Faster Together : Collective Quantum Search

Joining independent quantum searches provides novel collective modes of quantum search (merging) by utilizing the algorithm’s underlying algebraic structure. If n quantum searches, each targeting a single item, join the domains of their classical oracle functions and sum their Hilbert spaces (merging), instead of acting independently (concatenation), then they achieve a reduction of the search complexity by factor O(√n).


Introduction
The quantum search algorithm, from its initial conception [1][2][3], to the subsequent manifold of ongoing developments, see e.g., the various open research projects addressing the association of quantum search with e.g., quantum entanglement [4], quantum programming [5], error faultiness [6], fixed-point quantum search [7], and quantum walks [8], constitutes one of the pillars of the research area of quantum computing.Despite its simplicity and the manifested versatility in applications the algorithm remains a challenge to meet, especially when it is considered as a computational primitive that could be synthesized in non trivial ways with itself.
This point of view is put forward in this work, where utilizing the underlying algebraic structure of the search algorithm and its matrix representation theory [9], the algorithm is treated as a computational unit composed in two different ways, to be called merging and concatenation.Merging of two algorithms creates a computational advantage that reduces search complexity in contradistinction to non joint searches of simple concatenation.More accurately, it is shown that the merging of n single searches with database dimensions N k = 2 k , k = 1, ..., n, causes a complexity reduction proportional of square root of n.This main result of collective search is scrutinized in all intermediated joining schemes, where among n searches k are merged and the rest are left concatenated, via partitioning databases into distinct groups of merged algorithms and then concatenating the resulting groups.The logistics of joining schemes is carried out via Young diagrams and tableaux of partitions, as well as majorization theory [10].(Proofs and examples are placed in the second part of the paper).

Single Quantum Search
Find 1 ≤ k < N marked elements from the set ∆ = {1, 2, ...N}, by improving the classical complexity O(N) of the search.
In terms of oracle algebra generators the search operator reads

Collective Quantum Search: Merging and Concatenation
Considering joining of two searches in Hilbert spaces H r = span{|i } Nr i=1 , r = 1, 2, with dimensions N 1 , N 2 in the form of concatenation, we first need to embed their database vectors into a larger space H 1 ⊕ H 2 of dimension N 1 + N 2 , by padding in zeros into their components, on their top or on their tail, until their number becomes N 1 + N 2 .By convention, concatenating searches of dim N 1 with one of dim N 2 , would mean to form new basis vectors {|∅ N 1 ⊕ |i N 2 ; i = 1, ..., N 2 }, and {|i N 1 ⊕ |∅ N 2 , i = 1, ..., N 1 }, where we denote by |∅ N 1 , |∅ N 2 , the respective null vector with all their components being zero.These two new sets of basis vectors constitute the database of the jointed algorithms of dim N 1 + N 2 .The marked vector to be called |x conc will read Definition 1. l-merging and l-concatenation.Let l quantum search algorithms [1] U r (f r ) : its space, U merg (f merg ) = −J |smerg J |xmerg , its search unitary, and f merg : Z Nmerg → Z 2 its l-target oracle function, and also denoted by where U 1 , U 2 are the search operators in Hilbert spaces with dimensions N 1 , N 2 , respectively.

Collective Quantum Search: Joining Schemes and Young Diagrams
By convention we take the horizontal direction in a Young diagram (for notation c.f. [11]) to denote merging (the number of row boxes equals the number of merged searches), and in the vertical direction the number of rows denotes concatenated sets, i.e., final column: concatenation of l(λ) merged rows one merging per row l(λ) Recall the partial order of majorization between partitions [10].Let partitions π = (π 1 , ..., π s ) and ρ = (ρ 1 , ..., ρ t ); if s ≥ t then π weakly majorizes ρ, written as π w ≻ρ, if the following inequalities are satisfied, If the last relation above is only an equality, then π majorizes ρ , written as π ≻ ρ.Associating partitions to Young diagrams, i.e., π → Y (π), an equivalent definition of majorization of partitions is induced via In this way all, Young diagrams of given m are partially ordered in the poset {π ⊢ m, ≻}, via their associated Young diagrams as shown schematically below, In the context of collective search, we say equivalently that if diagram Y (π) of a partition π describing a joining scheme for a set of searches, has been obtained from some other Y (ρ) by merging some searches among them, i.e., then π ≻ ρ.

