes offer new , sometimes technologically advantageous , opportunities for quantum information processing .  toward this end , ahn ,doherty and landahl ( adl ) devised a scheme to implement general stabilizer quantum error correction using continuous measurement and feedback .unfortunately an exact implementation of the adl scheme is computationally demanding .for an -qubit code , the procedure requires one to time - evolve a -dimensional density matrix for the logical qubit alongside the quantum computation .this classical information - processing overhead must be performed to interpret the continuous - time error syndrome measurement data and determine how recovery operations , in the form of a time - dependent feedback hamiltonian , should be applied .while is a constant for any particular choice of code , even modest codes such as the five - qubit code and the seven - qubit steane code push classical computers to their limits . despite state - of - the art experimental capabilities, it would be extremely difficult to implement the adl bit - flip code in practice .consequently , ahn and others have devised alternate feedback protocols which are less demanding , but perform worse than the the original adl scheme .recently , van handel and mabuchi addressed the computational overhead of continuous - time error syndrome detection using techniques from quantum filtering theory .they developed an exact , low - dimensional model for continuous - time error syndrome measurements , but did not go on to treat continuous - time recovery .the complication is that any feedback hamiltonian suitable for correcting errors during the syndrome measurements violates the dynamical symmetries that were exploited to obtain the low - dimensional filter in ref .while one might address this complication by simply postponing error recovery operations until a point where the measurements can be stopped , there may be scenarios where it would be preferable to perform error recovery in real - time .for example , if the recovery operation is not instantaneous , responding to errors as they occur might outperform protocols where there are periods without any error correction .  in this paper , we extend the quantum filtering approach developed by van handel and mabuchi to include recovery operations .we consider an error - correcting feedback hamiltonian of the form devised by ahn , doherty and landahl , but our approach readily extends to other forms for the feedback .while an exact low - dimensional model for continuous - time stabilizer generator measurements in the presence of feedback does not appear to exist , we devise an approximate filter that is still low - dimensional , yet sufficiently accurate such that high - quality error correction is possible .for our purposes , a quantum error correcting code is a triple . the quantum operation _ encodes _ logical qubits in physical qubits . is a set of stabilizer generator observables with outcomes that define the _ error syndrome_. is the _ recovery operation _ , which specifies what correction should be applied to the physical qubits in response to the syndrome measurement outcomes .the particular choice of code is usually made with consideration for the nature of the decoherence affecting the physical qubits .for example , the bit - flip code ( considered by both adl and van handel and mabuchi ) improves protection against an error channel that applies the pauli operator to single qubits at a rate .here , we adopt the notation that represents the pauli operator on qubit , and similarly for and . in the bit - flip code , encodes qubits in qubits by the map .the stabilizer generators are and ; each extracts the parity of different qubit pairs .the recovery , given the outcomes of measuring , is defined by , , and .in this paper , we focus primarily on the five - qubit - code ( ) that increases protection against general separable channels , and in particular the continuous - time symmetric depolarizing channel that applies all three pauli operators to each of the physical qubits at the same rate .the five - qubit code has stabilizer generators .it is also a _ perfect _ code in that all 16 distinct syndrome outcomes indicate distinct errors : one corresponding to the no - error condition , and one syndrome for each of the three pauli errors on each of the five qubits .we defer to for the encoding and recovery procedures for this code .quantum error correction can be extended to continuous time by replacing discrete measurements of the stabilizer generators with a set of continuous observation processes .this creates measurement records }dt + dw_t^{(i)}\ ] ] obtained from the encoded qubit with state at time . here , is a constant called the measurement strength that depends upon the physical implementation of the continuous measurement and the are independent wiener processes , each with = 0 ] and \rho = g_l\rho  + \rho g_l - 2{\operatorname{tr}\bigl[g_l\rho \bigr]}\rho ] .the time evolution generated by a particular noise realization is generally called a _ trajectory_.  the final term in eq .( [ eq : belavkinfilter ] ) describes the action of the time - dependent feedback hamiltonian used to implement error recovery .following ahn , doherty and landahl , we choose the feedback hamiltonian to be of the form which corresponds to applying pauli operators to each qubit with a controllable strength .the policy for determining the feedback strengths at each point in time should be chosen optimally .ahn , doherty , and landahl obtained their feedback policy by defining the _ codespace projector _ onto the no error states ( states which are eigenvectors of all stabilizers ) and then maximizing the _ codespace fidelity _ } ] ) .this is far from adequate for use in an actual feedback controller even in state - of - the - art experiments .moreover , eq . ( [ eq : belavkinfilter ] ) is a nonlinear filter , and for such filters it is rarely possible to evaluate even qualitative properties analytically .one must then average over an appreciable number of trajectories to find the expected behavior of quantities such as the codespace fidelity as a function of time .for the five - qubit code , our integrator requires approximately 10 hours to simulate 1000 trajectories .   considering that the syndrome measurements yield information about correlations between qubits and not information about the individual states of the qubits, one can imagine that propagating the full density matrix is excessive .indeed , the adl scheme only makes use of the projection of onto the codespace , generating the same feedback policy regardless of which state in the codespace is initially chosen .it is reasonable to expect that a lower dimensional model could track solely the information extracted from the syndrome measurements .this is exactly the premise used by van handel and mabuchi to obtain a low - dimensional model of continuous - time stabilizer generator measurements ( in the absence of feedback ) .they formulate the problem as a graph whose vertices correspond to syndromes and whose edges reflect the action of the error model .the filtering problem is then reduced to tracking the node probabilities , i.e. , the likelihoods for the qubit to be described by each of the various syndrome conditions .dynamical transitions occur between the syndromes due to the error channel , and the filter works to discern these transitions from the stabilizer measurement data .    for an code, van handel and mabuchi define a set of projectors onto the distinct syndrome spaces . for the five - qubit code, there are 16 such projectors ; is the codespace projector as before and are projectors onto states with a syndrome consistent with a error on qubit .forming the probabilities }\ ] ] into a vector and computing from the full dynamics leads to the reduced filter with , the outcome of measuring on and ( eq . ( 4 ) in ref .the equations for are closed and encapsulate all the information that is gathered from measuring the stabilizer generators .equation ( [ eq : basicwonham ] ) is an example of a _ wonham filter _, which is the classical optimal filter for a continuous - time finite - state markov chain with an observation process driven by white noise .further discussion of the wonham filter and its use in conjunction with discrete - time error correction can be found e.g. , in ref .we now extend eq .