Quantum Search on the Spatial Grid

Transcription

1 Quantum Search on the Spatial Gri Matthew D. Falk MIT 2012, 550 Memorial Drive, Cambrige, MA (Date: December 11, 2012) This paper explores Quantum Search on the two imensional spatial gri. Recent exploration into the topic has evise a solution that runs in O( n ln n). This paper explores a new algorithm that gives promise for the O( n) result that is the lower boun off of the gri. I. INTRODUCTION Some classical solutions to problems achieve particular spee ups when we allow those algorithms to become quantum base. The most obvious result is the ability to search a group of n elements in sub linear time. Classically, it is require to look at all of the elements, one at a time, to ensure that the marke item is or is not present. In the quantum worl, thanks to Lov Grover [4], we can o this in O( n) steps an queries. In particular, this subroutine has prove useful for a number of other algorithms an achieving quantum lower bouns. In this paper we explore the complexity of performing this algorithm when reuce to movement on a two imensional spatial gri. We restrict our moel to a quantum robot walking along the two imensional gri. We begin by escribing the moel we aapt for out algorithm in Section II. We then procee to a iscussion of Grover s Algorithm in Section III an follow up with the latest results about search on a spatial gri in Section IV. Finally, we give our new algorithm for search on the spatial gri with results in Section V. We escribe the case when there are multiple marke items being search for an how it iffers from the non spatial version in Section VI an comment on the implications this gives for other problems in Section VII. We give some concluing remarks in Section VIII an iscuss what nees to be one. II. THE MODEL The majority of this paper refers to the two imensional gri. This is use to escribe a moel in which a quantum robot traverses a two imensional array of points through a quantum walk. That is, a quantum robot is initially place on the gri at an arbitrary location. They can then move to ajacent noes, an only ajacent noes in the gri uring one time step. At each noe, the robot is allowe to rea the information that is locate there. Our gri consists of all of the elements in X that we are intereste in looking at. The quantum robot, upon reaching a particular noe, can rea the value from the gri an perform a query on that element. The robot s movements through the gri are etermine by a quantum coin. Whenever the robot moves, it can simultaneously move in all irections (or whichever are escribe by the coin) in superposition. Thus, we can achieve a superposition of all states in X in n steps by performing a walk along the base of the gri an then performing a walk in the perpenicular irection from each of those noes in parallel. It is useful to note that each location on the gri can be classifie as a two imensional ket, i, j, an that the gri is cyclic in both irections. (i.e. i, j is connecte to n, j ). III. GROVER S ALGORITHM Grover s Algorithm can be viewe as acting on a completey connecte graph, G n, where every noe in the graph has an ege connecting it to every other noe in the graph. Thus, a quantum robot walking on the graph is able to get to absolutely any noe in a single time step. The algorithm is epenent upon this because at every iteration of the iffusion operator each noe communicates with every other noe in parallel to etermine how much amplitue will be transferre between them. Herein lies the problem when moving to the gri, noes can only talk to their irect neighbors. A lot of the eges have been remove from the graph. We can no longer calculate the mean (or invert about it) in a single time step. In orer to o this exactly as it is one in Grover s Algorithm, it woul require O( n) steps simply to get across the gri an fin this information. Thus, we nee some other way of performing this calculation. The search problem is escribe as fining a marke item in a given set of values. In this case, we let a ket, x, represent an element x X, our set of all items. We let the state s represent the equal superposition of all elements in X : s = 1 x n Grover s Algorithm is epenent on two subroutines that are repeate O( n) times. The first is the negation unitary. It negates the sole marke item in the set of elements we are looking at. If the marke item is w, then out negation unitary U w can be written as: where x U w = I 2 w w