Collective Quantum Search: Complexity
For the corresponding search complexities T π , T ρ we have the following lemma.
Lemma 3. The search complexity function T π (N 1 , ..., N n ), for a given joining scheme of n searches with dimensions N 1 , ..., N n , described by partition π, is a Schur concave function, for which it is valid that for any two weakly majorized partitions π w ≻ρ of n, the corresponding complexities are anti-isotonic, i.e., T π ≤ T ρ .
For simplicity's sake, hereafter and unless otherwise stated we consider that a single search algorithm has only one marked element, i.e., k = 1.Symbolism: We state the following lemma.Lemma 4. Let l searches with database Hilbert space dimensions {N 1 , ..., N l } , arranged in a Young tableau either as an l -box row, in case of merging, or as an l-box column, in case of concatenation.Denoting the corresponding complexities as Having introduced the main concepts and mathematical tools of collective quantum search we proceed to state and show the main result.
Consider the ratio of the extreme values of complexities T conc /T merg , i.e., "all concatenated" over "all merged".The sequence {N i } n i=1 of dimensions, can be of two distinct kinds : (i) {N i } n i=1 an unbounded sequence, e.g., N i 's are consecutive terms of sequence 2 i (a natural choice for database sizes), in this case we show that T conc /T merg = O( √ n); (ii) if the sequence {N i } n i=1 is bounded (e.g., N i = 2 b i ,where {b i } n i=1 is bounded), then the ratio Tconc Tmerg ∈ Θ(n), i.e., it is asymptotically linear in n, the number of databases (for "Big Theta" notation c.f. [12]).Next lemma and proposition provides an estimation for the search complexity for arbitrary database dimensions.i=1 N i , are the continuous analogues (continuous functions) for complexities T conc , T merg , then, i) Proposition 1.For arbitrary positive integers (database sizes) N i , i = 1, 2, ..., n it holds that (a) consecutive terms of the unbounded sequence (b) terms of a bounded sequence of positive integers with p = sup{N i } n i=1 , q = inf{N i } n i=1 , then : , for all i = 1, ..., n, and {b i } n i=1 is an increasing and bounded above sequence of positive integers, the statement of lemma remains valid.
(ii) Since lim n→∞ N n = 2 6 , database sizes N n are asymptotically equal to a constant number, and this is true since (R, |•|) is a complete metric space.Observe that the curve in Figure 1 is close to line y = x (i.e., the ratio T conc /T merg is close to n).In the special case of constant sequence {N j }, for the continuous versions merg of the complexities, we have that (iii) Since every sequence in R has a monotone subsequence, it follows that, given a bounded above sequence {N j }, we can always extract a monotone subsequence {N c j } necessarily bounded, and therefore convergent.(c.f.Bolzano-Weirstrass theorem, stating that each bounded sequence in R m has a convergent subsequence).Hence, even if {N j } is bounded above but not convergent, if using only {N c j } as database sizes, the ratio T conc /T merg will be close to database number.
(iv) A geometric interpretation of inequalities of the proposition, providing bounds for the complexity, is that asymptotically, the ratio Tconc Tmerg lies in the interior of an angle δ = arctan(λ) − arctan(λ −1 ) with vertex at point (0, 0) and sides along directions nλ −1 and nλ, symmetric wrt bisector y = x; it lies on the bisector if N i = N, i.e., all distances are equal, (in this case the search operator is A special case of minimum complexity is stated in the following lemma. x-axis: # of databases .Plots for T conc /T merg , for non decreasing and bounded above sequence of database sizes (blue curve), and an unbounded one (black curve).Here the bounded sequence , and the unbounded one N j = 2 j , N 1 = 2 are used.Dashed line: y = x .Lemma 6.The complexity of an l merging is minimum and independent of l if and only if all involved database dimensions are equal.

Collective Quantum Search: Threshold Cases
Summarizing the study so far by referring to sequences , we note that: the first sequence concerns the weak ordering of partitions ranging from total concatenation to total merging of n searches.The second one concerns the associated numerical ordering of these schemes via comparison between their corresponding complexities.We seek to clarify which are the generic threshold cases in the sequences according to some criteria, i.e., the cases in which merging gives no computational advantage in search, due to some circumstantial reasons to be determined.Two such criteria are, the conjugate partition criterion (CPC), and the threshold partition criterion (TPC).In case of CPC the * conjugation for partitions is used to single out as threshold cases the self-conjugate partitions π = π * for which T π = T π * , [13], under some specified database dimensions.In case of TPC the threshold cases are the so called threshold partition π, which hold a balanced number of boxes (searches) in the upper and lower parts of its Young diagram.