( [ eq : basicwonham ] ) to include a feedback hamiltonian suitable for error recovery . following van handel and mabuchi s lead, we see that eq .( [ eq : basicwonham ] ) was derived by taking } ] also results in a set of closed equations .however , that is not the case when using the basis of the sixteen syndrome space projectors .specifically ,  ] . if we then iterate the dynamics of the filter ( [ eq : belavkinfilter ] ) by calculating starting from the syndrome space projectors , we find that each feedback hamiltonian term generates pairs of feedback coefficient terms .for example , calculating the dynamics due to feedback on generates two feedback coefficient operators : and .we must then determine the dynamics for these first level feedback coefficients .this will include calculating the feedback on , which generates second level feedback coefficients and . continuing to iterate feedback coefficient terms, we find that an additional 1008 distinct terms are needed to close the dynamics and form a complete basis .adding in the initial 16 syndrome space projectors gives a 1024 dimensional basis clearly no better than propagating the full density matrix .however , it is now relatively easy to calculate the feedback strengths , which depend only on pairs of first - level feedback coefficients .for example , from eq .( [ eq : feedbackpolicy ] ) we find that , where } ] are first - level coefficients developed earlier in the paragraph .although the dimension of the alternate basis is no smaller than the dimension of the full density matrix , the structure of the filter represented in the alternate basis provides a manner for interpreting the relative importance of the feedback coefficients .this is best seen graphically in fig .[ fig : annotatedmatrix](a ) , which superimposes the non - zero matrix elements coming from the noise , measurement and feedback terms .both measurement and noise are block diagonal as expected ; it is the feedback that couples blocks together in a hierarchical fashion .this hierarchy can be parameterized by the number of `` feedback transitions '' which connect a given feedback coefficient to the syndrome space block .for example , the upper left block , which corresponds to the syndrome space projectors , is connected via feedback terms to the first level feedback block , whose feedback coefficients are each one feedback transition away from the syndrome space block . in turn , the first level block is then connected to a second level feedback block , whose feedback coefficients are two feedback transitions away from the syndrome block.  given that the initial state starts within the codespace and given that feedback is always on , the feedback coefficients that are more than one feedback transition from the syndrome space block should be vanishingly small . limiting consideration to these first two blocks , we also find that pairs of feedback coefficients couple identically to the syndrome space block .for example , we find that and couple to syndrome space projectors identically .this is not surprising , as these two terms comprise the commutator that results from the feedback hamiltonian .however , outside the first level of feedback transitions , the matrix elements of these feedback coefficients differ .additionally , feedback coefficients involving feedback hamiltonians which correspond to a syndrome error on the codespace projector are related as for the feedback coefficient examples just mentioned , this relation is . truncating the dynamics to include only the first level of feedback and combining distinct feedback coefficients which act identically within this block results in the matrix of fig .[ fig : annotatedmatrix](b ) over only 136 basis elements .note that the controller now only needs to reference a single basis element for calculating a given feedback strength .our truncation scheme generalizes for reducing the dimensionality of the quantum filter for an arbitrary code .such a filter for an \!] ] , which is a measure relevant for a quantum memory .since error correction is independent of the encoded state , we choose the encoded state as a fiducial initial state .given that the five - qubit code protects against only single qubit errors , we find that after error correction at time , the codeword fidelity for discrete - time error correction is which asymptotes to .this limit arises because prior to the stabilizer generator measurements , the noise pushes the state to the maximally mixed state , which is predominately composed of the through terms .the feedback codeword fidelity was calculated by integrating both the full quantum filter ( [ eq : belavkinfilter ] ) , representing the underlying system of qubits , and the truncated filter , representing the feedback controller .again , we chose , and and used the same sde integrator described above .figure [ fig : fivequbitperformance ] shows the average of over 2000 trajectories , demonstrating that there are regimes where feedback - assisted error correction can significantly outperform discrete - time error correction .feedback - assisted error correction appears to approach an asymptotic codeword fidelity greater than what would be obtained by decoherence followed by discrete - time error correction . due to the nonlinear feedback, we were unable to calculate an analytic asymptotic expression for the continuous - time strategy .   and .( color online . )[ fig : fivequbitperformance ] ]  nonetheless , the improved performance for the timespan considered suggests that better quantum memory is possible using the feedback scheme .extending control theory techniques introduced by van handel and mabuchi , we have developed a computationally efficient feedback controller for continuous - time quantum error correction . for our truncation scheme , the dimension of the filtering equations grows as in the number of physical qubits , rather than for the original ahn , doherty and landahl procedure . by numerical simulation of the five - qubit code, we have demonstrated the viability of such a filter for a quantum memory protecting against a depolarizing noise channel .moreover , in all our simulations , this performance is indistinguishable from that of the computationally more demanding filter of the adl style .  in systems where recovery operations are not instantaneous relative to decoherence, consideration suggests that it is desirable to perform syndrome measurement , recovery , and logic gates simultaneously .however , it is not immediately clear how gates impact the feedback controller . indeed ,if a hamiltonian is in the code s normalizer , the continuous - time feedback protocol and its performance are unchanged .though a universal set of such hamiltonians can be found , it might be desirable to find universal gates which have physically simple interactions .future work involves finding such gate sets and developing a framework for universal quantum computing .additional issues of fault - tolerance and robustness could then be explored within such a universal setup .exploring feedback error correction in the context of specific physical models will provide opportunities to tailor feedback strategies to available control parameters and salient noise channels .such systems might allow the calculation of globally optimal feedback control strategies .we thank ramon van handel for helpful comments on the manuscript and daniel lidar for his input on how one might combine continuous - time stabilizer generator measurements and quantum gates .this work was supported by the doe nine program under a contract with sandia national laboratory ( 60071 - 699182 ) .ajl was also supported in part by the nsf under grant ( phy-0555573 ) .jmg was also supported in part by the afosr under grant ( fa9550 - 06 - 1 - 0178 ) .