2 Quantum Search on the Spatial Gri 2 an U wu w = (I 2 w w ) (I 2 w w ) = I 4 w w + 4 w w w w = I U w w = (I 2 w w ) w = w 2 w = w U w x = (I 2 w w ) x = x The secon subroutine is the iffusion operator: U s. U s = 2 s s I where unitarity is easily confirme an an U s s = (2 s s I) s = 2 s s = s U s w = (2 s s I) w = 2 n s w The algorithm consists of repeate applying U w U s to the starting state an having the amplitue of the marke state grow with each iteration. While this is an optimal algorithm, running as asymptotically fast as it can, the algorithm cannot be translate irectly to the spatial gri, ue the missing eges of the graph. IV. QUANTUM SEARCH ON THE GRID It was shown by Benioff that Grover s search algorithm took a serious hit when applie to the two imensional spatial gri. The application still requires O(n 1/2 ) queries; however, in between each of those queries, the quantum robot may have to move a istance equal to the iameter of the graph, also O(n 1/2 ). Thus, the total running time was O(n) [3]. Aaronson an Ambainis fixe this by giving an algorithm that searches a gri for a single marke item in O( n log 2 n) total steps (queries an walking) [1]. Their algorithm is the main breakthrough in this area an uses Grover s Algorithm recursively on smaller an smaller subcases combine with amplitue amplification. Later, the search problem on the 2D gri was reuce to a O( n ln n) solution by Ambainis et. al [2]. In 2008, Avatar Tulsi evise a way of improving on the leaing algorithm for two-imensional spatial search using an ancillary qubit. His results lea to an O( n ln n) solution to fin a marke item out of a list of n elements arrange on the vertices of a two-imensional lattice [5]. It remains an open problem whether or not this can be improve upon an the optimal non-lattice solution of O( n) reache. V. SPATIAL QUANTUM SEARCH Our algorithm for search on the gri consists of 4 pieces. The first is the builing of the superposition on the gri. A quantum robot starting at any location walks along any horizontal in the gri, creates a superposition of all states an then repeats that process along the vertical to create the state s, state below for convenience: s = 1 x n Once the superposition is create, our robot can act on each noe or state simultaneously however information cannot be transferre from one noe to another if they are more than a constant number of eges apart. That is, noes that are O( n) apart cannot communicate an thus we lose the ability to invert about the mean. Our next piece of machinery is to use the negation operator, as Grover i, U w. The next two final pieces of the algorithm are the Local Diffusion operator, UL, an the Amplitue Dispersion operator, UA ( is not a power but a paramter), which are escribe in the following sections. The algorithm consists of repeate applications of U w UL 4U wua 4. A. Local Diffusion Operator Even though we cannot talk to all of the noes on the gri, we are still allowe to learn about our immeiate neighbors. We can travel to an communicate with any noes that live within a constant istance of the current noe. However, we cannot talk with a noe that is further in communication with a noe that we cannot reach. That is, the set of noes that we are communicating with uring a particular time step plus all the noes that are transitively communicating with us via its communicators must form a constant size set; this to preserve unitarity. We are free to perform any unitary on this subset of noes. Thus, we can ivie the gri into equally size pieces that tessellate an cover the gri an then perform Grover s Diffusion on each of them. Our iffusion operator must act locally, cover the gri, an preserve unitarity. We constructe our operator in the following form, which allows us to locally sprea the amplitues. It is built up of a superposition of states making small squares that tessellate the entire gri. x

3 Quantum Search on the Spatial Gri 3 u L (i, j) = x,y=0 4i + x, 4j + y u L (i, j) u L(m, n) = δ i,m δ j,n It is useful to notice that each of the tessellation states are orthogonal to each other an that the states are properly normalize, mainly: u L (i, j) u L (m, n) = δ i,m δ j,n Using this, we can buil our unitary operator. U 4 L = 2 u L (i, j) u L (i, j) I Knowing this, it is fairly trivial to show that U L is unitary. U 4 L U 4 L = (2 = 4 = 4 = 4 (2 i,j,m,n=0 u L (i, j) u L (i, j) I) u L (m, n) 4 i,j,m,n=0 4 4 u L (i, j) u L (i, j) I) u L (i, j) u L (i, j) u L (m, n) u L (i, j) u L (i, j) + I u L (i, j) δ i,m δ j,n u L (m, n) u L (i, j) u L (i, j) + I u L (i, j) u L (i, j) u L (i, j) u L (i, j) + I UL 4 U 4 L = I UL 4 is unitary an therefore we can use it as an operation in our algorithm. The point of