Quantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search

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1 Quantu algoriths (CO 781, Winter 2008) Prof Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search ow we begin to discuss applications of quantu walks to search algoriths We start with the ost basic of all search probles, the unstructured search proble (that Grover s algorith solves optially) We discuss both discrete- and continuous-tie quantu walk algoriths for this proble, and we prove that the running ties of these algoriths are optial Finally, we discuss quantu walk algoriths for the search proble under locality constraints Unstructured search In the unstructured search proble, we are given a black box function f : S {0, 1}, where S is a finite set of size S = The inputs x M, where M := {x S : f(x) = 1}, are called arked ites In the decision version of the proble, our goal is to deterine whether M is epty or not We ight also want to find a arked ite when one exists It is quite easy to see that even the decision proble requires Ω() classical queries, and that queries suffice, so the classical query coplexity of unstructured search is Θ() You should already be failiar with Grover s algorith, which solves this proble using O( n) quantu queries Grover s algorith works by starting fro the state S := x S x / and alternately applying the reflection about the set of arked ites, x M 2 x x 1, and the reflection about the state S, 2 S S 1 The forer can be ipleented with two quantu queries to f, and the latter requires no queries to ipleent It is straightforward to show that there is soe t = O( / M ) for which t steps of this procedure give a state with constant overlap on M (assuing M is non-epty), so that a easureent will reveal a arked ite with constant probability Discrete-tie quantu walk algorith Consider the discrete-tie rando walk on the coplete graph represented by the stochastic atrix P = (1) = 1 S S 1 I (2) 1 It has eigenvalues 1 (which is non-degenerate) and 1/( 1) (with degeneracy 1) Since the graph is highly connected, its spectral gap is very large: we have δ = = 1 This rando walk gives rise to a very siple classical algorith for unstructured search In this algorith, we start fro a uniforly rando ite and repeatedly choose a new ite uniforly at rando fro the other 1 possibilities, stopping when we reach a arked ite The fraction of arked ites is ɛ = M /, so the hitting tie of this walk is O ( ) 1 δɛ = ( 1) M 1 = O(/ M ) (3)

2 (this is only an upper bound on the hitting tie, but in this case we know it is optial) Of course, if we have no a priori lower bound on M in the event that M is non-epty, the best we can say is that ɛ 1/, giving a running tie O() The corresponding quantu walk search algorith has a hitting tie of ( ) 1 O = O( / M ), (4) δɛ corresponding to the running tie of Grover s algorith To see that this actually gives an algorith using O( / M ) queries, we need to see that a step of the quantu walk can be perfored using only O(1) quantu queries In the case where the first ite is arked, the odified classical walk atrix is P = , (5) so that the vectors ψ j are ψ 1 = 1, 1 and ψ j = j, S \ {j} = 1 j = 2,, With a general arked set M, the projector onto the span of these states is Π = j M j, j j, j + j / M j, S 1 1 j, j for j, S \ {j} j, S \ {j}, (6) so the operator 2Π 1 acts as Grover diffusion over the neighbors when when the vertex is unarked, and as a phase flip when the vertex is arked (ote that since we start fro the state ψ = j / M ψ j, we stay in the subspace of states span{ j, k : (j, k) E}, and in particular have zero support on any state j, j for j V, so 2Π 1 acts as 1 when the first register holds a arked vertex) Each such step can be ipleented using two queries of the black box, one to copute whether we are at a arked vertex and one to uncopute it; and the subsequent swap operation requires no queries Thus the query coplexity is indeed O( / M ) This algorith is not exactly the sae as Grover s; for exaple, it works in the Hilbert space C C instead of C evertheless, it is clearly closely related In particular, notice that in Grover s algorith, the unitary operation 2 S S 1 can be viewed as a kind of discrete-tie quantu walk on the coplete graph, where in this particular case no coin is necessary to define the walk The algorith we have described so far only solves the decision version of unstructured search To find arked ite, we could use bisection, but this would introduce an overhead of O(log ) Alternatively, we could show that the final state of the quantu walk actually encodes a arked ite when one exists We will now show that this is in fact the case for the continuous-tie version of this algorith; the analysis of the discrete-tie case is left as an exercise Continuous-tie quantu walk algorith ow let us fully analyze the behavior of a corresponding continuous-tie quantu walk algorith for unstructured search, assuing for siplicity that there is a unique arked ite, and our goal is to find it Clearly this is sufficient to solve the decision proble with the proise that there are either 0 or 1 arked ites, which is essentially the hardest case 2

3 This algorith is defined by a Hailtonian given by the adjacency atrix of coplete graph plus a arking ter, naely H = ( S S 1 ) I + (7) Since the ter proportional to I just generates a global phase, we can drop it to give H = S S + (8) Suppose we start fro the state S (the only sensible starting state given the syetry of the proble) and choose and evolution tie so that we have a substantial probability of observing if we easure in the vertex basis The calculation of the walk dynaics is particularly straightforward since the walk is confined to the two-diensional subspace span{, S } Let us write H in an orthonoral basis coposed of and the orthogonal state S α = (9) 1 α 2 where α := S = 1 Then Thus in the basis {, }, we have ( α H = 2 α ) 1 α 2 α 1 α 2 1 α 2 + Finally, we can calculate the evolution as ψ(t) := e iht S S = α + 1 α 2 (10) ( ) (11) = I + α( 1 α 2 σ x + ασ z ) (12) = cos αt S i sin αt( 1 α 2 σ x + ασ z ) S (14) = cos αt S i sin αt( 1 α 2 σ x + ασ z )(α + 1 α 2 ) (15) = cos αt S i sin αt(α 1 α 2 + (1 α 2 ) + α 2 α 1 α 2 ) (16) = cos αt S i sin αt (17) The probability of observing if we stop the walk and easure in the vertex basis after tie t is (13) ψ(t) 2 = α 2 cos 2 αt + sin 2 αt (18) In particular, when t = π 2α = π 2, we observe the arked ite with probability 1 Lower bound In fact, Grover s algorith is optial (up to a constant factor): any algorith for unstructured search, even with the proise that there are either 0 or 1 arked ites, requires Ω( ) queries This is one of the ost iportant results in quantu coputing, so let s prove it Suppose we can eploy an arbitrary tie-dependent Hailtonian K(t) in the case where there is no arked ite, and H (t) = K(t) + (19) 3