Abstract: we present an efficient approach to continuous - time quantum error correction that extends the low - dimensional quantum filtering methodology developed by van handel and mabuchi [ quant - ph/0511221 ( 2005 ) ] to include error recovery operations in the form of real - time quantum feedback . we expect this paradigm to be useful for systems in which error recovery operations can not be applied instantaneously . while we could not find an exact low - dimensional filter that combined both continuous syndrome measurement and a feedback hamiltonian appropriate for error recovery , we developed an approximate reduced - dimensional model to do so . simulations of the five - qubit code subjected to the symmetric depolarizing channel suggests that error correction based on our approximate filter performs essentially identically to correction based on an exact quantum dynamical model .


in this paper , we present results of our analysis of a large organizational email dataset , comprising nearly complete email traffic records for los alamos national laboratory ( lanl ) over a period of several months .very few organizational communication networks of this scale have been analyzed in the literature .analyzing such large email datasets from complex organizations poses a number of challenges .first , considerable work is required to parse large quantities of raw data from network logs and convert it into a format suitable for network analysis and visualization .second , a great deal of care is required to analyze and visualize network data in a way that makes sense of complex formal organizational structures - in our case , 456 organizational units that are connected through diverse organizational hierarchies and management chains .finally , it can be difficult to sort out the effects of email traffic generated by mass announcements and communications along management chains from the more chaotic , less hierarchical traffic generated by everyday interactions among colleagues .this paper addresses these complexities in two ways .first , we demonstrate methods for understanding large - scale structural relationships between organizational units by using carefully thought - out visualization strategies and basic graph statistics .second , we propose a power law model for predicting the degree distribution of email traffic for nodes of large degree that engage in mass emails along hierarchical lines of communication .this likely characterizes a significant portion of email traffic from managers ( and their agents ) to employees under their supervision .while many analysts have examined ways of extracting structural features from corporate email exchange networks , they have typically focused at the level of email exchanges between individuals ( albeit sometimes large numbers of individuals ) , bringing little or no information about formal organizational structures into their analysis .aggregating relationships based on formal organizational structures offers another important level of insight , which can be particularly useful for managers and analysts interested in interactions among business units , capabilities , or functions rather than individuals . automatically collected email datahas significant advantages for capturing interactions among organizational units : although email does not capture all relevant interactions , it provides comprehensive coverage across the entire organization without the overhead involved in large - scale survey - based studies . in order to locate individuals within organizational structures , we used organizational telephone directory data to associate email addresses with low - level organizational units , and information from organization charts to generate mappings of these units to higher - level ones .our analysis of structural relationships within lanl focuses on two broad , cross - cutting distinctions : program vs. line organizations , and technical research and development functions vs. operations functions ( safety , physical plant , etc . )+   lanl is a hybrid matrix management organization . in a fully matrixed organization , each employee has two managers : a line manager and a program or project manager ( fig .[ matrix]a ) .the employee is assigned to a line management unit based on their skill set and capabilities .for example , a computer scientist might be assigned to a computational modeling group , or an engineer to a structural engineering group .line management plays little or no role in guiding the day - to - day work of employees , however .instead , the employee is assigned to work on one or more projects , each of which is supervised by a program or project manager .a project is generally directed toward a specific product or deliverable , such design of a particular model of aircraft or completion of a particular research task .the day - to - day work of the employee toward these particular goals is directed by the program or project manager .both line and program managers usually report , through some management chain , to upper level general managers .the idealized communication pattern that results is one in which program and line managers communicate primarily vertically , interacting with both upper management and employees ( fig .[ matrix]b ) . in order to keep things running smoothly , however, program and line managers must also periodically communicate laterally , to ensure a good fit between capabilities and projects .the matrix management model became popular in the aerospace industry with the rise of program management in the 1950s , and was in part influenced by the organizational structure of the manhattan project , in which los alamos played a major role . at lanl today , line and program organizations play less distinct roles .the base - level line units that house most employees are called groups , which may be built around programs or capabilities . in our analysis, we draw a distinction between groups and higher - level line management organizations , which are nt directly involved in technical or operations work .program organizations play a variety of coordinating roles among groups , management , and outside organizations , and sometimes conduct technical or operations work as well . despite this flexible definition, our analysis reveals that technical program organizations occupy a very well - defined structural space within the organization as a whole .[ biggraph ] shows email traffic between all organizational units at lanl over a period of 25 days , laid out using a force - vector algorithm .the units are colored according to the classification described above , and their sizes represent betweenness centrality .there are some visible patterns in this layout .first , a number of operations groups have the highest centrality , probably because they provide services to most of the other organizational units at the laboratory . ranking the nodes by betweenness centrality confirms this : 17 of the top 20 nodes are operations organizations .in addition , operations units and technical units occupy distinct portions of the graph ; this indicates that there is generally more interaction within these categories than between them . the highly central operations groups appear to play a bridging role between the two categories .administration units appear to be somewhat more closely associated with technical units than operations units , although this is difficult to state with certainty .some of the ambiguities in interpretation can be clarified by grouping all units in a given category into a single node , resulting in the 7-node graph shown in fig .