this operator is to take local pieces of the gri an trae their amplitues. That is, if a particular piece of the gri contains the marke item, it will act like Grover s Algorithm on that piece an start sening amplitue to the marke item s noe. Otherwise, if there is no marke item in the piece, this operator will actively try to level off the amplitue in that region. At this point we introuce the choice of region. We can choose any region that we can tesselate the gri with. For instance, the above unitary operator is base on a region that is a local square. If we let be the length of the square, we can efine a more general, still unitary, operator that acts on the gri: u L (i, j) = 1 1 x,y=0 where we state without proof that i + x, j + y an U L = 2 n/ u L (i, j) u L(i, j) I with our operator being efine for = 4 an (UL ) UL = I. These are just very basic examples of regions that tessellate, the gri. We measure the amount of work a robot has to o for each particular iteration of this operator in terms of. That is, a square region of size requires the robot to visit 2 noes on the gri an it woul take O() steps for the robot to o this an perform the unitary on this region. However, this parameter is chosen ahea of time (even though it is left as a parameter of the system) an is therefore a constant in the analysis of the algorithm. In our version, we choose = 4 for the Local Diffusion Operator, which gives only a constant O(4) = O(1), an negligible, slow own. The Local Diffusion Operator acts to level out each local region of the gri, unless it contains the marke item. Together with the Negation Operator, U w, this buils the amplitue of the marke item, locally. B. Amplitue Dispersion Operator Simply increasing the amplitue of the marke item within a local region of the gri is not enough. As n may be very large, increasing w s amplitue over a finite region will place a very small upper boun on the amplitue that this noe can reach. Therefore, we nee some way of pulling the amplitue from other regions of the gri. First we notice that by requiring amplitue to move from everywhere on the gri towars our marke item, this puts a lower boun of O( n) on the algorithm, because the robot will nee that many steps to reach all other noes. Our trick here is to perform another iffusion. The secon pass over the gri acts very similarly to the previous iffusion operator, except that instea of working on a strictly local region, the new regions are sprea out to cover multiple of previous regions. For our algorithm, after trying many regions, we foun it extremely successful to make our Amplitue Dispersion Operator the same as the Local Diffusion Operator with a slight shift. The region is a square of size 4 4 so we ecie to move the starting location over 2 noes in each irection (right an own), in orer to get the maximal overlap. Our new states are: u A (i, j) = 1 1 x,y=0 i + x + 2, j + y + 2

4 Quantum Search on the Spatial Gri 4 with the operator efine the same way: U A = 2 n/ u A (i, j) u A(i, j) I again with (UA ) UA = I, an = 4 for our algorithm. Alreay unitary by efinition of our previous operator, we now have two new tools at our isposal, UL an U A, which provie a slowown of 2 an act to iffuse an isperse the amplitue respectively. C. Other Local Regions Here we escribe some of the other regions that we explore. We measure each each with respect to. 1. Four Corners This region acts by looking at aset of corners of istance apart. Similar to the square this takes O() steps. These corners provie a smaller amount of computation that nees to be one by the robot, but because it only looks at 4 separate noes, oes not o as well as the 4 4 square region. However, in our experiments, it performe the same as the 2 2 square. Our states for 4 corners are: u c(i, j) = i, j x,y {0,} i + x, j + y We leave it as an open question to fin an/or prove the optimal region given a restriction on. There are many other regions you coul choose. However, the larger you go the slower the algorithm performs. It is useful to note that the best performing iffusers are the ones that have the largest number of overlap between the two iffusions. Our algorithm has an average overlap of 4 regions allowing the probability to isperse throughout the gri polynomially fast. We now introuce a restriction on the choice of, or rather how that choice restricts the rest of the algorithm. In orer for the chosen region to properly tessellate the gri, we nee both the length an with to be multiples of, or mainly 2 n. D. Algorithm As mentione previously, our algorithm is similar to that of Grover s; it relies on the repeate application of the unitary. 