4 in the case where there is a unique arked ite (If the algorith uses additional workspace, then the arking ter siply acts as the identity on that part of the space) Furtherore, we let the initial state ψ(0) be any fixed, -independent state We will show that the tie to decide whether soe vertex is arked is Ω( ) ote that this will iply a lower bound of Ω( ) queries in the usual quantu query odel, since by letting K(t) = 0 and evolving for tie π, we can perfor a phase flip; and by letting K(t) be arbitrarily large for a short tie period, we can perfor a unitary gate without the oracle acting ow let us copare the evolution under H (t), giving a state ψ (t), to the evolution under K(t) alone, giving a state φ(t) Define d := ψ (t) φ(t) 2 (20) = 2(1 Re ψ (t) φ(t) ) (21) To be able to distinguish the two possibilities, we need d ɛ for soe ɛ > 0 Since this ust hold for each possible arked ite, we have d ɛ (22) ow dd dt = 2 d dt Re ψ (t) φ(t) (23) = 2 Re( ψ (t) ik(t) φ(t) ψ (t) ih (t) φ(t) ) (24) = 2 I( ψ (t) φ(t) ) (25) 2 ψ (t) φ(t) (26) 2 φ(t) (27) (Here our notation assues that there is no extra workspace; the reader is invited to check that the conclusions are not affected by this assuption) Suing on, we have d d d dt dt d (28) 2 φ(t) (29) 2 (30) where in the last step we have used Cauchy-Schwarz Finally, integrating d with the initial condition d = 0 at t = 0 (and using the triangle inequality), we find d 2 t (31) Coparing to (22), we have t ɛ 2, which shows that t = Ω( ), as claied 4

5 Search on graphs ow let s consider a variant of unstructured search with additional locality constraints We will view the ites in S as the vertices of a graph G = (S, E), and we require the algorith to be local with respect to the graph More concretely, the algoriths alternates between queries and unitary operations U constrained to satisfy U j, ψ = k j (j) α k k, φ k for any j S (where the second register represents possible ancillary space, and recall that (j) denotes the set of neighbors of j in G) Since we have only added new restrictions that an algorith ust obey, the Ω( ) lower bound fro the non-local version of the proble still applies However, it is iediately clear that this bound cannot always be achieved For exaple, if the graph is a cycle of vertices, then siply propagating fro one vertex of the cycle to an opposing vertex takes tie Ω() So we would like to know, for exaple, how far fro coplete the graph can be such that we can still perfor the search in O( ) steps First, note that any expander graph (a graph with degree upper bounded by a constant and second largest eigenvalue bounded away fro 1 by a constant) can be searched in tie O( ) Such graphs have δ = Ω(1), and since ɛ 1/ when there are arked ites, the quantu hitting tie is O(1/ δɛ) = O( ) (whereas the classical hitting tie is O(1/δɛ) = O()) Indeed, a randoly chosen d-regular graph for constant d is such an expander with high probability There are also any cases in which a quantu search can be perfored in tie O( ) even though the eigenvalue gap of P is non-constant For exaple, consider the n-diensional hypercube (with = 2 n vertices) Recall that since the adjacency atrix acts independently as σ x on each coordinate, the eigenvalues are equally spaced, and the gap of P is 2/n Thus the general bound in ters of the eigenvalues of P shows that the classical hitting tie is O(n) = O( log ) In fact, this bound is loose; the hitting tie is actually O(), which can be seen by directly coputing P M with one arked vertex So there is a local quantu algorith that runs in the square root of this tie, naely O( ) Perhaps the ost interesting exaple is the d-diensional square lattice with sites (ie, with linear size 1/d ) This case can be viewed as having ites distributed on a grid in d-diensional space For siplicity, suppose we have periodic boundary conditions; then the eigenstates of the adjacency atrix are given by k := 1 e 2πik x/ 1/d x (32) x where k is a d-coponent vector of integers fro 0 to 1/d 1 The corresponding eigenvalues are 2 d j=1 cos 2πk j (33) 1/d oralizing to obtain a stochastic atrix, we siply divide these eigenvalues by 2d The 1 eigenvector has k = (0, 0,, 0), and the second largest eigenvalue coes fro (eg) k = (1, 0,, 0), with an eigenvalue ( 1 d 1 + cos d ) 2π 1/d 1 1 ( ) 2π 2 2d 1/d (34) 2π Thus the gap of the walk atrix P is about 2 = O( 2/d ) This is another case in which 2d 2/d the bound on the classical hitting tie in ters of eigenvalues of P is too loose (it gives only O( 1+2/d )), and instead we ust directly estiate the gap of P M One can show that the classical 5

6 hitting tie is O( 2 ) in d = 1, O( log ) in d = 2, and O() for any d 3 Thus there is a local quantu walk search algorith that saturates the lower bound for any d 3, and one that runs in tie tie O( log ) for d = 2 We already argued that there could be no speedup for d = 1, and indeed we see that the quantu hitting tie in this case is O() ote that siilar results for spatial search can be obtained in the fraework of continuous-tie quantu walk 6

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