[ topemail ] .this view , which uses a simple circular layout , reveals that there is a large amount of email traffic ( in both directions ) on the technical side of the organization along the path administration - management - program - group , and relatively little traffic between these entities along any other path .the operations side of the organization does not display this pattern , indicating that relationships between groups , programs , and management are more fluid there .this suggests that technical program organizations at lanl , rather than representing an independent chain of command ( as in a true matrix organization ) have instead evolved to play an intermediary role between technical groups and technical management .the structure of this relationship at lanl is depicted in fig .[ matrix]c .  another way of understanding the roles different types of organizational units play is in terms of their relationships with outside entities .[ comnoncom ] plots the number of emails each type of organization sends and receives to / from commercial vs. non - commercial domains .this indicates that all types of operational units communicate significantly more with commercial entities , which is probably driven by relationships with suppliers and contractors .technical groups , technical management , and administration communicate about equally with commercial and non - commercial domains .the outlier here is technical programs , which are much more highly connected to non - commercial domains , particularly .gov addresses .this further expands on the role of technical programs , suggesting that they are a nexus for coordination of technical work both internally , among line management organizations , and externally , between lanl and outside funding agencies .this is a potentially important finding , with implications for how program organizations should be supported and managed .email network maps can also be used to visualize relations among members of an organizational unit. figures [ gsmall ] and [ gbig ] show email networks that were obtained from email exchange records among the members of two lanl groups over a period of two weeks .we intentionally chose groups that do similar work ( theoretical research ) . in the smaller group in fig .[ gsmall ] , the two nodes with highest betweenness centrality are group managers , and the third is technical support staff .thus , the group has a relatively unified hierarchical structure with management and support staff at the center . in the larger group ,managers were still among the most central nodes , but many other nodes had similar betweenness centrality ( fig  [ gbig ] ) .these include administrative assistants , seminar organizers , and several project leaders .this indicates a flatter , less centralized organizational structure .application of a community detection algorithm to this graph reveals two main communities .as it happens , this group was created recently by merging two previously existing groups , and the detected communities generally correspond to those groups .several network types , including biological metabolic networks , the world wide web , and actor networks , are conjectured to have power law distributions of node connectivity . in the case of metabolic networks ,the interpretation of scale free behavior is complicated by the lack of complete knowledge and relatively small sizes ( nodes ) of such networks , while the mechanisms of self - similarity in many large social networks are still the subject of debate .however , organizational hierarchy has been shown to generate degree distributions for contacts between individuals that follow power laws .managers prefer to use email to communicate with subordinates in many different communication contexts .we propose that node connectivity patterns in the email networks of large , formal organizations are driven , in part , by management hierarchy and specific patterns of email use by managers , in particular the mass broadcast of email announcements .based on this observation , we develop a scale - free behavioral model that takes into account features specific to email communications in organizations . in this model ,the self - similarity of the connectivity distribution of the email network is a consequence of the static self - similarity of the management structure , rather than resulting from a dynamic process , such as preferential attachment or optimization strategies .more specifically , self - similarity is due to the ability of a manager to continuously and directly communicate only with a relatively small number of people , while communications with other employees have to be conveyed in the form of broad announcements .suppose that the top manager in an organization sends emails to all employees from time to time .this manager must correspond to the node in the email network that has highest connectivity .suppose that the top manager also talks directly ( in person ) to managers that are only one step lower in the director s hierarchy ( let s call them 1st level managers ) .each of those 1st level managers , presumably , control their own subdivisions in the organization .assuming roughly equal spans of managerial control , we can expect that , typically , one 1st level manager sends emails to people . in reality, each manager also has a support team , such as assistants , administrators , technicians , etc . who also may send announcements to the whole subdivision .let us introduce a coefficient which says how many support team employees are involved in sending global email announcements in the division on the same scale as their manager .we can then conclude that at the 1st level from the top there are persons who send emails to employees at a lower level .each 1st level manager controls 2nd level ones and we can iterate our arguments , leading to the conclusion that there should be managers on the 2nd level who should be connected to people in their corresponding subdivisions . continuing these arguments to the lower levels of the hierarchy , we find that , at a given level , there should be managers ( or their proxies ) who write email announcements to people in their subdivision .  consider a plot that shows the number of nodes vs. the weight of those nodes , i.e. their outdegree . considering previous arguments , we find that the weight should correspond to nodes .excluding the variable , we find where is the natural logarithm .( [ lognw ] ) shows that the distribution of connectivity , , in a hierarchical organizational email network should generally be a power law with exponent . obviously ,at some level , this hierarchy should terminate around the point at which , because the number of managers should not normally exceed the number of employees .hence the power law ( [ lognw ] ) is expected to hold only for nodes with heavy weights , e.g. , i.e. for nodes that send announcement - like one - to - many communications , and at lower this model predicts a transition to some different pattern of degree distribution . at this level , it is likely that non - hierarchical communication patterns begin to dominate in any case .   for .red line is a linear fit corresponding to .,width=288 ]   for .red line is a linear fit corresponding to .,width=288 ]  in order to compare this model to actual network data , we analyzed the statistics of node connectivity in email records at lanl during a two - week time interval ( fig .[ triv1 ] ) .we removed nodes not in the domain and cleaned the database of various automatically generated messages , such as bouncing emails that do not find their target domain .however , we kept domains that do not correspond to specific employees , such as emails sent from software support services .our remaining network consisted of nodes , which is still about three times the number of employees at lanl .this is partially attributed to the fact that we did not exclude domains that are not attached to specific people , and also the fact that a significant fraction of employees have more than one email address for various practical reasons .  