1. Buil the superposition over the gri by walking along a horizontal an then a vertical. This takes a one time cost of O(n 1/2 ). 2. Apply the operator U w UL 4U wua 4 to the system O(n 1/2 ) times. Each iteration takes O( ) steps, as the negation operator is a single step an the others are on the orer of. 3. Measure the state an obtain the marke item with high probability. 2. Crosses This is the region built up of a noe s nearest neighbors in the four carinal irections. For this region we get = 5, but it only incurs a slowown proportional to 1 as it escribes the total number of noes hit, not the iameter of the region an each noe is one step away. Our states for the crosses are: u (i, j) = i, j ( x=±1 i + x, j + y=±1 i, j + y + i, j This covers yourself an your nearest neighbors. When tessellation the centers of the next region occur (2, 1) away an this ensures that we on t overlap any noe more than once on the same pass. 3. Restrictions One can use any combination of the above regions or efine their own. Results will vary base on the tessellation pattern an size of those regions that you choose. ) E. Results As we show next, experiments run with this operator show that the marke item s probability peaks aroun n 1/2 iterations; however, we o not get amplitues arbitrarily close to 1. Most of our amplitues are on the orer of which correspons to a 1 2 probability or better of measuring the marke item. We are still working on ways to increase this as well as to etermine exactly when this maximum amplitue is hit. Either one on its own woul prove effective, as once we know how many iterations it takes, we can perform a Local Diffusion Operator on the gri without ispersing. This will buil up the marke item as the amplitue that has not reache it yet is store in the neighboring noes. This also means that if we measure the noe incorrectly, we have been given information as to where the noe is an can rerun the algorithm on a smaller portion of the gri in orer to fin the marke item with even better probability. Our results are all base on the tessellation of shifte squares of size 4 4. Below is a graph relating the size of our ata n to the number of iterations neee to reach the maximum amplitue an below that the maximum amplitue that is reache.

5 Quantum Search on the Spatial Gri 5 In every case, the number of require iterations was roughly equal to the square root of the number of items being searche. Thus, we have conclue that the algorithm hits its first peak somewhere within the first O(n 1/2 ) iterations; however, we have not proven this rigorously, perhaps it can be one through the use of a geometric isplay. The amplitues ecrease as the input size gets larger an larger, which is to be expecte as the amplitue is isperse aroun a greater number of incorrect noes. In fact, this algorithm, as the reaer can see below, works so as to buil up the amplitue aroun the correct item. That is, it acts as a sink on the gri an pulls amplitues towars itself. Therefore, the noes surrouning the marke item will be the most likely options for incorrect measurements. FIG. 2: This shows the max value of the amplitue of the marke item at the above iteration count. n Max Amplitue Iterations n FIG. 1: This shows the number of iterations require for the marke item to reach its maximum amplitue. Clearly visible in this plot is the tren of the iterations neee to follow n 1/2. It is clear that the number of iterations require to hit the maximum aplitue is on the orer of n 1/2. The graph shows that the number of iterations require to get to the maximum aplitue closely follows, while slightly overshooting, n 1/2. As we can see the maximum amplitue is approaching an asymptote at 2, which 1 correspons to a 1 2 probability of measurement, at the ieal iteration. We suspect that this value can be improve upon, but we will see in the next section the length of time we have in orer to try an measure this value. We suspect that there is a way to perform a final iffusion step on the gri at this final location in orer to boost the amplitue of the marke noe, but have not foun it yet ue to the uncertainty in the max amplitue s iteration. Figures 1 an 2 show the results of simulating on our algorithm for increasing powers of 4. We present the actual values obtaine from out simulations in the following chart: F. Amplitue Propagation While a large maximum probability is nice, it is ieal to have relatively large amplitue for a long perio of time (over many consecutive iterations). That way, without knowing exactly how many iterations are neee, one is still likely to measure the actual marke item with high probability. Below we show two ifferent images. They each epict, graphically, the amplitue values for the noes on a 400 element set. The first epicts Grover s Algorithm running on this set. The noes start out in an