numerical analysis , in principle , should allow us to obtain information about parameters , and , from which one can make some very coarse - grained conclusions about the structure of the organization .such an analysis should , of course , always be applied with a certain degree of skepticism due to potential issues with data quality , the simplicity of the model , and logarithmic dependence of the power law on some of these parameters .we found that our data for could be well fitted by ( fig .[ triv2 ] ) . if , e.g. , we assume , then , i.e. each manager has the support of typically people , who help her post various announcements to her domain of control .the power law should terminate at the level of hierarchy given by , which corresponds to , i.e. the email network data suggest that there are typically managers of different ranks between the working employee and the top manager of the organization .the typical number of email domains to which the lowest rank manager sends announcements is .this should also be the degree of the nodes at which the power law ( [ lognw ] ) should be no longer justified .indeed , we find the breakdown of the power law ( [ lognw ] ) at .this estimate also predicts that a typical working employee receives emails from managers or their support teams .comparing these results to the actual organizational structure of the organization is very difficult due to the large excess of email addresses over the number of actual employees , and the lack of empirical data on many of the model parameters . keeping in mind these difficulties ,the estimated model parameters seem to be generally consistent with the actual organizational structure .in reality , lanl has 5 possible layers of line management between an employee and the laboratory director , but this is complicated by the facts that the lowest layer is often not used , and some employees work for organizations that report directly to a higher - level manager .so the estimate of given above might be consistent with the actual organization structure .the average group size at lanl is difficult to determine quantitatively from available data , but appears to be generally in the 20 - 40 person range , which is somewhat lower than the number of domains ( 48 ) to which the lowest - level manager sends emails based on model estimates .again , although these results might suggest possible conclusions about the accuracy of the model , we do not currently have data of sufficient quality to make a rigorous comparison between model estimates and real - world organizational structure in this case .fig . [ time1 ] shows total email traffic and number of addresses sending email over one week with a one minute resolution .working days have a bi - modal distribution with heaviest activity at the beginning and end of the day . the lower level of activity on friday is related to an alternative work schedule that most lanl employees follow .this schedule enables employees to take every other friday off in exchange for working longer hours monday - thursday . as a consequence ,only slightly more than 50% of the workforce is at work on a given friday .this is directly reflected in the amount of email traffic on fridays .visualizing and modeling email traffic in complex organizations remains a challenging problem .visualizing email data in terms of formal organizational units reduces complexity and provides results that are more intelligible to organization members and analysts interested in understanding organizational structure at a macro level . for predicting the degree distribution of high - degree nodes in an organization, we find that it is useful to take into account both organizational hierarchy and email - specific behavior ( in particular , the use of mass emails within line management chains ) .these findings suggest that considering information about formal organizational structures alongside email network data can provide significant new insights into the functioning of large , complex organizations .100 sims , b.h . , sinitsyn , n. , eidenbenz , s.j . : visualization and modeling of structural features of a large organizational email network . in : proceedings of the 2013 ieee / acm international conference on advances in social networks analysis and mining , pp .787 - 791 .acm , new york ( 2013 ) j. diesner , t. l. frantz , and k. m. carley , `` communication networks from the enron email corpus ` it s always about the people .enron is no different . ' '' _ computational and mathematical organizational theory _ , vol .201 - 228 , oct . 2005 .a. chapanond , m. s. krishnamoorthy , and b. yener , `` graph theoretic and spectral analysis of enron email data , '' _ computational and mathematical organizational theory _ ,265 - 281 , oct . 2005 .t. karagiannis and m. vojnovic .( 2008 , may ) ._ email information flow in large - scale enterprises _ [ online ] .available : http://research.microsoft.com/pubs/70586/tr-2008-76.pdf .g. e. bugos , `` programming the american aerospace industry , 1954 - 1964 : the business structures of technical transactions , '' _ business and economic history _ , vol . 22 , no . 1 , pp .210 - 222 , fall 1993 .d. l. hansen , b. shneiderman , and m. a. smith , _ analyzing social media networks with nodexl : insights from a connected world_. burlington , ma : elsevier , 2011 .m. girvan and m. e. j. newman , `` community structure in social and biological networks , '' _ pnas _ , vol .2 , pp . 7821 - 7826 , apr .e. ravasz , a. l. somera , d. a. mongru , z. n. oltvai , and a .-barabasi , hierarchical organization of modularity in metabolic networks , " _ science _ vol .1551 - 1555 , aug .e. ravasz and a .-barabasi , `` hierarchical organization in complex networks , '' _ phys .e _ , vol .67 , no . 2 , 026112 ,barabasi , e. ravasza , and t. vicsek , deterministic scale - free networks , " _ physica a _ , vol .299 , no . 3 - 4 ,559 - 564 , oct . 2001 .m. l. markus , electronic mail as the medium of managerial choice , " _ organization science _, vol . 5 , no .502 - 527 .m. mitzenmacher , a brief history of generative models for power - law and lognormal distributions , " _ internet mathematics _ ,f. papadopoulos , m. kitsak , m. a. serrano , m. boguna , and d. krioukov , popularity versus similarity in growing networks , " _ nature _ vol .489 , p. 537a. clauset , c. r. shalizi , and m. e. j. newman , `` power - law distributions in empirical data , '' _ siam review _ , vol .661 - 703 , 2009 .

Abstract: this paper presents findings from a study of the email network of a large scientific research organization , focusing on methods for visualizing and modeling organizational hierarchies within large , complex network datasets . in the first part of the paper , we find that visualization and interpretation of complex organizational network data is facilitated by integration of network data with information on formal organizational divisions and levels . by aggregating and visualizing email traffic between organizational units at various levels , we derive several insights into how large subdivisions of the organization interact with each other and with outside organizations . our analysis shows that line and program management interactions in this organization systematically deviate from the idealized pattern of interaction prescribed by `` matrix management . '' in the second part of the paper , we propose a power law model for predicting degree distribution of organizational email traffic based on hierarchical relationships between managers and employees . this model considers the influence of global email announcements sent from managers to all employees under their supervision , and the role support staff play in generating email traffic , acting as agents for managers . we also analyze patterns in email traffic volume over the course of a work week .


we are given a subroutine which for every one of possible inputs gives us a or a as output .the subroutine is in a `` black box '' , so we are not allowed to see what algorithm it uses .the problem is to find an input for which the subroutine gives ( a `` marked '' input or `` element '' ) or to say that there is no such input .classically there is no other way than to try out many inputs on the subroutine . on the other hand ,if the subroutine also works for superpositions of inputs    we can use grover s quantum searching algorithm to find a `` marked element '' ( an input that yields ) which gives us a quadratic speedup relative to the classical method .in particular when we know the number of marked elements , grover s algorithm will find one of these elements with high probability using only about queries to the subroutine .a slight modification of this algorithm can actually increase the success probability to while using at most one more query ( see below ) .all this also works when we are only told that the number of marked elements is either or .on the other hand , when we do nt know the number of marked elements , grover s algorithm does nt perform that well . in particular to get a certainly correct answer any algorithm will use on the order of queries as for the classical case .if we allow for some error probability , a grover type quantum search still helps . note that when the quantum algorithm outputs a candidate marked element, we can check it with just one more query .thus the only error that can happen , is that the algorithm wrongly says that there are no marked elements .thus the error is so - called one sided and we have not only a quantum monte carlo algorithm ( bqp ) but actually a quantum - rp algorithm .so the problem now is : given and some upper bound on the error probability , find an algorithm that uses as few queries to the black box as possible .note that by error probability we mean the error probability for the worst case , which here in particular means the number of marked elements for which the algorithm performs worst .a simple , but not optimal , solution to this problem is to run grover s algorithm many times for some random number of iterations (= number of queries ) between and , which is about the optimum for just one marked element .we choose the number of iterations of each run uniformly at random from the given range . from the evolution of the state vector ( of the qc ) in grovers algorithm ( see below ) it is easy to see that in each such run and for any number of marked elements ( except ) the probability of finding a marked element is about .actually a careful analysis ( see below ) shows that for the worst - case number of marked elements , it s about . by repeating thismany times we get asymptotically at most about the following number of queries :     the proposed algorithm again basically consists of many runs of grover s algorithm , each for some numbers of iterations ( = number of queries ) .it consists of 2 parts plus possibly a third one to improve it a bit .the first part checks for all numbers of marked elements from to some maximum , each time running grover s algorithm for the appropriate number of iterations , which is about for marked elements .if the maximal error probability which we allow is below about we have to use the exact version of grover which is guaranteed to work for a given known number of marked elements .the total number of queries of this first step is about :    which i ve obtained by approximating the sum with an integral .the second part consists of a number of grover runs for a random number of iterations smaller than the last grover run of the first part .thus we choose the number of iterations uniformly at random from the range .this i propose to do times .thus the second part consists of at most the following number of queries :    it turns out that this choice of the number of grover runs in the second part which gives equal number of queries for the first and second parts is optimal .  in the second part we have probability about to find a marked element in every run ,provided the number of marked elements is larger than .say the actual ( worst case ) probability is some ( which is a bit smaller than 1/2 and which we will determine below ) , then and we get the following number of queries for the total algorithm as a function of and :    note the ( admittedly small ) improvement relative to the simpler algorithm described above .note also that as a function of for a fixed goes as    which is of course better than the exponential we could achieve by just running the same probabilistic algorithm over and over .it turns out that the worst case detection probability actually occurs for a number of marked elements close to .so if we want we can add a third part to the algorithm where we take care of this worst case by just classically checking a number of random inputs to the black box . to achieve the maximum allowed error probability only takes a number of queries that is of a smaller order than the number of queries used in the first two parts , thus we can neglect the cost of this third part .  in the remainder of the paperi will determine the maximal error probability for a grover run with a random number of iterations and i will also give a simple argument that an exact version of grover s algorithm can be constructed for a known number of marked elements , which has been known before ( i think it s by peter hoyer , but ca nt find a reference ) .first we have to review grover s algorithm :   each iteration of grover s algorithm consists of the following four steps :    the initial state is the uniform amplitude superposition of all possible inputs to the black box .the first step involves querying the oracle , while the second and fourth steps involve hadamard transforming each of the qubits ( thus here we have ) .we can write the initial state as    where is the set of marked elements while and .it turns out ( and is easy to see ) that after any number of applications of the 4 steps of grover s algorithm the state remains of the above form , thus a superposition with real coefficients of the uniform amplitude superposition of all unmarked ( basis- ) states and of the uniform amplitude superposition of all marked states .the actual calculation consists simply of applying the 4 above steps to each of these 2 states . in every iterationthe state vector gets rotated by some angle , so after iterations we have :    where is given by    it is easy to check that which will facilitate the subsequent calculations .the probability of finding a marked element after steps is . if we choose the number of iterations uniformly at random from the range we get the success probability    where the summation is easily accomplished because we can write the trigonometric functions in terms of exponentials which gives us geometric series .also we used . note that , as stated above , .now we have to look for the minimum of ( = worst case ) over the range of marked elements .the lower end of this range corresponds to .the following plot shows as a function of ( here for ). note that while .the point is where first reaches .significant for us is the next local minimum and the last local minimum , where i have taken the limit to get these values .these two values are the absolute minimum ( in our range ) and the next larger local minimum . if we include the third part of the algorithm ( classical checking ) we do nt have to worry about , thus . then asymptotically the number of queries becomes ( from equation [ res ] ) :  the second graph above shows how the success probability evolves in grover s algorithm for 3 cases of interest .the range shown in the plot is the range which we use in the second part of the algorithm , thus from 0 to about iterations .the thick solid line is for marked elements , for which the average probability over this range is .the dashed graph corresponds to the first local minimum in the first plot , whereas the dotted graph gives the last local minimum which occurs for only few unmarked elements .here i give a simple argument why grover s algorithm can easily be modified to give a correct answer for a known number of marked elements . the problem with standardgrover is that never exactly becomes .our task now is to modify the 4 steps in grover s algorithm so that we get a smaller rotation angle .imagine we apply these ( modified ) 4 steps to the initial ( uniform amplitude ) state .usually in the 1 .step we change the phase of the marked states by which allows us to increase the amplitude of the marked states to . if we do nt change the phase it is easy to see that the 4 steps of grover s algorithm do nt change the initial state at all , thus the amplitude of the marked states remains . by continuityit is now clear that we can adjust the absolute value of the amplitude of the marked states to any value between these extremes . to get the amplitude back to real and positivewe then call the black box once more to rotate the phase of the marked elements by the right amount.  actually one can avoid this last ( additional ) call to the black box by also choosing a different phase change in step 3 , but this is not so easy to explain .of course there are also various other ways to make grover s algorithm exact .the algorithm we have constructed is clearly not optimal .it s order _ is _ optimal , as stated in corollary 3 on page 6 of , but the performance can obviously still be improved by a multiplicative constant .my guess is that the number of queries can be reduced by at least a factor of , but probably by a factor of or more .unfortunately no bound on the multiplicative constant is specified in .  in the second part of the algorithm we choose the number of iterations of grover at random .this is not very elegant and certainly not optimal , but it makes the algorithm and the assessment of its performance easy .a general observation is that zero error algorithms are a rather academic and unphysical concept .any computer has some failure probability , this is especially true for quantum computers .fortunately fault tolerant techniques allow us to greatly increase the reliability .essentially the error probability can be reduced exponentially in the resources we invest into fault tolerance which allows us to attain a reliability that is good enough for all practical purposes .so what about my use of the exact variant of grover s algorithm ?actually we do nt really need the exact version , we merely need to be able to greatly reduce the error without using many more queries . in a faulttolerant implementation we actually anyways ca nt really apply phase rotations by any amount because we only can use a finite `` universal '' set of gates .the better we want to approximate a given phase rotation the more such gates we have to use .fortunately the number of gates necessary typically only increases as the logarithm of the precision of the approximation .