6 Quantum Search on the Spatial Gri 6 equal superposition an on each iteration the amplitues of the wrong elements go own while the marke item s amplitue increases. The cycle is perioic an eventually repeats. The marke element is clearly visible after the very first iteration an stays ominant for most of the progression. Unlike in our algorithm, all of the other noes ecrease equally. Each row of ata epicts three separate iterations overlai next to each other. The secon image epicts our algorithm working on the same set of 400 elements once those elements have been shifte to a spatial gri. Again the noes start in an equal superposition but now the amplitue travels towars the marke noe in waves an also buils up on those noes that are closest to the marke noe. We see a few noes (unmarke) that have relatively high amplitues when the marke item peaks, this is not completely ba. If we o in fact measure the wrong item we know that the actual marke item resies fairly close by on the gri an we can repeat the algorithm on a much smaller section of the gri to fin the marke item. The marke element is clearly visible after the very first iteration an stays ominant for most of the progression. Unlike with Grover, amplitue travels in waves to the marke noe an creates a sink or pyrami aroun the marke noe. First, however, we show some images an escribe what is being seen in each of these plots. of the marke noe, is forming. We have successfully create a sink/pyrami of amplitue aroun the marke item. Although we have not explore this in too much epth, an thus o not have exact results, this means that when measuring the state, there is an even higher probability (than just that of the marke item) that if we measure a state, it is either correct OR it is a noe that is extremely close the esire noe. We o not have the numbers corresponing to how close the marke item is, but it is an interesting topic to research. We are very intereste in knowing how far away you nee to travel from the marke noe in orer to get a max amplitue that matches Grover s Algorithm off of the gri. While we i not o the stuy, out intuition is that the istance is irectly proportional to, the parameter we use to classify a tesselation s size. Below is another image of a ifferent iteration showing more clearly the buil up of amplitue that clusters aroun the marke item. Figure 4 epicts an earlier iteration than that of Figure 3, before that amplitue has ha enough time to travel all the way to the marke noe. FIG. 3: This is a close up view of a single iteration of our algorithm. In Figure 3 we see a close up shot of a single iteration of our algorithm being run on a gri. The ifferent colors represent.15 histogram groupings. That is, everything that is a particular color falls within a.15 range of follows. As the graph is of amplitues, the maximum value is 1. Because none of our values ippe, negatively, below 0.5, we ecie to cut off the gri there in orer to have a closer view of the plot. As we will see on the next page, we have lai out multiple of these images next to each other for convenience, but o not want to confuse what a single iteration looks like. The iteration epicte is very close to the ieal iteration. We can see that there is a very nice peak inicating the noe of the marke item. Aitionally, we see a small region where some noes have a higher amplitue than the rest (the are colore lime green). Here is where most of the amplitue, that is not alreay part FIG. 4: This is a close up view of a single iteration of our algorithm, more clearly epicting the amplitue buil up aroun the marke noe. In the above figure we see the tall spike just as before. However, in this plot in only goes up to 0.6 as it is an early stage iteration. What we o see here is that there is a nice sink of light blue noes surrouning the spike, but only in the irection of its own local tessellation. We also see the wave of amplitue moving horizontally an vertically towars the marke noe. Now we show the actual plots from the simulation of our s an Grover s algorithm sie by sie.

7 Quantum Search on the Spatial Gri 7 FIG. 5: This shows the amplitue progression of Grover s Algorithm on a gri of 400 elements. FIG. 6: This shows the amplitue progression of running our algorithm on a gri of elements.

8 Quantum Search on the Spatial Gri 8 G. Comparison As seen in Figures 5 an 6 above, both algorithms o a great job of isolating the marke item an increasing its amplitue. Our algorithm oes it in fewer iterations, which is largely ue to our unitary operator which oes two rouns on each iteration (one for iffusion an one for ispersal). Each shows the marke item staying prevalent for most of the iterations. While Grover s gets the amplitue to be almost completely singular, our simulation shows that we get the marke item to a high enough amplitue to be measure over a similar time perio with high constant probability. VI. MULTIPLE MARKED ITEMS Here we show the results from running the algorithm when there are multiple marke items. Below we show the results of running the algorithm when there are two marke items. Other than that, the etails are the same: same iffusion an ispersion tessellations an still a gri. As expecte, the amplitues on t peak as high, but they shoot up very quickly, much more quickly than when there was a single marke item. This type of search coul be affecte ifferently then the regular search because the marke items proximity to each other actually matters in the running of out algorithm. Our simulation results show that the closer the marke items are to each other the worse the algorithm oes. It still fins an isolates the marke items, but the amplitues interfere with each other builing a larger pyrami of amplitue in the surrouning noes. When the items are very far apart they each quickly act as if they are the only marke item on their half of the gri. The amplitue ifferences in these cases were relatively small compare to the rest of the noes (i.e vs. 0.58). We have seen that the total combine probabilities of marke noes oes not stay constant as you increase the number of marke items. In fact, in our above case, it actually ecreases. The two noes ha a combine probability of measurement of roughly 72%, while in the single marke item case, at the peak of the algorithm the item ha a probability of 79%. We have not explore using ifferent tesselations to boost the multiple marke items avenue, but conjecture that there might be a better tessellation that is inepenent of the locality of the marke items. This is a great area to explore. We also conjecture that as the number of marke elements (percentage of total elements) increases the algorithm runs in fewer iterations but the combine probability remains roughly equivalent. FIG. 7: This shows the amplitue progression of running our algorithm on a gri of elements when there are two marke elements. VII. OTHER PROBLEMS If this pans out an proves to be a O(n 1/2 ) solution for Search on the Spatil Gri that has many implications for other problems that can be move to the gri. Mainly, any quantum that relies on searching as its lower boun (with some exceptions) woul now be able to be solve on the gri, as well, in the same running time. For instance, Collision, which has a O(n 1/3 ) solution off of the gri woul not be affecte by moving to the gri. We can still perform the same algorithm: look up n 1/3 elements an perform a search over a set of n 2/3 elements. Both on an off the gri woul then yiel the same complexity, O(n 1/3 ). VIII. CONCLUSION This paper provies the initial outline for what we believe to be the possibility of an optimal Quantum Search algorithm on the Spatial Gri. An interesting question is whether this iffuse an isperse algorithm works in higher imensions. As we are limite in the above sce-

9 Quantum Search on the Spatial Gri 9 nario by the iameter of the gri, perhaps in higher imensions we can improve upon this result. This algorithm woul also benefit from Amplitue Amplification, an iea that has not been explore in this paper an etermining exactly what the number of iterations neee is. Lastly, we amit that our tessellation pattern is not prove to be optimal, that is, there may be a better ispersion tessellation that couples more nicely with our Local Diffusion Operator. We have presente the first application of search on a spatial gri that oes not rely on Grover s Diffusion Operator in its entirity. We have introuce many new areas for exploration. Mainly: 1. Amplitue amplification from the pyrami of amplitue surroun the marke item 2. Tessellation patterns that increase ispersion 3. The application into higher imensions 4. Tessellation patters that work equally regarless of multiple item locality These are just some of the new irections that can be explore an shoul prove some pretty nice results. IX. ACKNOWLEDGMENTS I woul like to thank Scott Aaronson for the numerous iscussion we ha about Quantum Algorithms on the Gri an helping me work through new ieas about them. I woul also like to thank Vlaimir Sobes for helpful iscussions about the tessellation patterns an the types of unitary operators we can get on the spatial gri. [1] Scott Aaronson an Anris Ambainis. Quantum search of spatial regions. Proc. 44th Annual IEEE Symp. on Founation of Computer Science, pages , [2] Anris Ambainis, Julia Kempe, an Alexaner Rivosh. Coins make quantum walks faster [3] Paul Benioff. Space searches witha quantum robot. Quantum Computation an Information: Contemporary Mathematics, 305:1 12, [4] Lov Grover. A fast quantum mechanical algorithm for atabase search. STOC 96 Proceeings of the twentyeighth annual ACM symposium on Theory of computing, pages , [5] Avatar Tulsi. Faster quantum walk algorithm for the two iminsional spatial search. The American Physical Society, 2008.