Abstract: we want to find a marked element out of a black box containing elements . when the number of marked elements is known this can be done elegantly with grover s algorithm , a variant of which even gives a correct result with certainty . on the other hand , when the number of marked elements is not known the problem becomes more difficult . for every prescribed success probability i give an algorithm consisting of several runs of grover s algorithm that matches a recent bound on the order of the number of queries to the black box . the improvement in the order over a previously known algorithm is small and the number of queries can clearly still be reduced by a constant factor .


models constructed with -stable distributions possess several useful properties , including infinite variance , skewness and heavy tails (; ; ; ) .-stable distributions provide no general analytic expressions for the density , median , mode or entropy , but are uniquely specified by their characteristic function , which has several parameterizations . considered as generalizations of the gaussian distribution ,they are defined as the class of location - scale distributions which are closed under convolutions .-stable distributions have found application in many areas of statistics , finance and signal processing engineering as models for impulsive , heavy tailed noise processes (; ; ; ; ; ) .the univariate -stable distribution is typically specified by four parameters :  ] determining the degree and sign of asymmetry ( skewness ) ; the scale ( under some parameterizations ) ; and the location . the parameter is termed the characteristic exponent , with small and large implying heavy and light tails respectively .gaussian ( ) and cauchy ( ) distributions provide the only analytically tractable sub - members of this family . in general , as -stable models admit no closed form expression for the density which can be evaluated pointwise ( excepting gaussian and cauchy members ) , inference typically proceeds via the characteristic function .this paper is concerned with constructing both univariate and multivariate bayesian models in which the likelihood model is from the class of -stable distributions .this is known to be a difficult problem .existing methods for bayesian -stable models are limited to the univariate setting (; ; ; ; ; ) .inferential procedures for -stable models may be classified as auxiliary variable methods , inversion plus series expansion approaches and density estimation methods .the auxiliary variable gibbs sampler increases the dimension of the parameter space from ( and ) to , where is the number of observations . as strong correlations between parameters and large sample sizesare common in the -stable setting , this results in a slowly mixing markov chain since gibbs moves are limited to moving parallel to the axes ( e.g. ) .other markov chain monte carlo ( mcmc ) samplers ( ) adopt inversion techniques for numerical integration of the characteristic function , employing inverse fourier transforms combined with a series expansion to accurately estimate distributional tails .this is performed at each iteration of the markov chain to evaluate the likelihood , and is accordingly computationally intensive .in addition the quality of the resulting approximation is sensitive to the spacing of the fast fourier transform grid and the point at which the series expansion begins ( ) .univariate density estimation methods include integral representations , parametric mixtures (; ) and numerical estimation through splines and series expansions (; ) .approximates symmetric stable distributions using a mixture of gaussian and cauchy densities . and approximate the -stable density through spline polynomials , and via a mixture of gaussian distributions .parameter estimation has been performed by an expectation - maximization ( em ) algorithm and by method of ( partial ) moments ( ) . implemented within an mcmc sampler , such density estimation methods would be highly computational .none of the above methods easily generalize to the multivariate setting .it is currently only practical to numerically evaluate two dimensional -stable densities via inversion of the characteristic function . herethe required computation is a function of and the number and spread of masses in the discrete spectral representation (; ) . beyond two dimensions this procedure becomes untenably slow with limited accuracy .  in this articlewe develop practical bayesian inferential methods to fit univariate and multivariate -stable models . to the best of our knowledge, no practical bayesian methods have been developed for the multivariate model as the required computational complexity increases dramatically with model dimension .the same is true of classical methods beyond two dimensions .our approach is based on recent developments in `` likelihood - free '' inference , which permits approximate posterior simulation for bayesian models without the need to explicitly evaluate the likelihood .  in section [ sec : abc ] we briefly introduce likelihood - free inference and the sampling framework used in this article .section [ sec : alpha - stable ] presents the bayesian -stable model , with a particular focus on summary statistic specification , a critical component of likelihood - free inference .we provide an evaluation of the performance of the proposed methodology in section [ sec : examples ] , based on controlled simulation studies in 1 , 2 and 3 dimensions .finally , in section [ sec : realdata ] we demonstrate an analysis of real daily currency data under both univariate and multivariate settings , and provide comparisons with existing methods .we conclude with a discussion .computational procedures to simulate from posterior distributions , , of parameters given observed data , are well established ( e.g. ) .however when pointwise evaluation of the likelihood function is computationally prohibitive or intractable , alternative procedures are required .likelihood - free methods ( also known as _ approximate bayesian computation _ ) permit simulation from an approximate posterior model while circumventing explicit evaluation of the likelihood function (; ; ; ; ) .assuming data simulation under the model given is easily obtainable , likelihood - free methods embed the posterior within an augmented model where , , is an auxiliary parameter on the same space as the observed data .the function is typically a standard smoothing kernel ( e.g. ) with scale parameter , which weights the intractable posterior with high values in regions when the observed data and auxiliary data are similar .for example , uniform kernels are commonplace in likelihood - free models ( e.g. ) , although alternatives such as epanechnikov and gaussian kernels provide improved efficiency .the resulting approximation to the true posterior target distribution \ ] ] improves as decreases , and exactly recovers the target posterior as , as then becomes a point mass at .posterior simulation from can then proceed via standard simulation algorithms , replacing pointwise evaluations of with monte carlo estimates through the expectation ( [ eqn : marginal - posterior ] ) , based on draws from the model ( e.g. ) .alternatively , simulation from the joint posterior is available by contriving to cancel the intractable likelihoods in sample weights or acceptance probabilities . for example , importance sampling from the prior predictive distribution results in an importance weight of , which is free of likelihood terms .see for a discussion of marginal and joint - space likelihood - free samplers .  in general, the distribution of will be diffuse , unless is discrete and is small . hence , generating with is improbable for realistic datasets , and as a result the degree of computation required for a good likelihood - free approximation ( i.e. with small ) will be prohibitive . in practice , the function is expressed through low dimensional vectors of summary statistics , , such that weights the intractable posterior through ( [ eqn : joint - posterior ] ) with high values in regions where .if is sufficient for , then letting recovers as before , but with more acceptable computational overheads , as . as sufficient summary statistics are generally unavailable ,the use of non - sufficient statistics is commonplace .the effect of less efficient estimators of in ( [ eqn : marginal - posterior ] ) is a more diffuse approximation of .hence the choice of summary statistics in any application is critical , with the ideal being low - dimensional , efficient and near - sufficient .  in this article, we implement the likelihood - free sequential monte carlo sampler of , detailed in appendix a. as the class of particle - based algorithms is the most efficient currently available in likelihood - free computation ( e.g. ) , and within this class , the sampler of is the only one to allow non - uniform functions , this sampler provides the best combination of efficient simulation and flexible modelling .we now develop univariate and multivariate bayesian -stable models . unlike existing methods ,likelihood - free inference is independent of model parameterization .denote the characteristic function of i.i.d .univariate -stable distributed random variables by a popular and convenient parameterization is \right ) & \text { \ if } \alpha \neq1\\ \exp\left (  i\delta t-\gamma\left\vert t\right\vert \left [ 1+i\beta\frac { 2}{\pi}\text{sgn}\left (  t\right )  \ln\left ( \gamma\left\vert t\right\vert \right ) \right ]  \right ) & \text { \ if } \alpha=1 , \end{array } \right.\ ] ] where and ( e.g. ) .many alternative parameterizations are detailed in and . under ( [ alphastablecharacteristicfn ] ) ,the intractable stable density function is continuous and unimodal , taking support on if ; if and otherwise .efficient simulation of auxiliary data , , under the model is critical for the performance of likelihood - free methods ( section [ sec : abc ] ) . here , it is straightforward to generate -stable variates under the model defined by the characteristic function ( [ alphastablecharacteristicfn ] ) ( e.g. ; ) .this approach is provided in appendix b.   a key component of likelihood - free inference is the availability of low - dimensional , efficient and near - sufficient summary statistics . since -stable models can possess infinite variance ( ) and infinite mean ( ) , this choice must be made with care .here we present several candidate summary vectors , , previously utilized for parameter estimation in the univariate -stable model . in section [ sec : examples ] we evaluate the performance of these vectors , and provide informed recommendations for the choice of summary statistics under the likelihood - free framework .  0.5 cm _mcculloch s quantiles _ + and estimate model parameters based on sample quantiles , while correcting for estimator skewness due to the evaluation of , the quantile of , with a finite sample . here, the data are arranged in ascending order and matched with , where .linear interpolation to from the two adjacent values then establishes as a consistent estimator of the true quantiles .inversion of the functions then provides estimates of and . note that from a computational perspective , inversion of or is not required under likelihood - free methods .finally , we estimate by the sample mean ( when ) .hence   0.5 cm _zolotarev s transformation _ + based on a transformation of data from the -stable family , ( p.16 ) provides an alternative parameterization of the -stable model with a characteristic function of the form + \mathbb{c}\left (  \nu^{-\frac{1}{2}}-1\right )  \right\},\ ] ] where is euler s constant , and where , and this parameterization has the advantage that logarithmic moments have simple expressions in terms of parameters to be estimated . for a fixed constant ( recommends ) and for integer , the transformation is defining and , estimates for and are then given by where , using sample variances and .as before , is estimated by ( for ) , and so .  0.5 cm _ press s method of moments _ + for and unique evaluation points , the method of moments equations obtained from can be solved to obtain (; ) where $ ] .we adopt the evaluation points and as recommended by , and accordingly obtain   0.5 cm _ empirical characteristic function _ + the empirical characteristic function , for , can be used as the basis for summary statistics when standard statistics are not available .e.g. this may occur through the non - existence of moment generating functions .hence , we specify where .   , , and under the univariate -stable model and likelihood - free sampler ( summary statistics ) , based on 10 sampler replicates .traces are shown as a function ( -axis ) of sampler progression ( ) and scale parameter reduction .parameter values used for data generation are , , and . ]spectral mass angles ( angle ) and weights ( radius ) for bivariate -stable model , with , and . plots( a)(d ) demonstrate evolution of the estimates for decreasing scale parameter values , based on 10 sampler replicates . ]   discrete spectral masses , , for the trivariate -stable model .true values of the first spectral mass are ( 70% ) and .point shading indicates mmse value of as a percentage .the plots demonstrate the evolution of the estimates for decreasing scale parameter values , based on 200 sampler replicates . ]posterior credibility intervals , based on discrete spectral masses , 20 randomly placed projection vectors and 10 replicate samplers .broken lines denote estimate of spectral mass using j. p. nolan s _ mvstable _ software , available at academic2.american.edu/ , with ( dashed line ) 40 deterministic projection locations and ( dash - dot line ) 80 projection locations . ]

Abstract: -stable distributions are utilised as models for heavy - tailed noise in many areas of statistics , finance and signal processing engineering . however , in general , neither univariate nor multivariate -stable models admit closed form densities which can be evaluated pointwise . this complicates the inferential procedure . as a result , -stable models are practically limited to the univariate setting under the bayesian paradigm , and to bivariate models under the classical framework . in this article we develop a novel bayesian approach to modelling univariate and multivariate -stable distributions based on recent advances in `` likelihood - free '' inference . we present an evaluation of the performance of this procedure in 1 , 2 and 3 dimensions , and provide an analysis of real daily currency exchange rate data . the proposed approach provides a feasible inferential methodology at a moderate computational cost .   * keywords : * -stable distributions ; approximate bayesian computation ; likelihood - free inference ; sequential monte carlo samplers .