Conjugate Partition Criterion
The complexity of any joining scheme is determined both by the partition shaping its Young diagram and by filling of partition's boxes by the respective Hilbert space dimensions N i of quantum databases.A simplification is the standard tableau and particularly the physically motivated choice N i = 2 i .Consider n jointed searches interpolating between full concatenation with partition (1 n ) and full merging with partition (n) .Consider the conjugation of partition π → π * , which produces partition π * by turning rows into column and vice versa and then assign dimensions N ij to each box (search), i.e., (π i , j) → N ij , and seeks values for N ij , so that the ensuing complexities are equal, i.e., T π = T π * .This equality is achieved by any intermediate joining scheme ) is self-conjugate and the next choice of dimensions gives equal complexity The indicated filling with dimensions p, q, r, s fulfils condition, i.e., T (3,2,1) = T (3,2,1) * .

Threshold Partition Criterion
Proceeding from full concatenation to full merging of n searches by moving up one box at a time (merging one more search), creates diagrams that majorize all preceding ones, as explained.Explicitly, let of division of a Y (π) into two disjoint pieces, Y u (π) with boxes lying on and to the right of the diagonal, and Y d (π) be the rest piece, i.e., Y (π) = Y u (π)∪Y d (π).E.g. for partition π = (6, 5, 3, 3, 2, 2, 1) Let next y u (π) be a partition whose parts are the lengths of the rows of the shifted shape Y u (π), and y d (π) be the partition whose parts are the lengths of the columns of Y d (π).If n is even and π ⊢ n, then partition π is called graphic partition iff y u (π) w ≺y d (π) and it is called threshold partition π th iff y u (π th ) = y d (π th ) [13].In the case of threshold partition it follows that T yu(π) = T y d (π) and that half of the number of merging responsible for crossing the diagonal have already happened (i.e., ).This threshold relation landmarks the midway situation before the onset of total merging.For the example above the pairs y u (π) = (6, 4, 1), y d (π) = (6, 4, 1), and y u (π) = (7, 3, 1), y d (π) = (6, 3, 2) satisfy the TPC.

Oracle Algebra for Collective Quantum Search
Let database Hilbert spaces and let the marked items be the first, the third, and the second elements in H N 1 , H N 2 , H N 3 , respectively.To the partitions (111), ( 21), (3), of 3, correspond the joining (i) a 3−merging in database

and concatenation between them, and finally (iii) concatenation of searches in
The marked items are |1 , |7 , |10 , so x (3) 12 The marked items are , the generators decompose (iii) Single searches in H N 1 , H N 2 and H N 3 and concatenation between them; The marked items are ), etc, so for a = 1, 2, 3, 0, the following decomposition is obtained, .
Having the oracle algebra matrix generators we compute the unitary search operators for the three corresponding partitions, U 2 ) , U , By means of a similar search unitary, the collective quantum search complexity measures can be computed.
preserves the projection of density matrix ρ along the collective marked vector |x x| := N l a=N 1 |x a x a | , or equivalently preserves the N l a=N 1 π a (Σ 3 ) component of the collective density matrix [9].This implies the search complexity remains invariant under the action of V 3 .
This equality of complexities is expressed in terms of the minimization of the projection of time-evolved collective density matrix on the collective marked item, i.e., which is a generalization of an analogues formula for l = 1, describing the azimuthal symmetry of single search algorithm [9].To any partition τ ⊢ N there corresponds a symmetry group M τ = G/K for the collective quantum search.

Proofs, Examples, and Discussion
In this second part of the paper we have put together a number of items : 1. "Collective quantum search: Merging and Concatenation", with proofs of lemmas and numerical examples; in the following section.
2. "Collective quantum search: Joining Schemes and Young diagrams" we have placed the proof of the main proposition and of the auxiliary lemmas, together with numerical examples that demonstrate the workings of collective quantum search; in the final section.
3. "Oracle algebra and representations" we introduce the mathematical details of the oracle algebra and some examples from its matrix representations.

Merging and Concatenation
Proof.(Lemma 1) The target vector decomposes in Let the initial vectors |x conc , |s conc and the corresponding projection operators Then Similarly and So the search operator by means of the previous decomposition splits into a direct sum, i.e.
Similarly, for an l-concatenation it is valid that U conc = l j=1 U j .Symmetries of U conc and U merg .For concatenation, the search operator is determined up to a V 1 ⊕ V 2 unitary, i.e.
. By induction on l, a l-concatenation algorithm, has l i=1 V (N l )-symmetry, which is the diagonal subgroup of U(N merg ).Grover [2] showed that for a single search algorithm with one target vector, the unitary search operator U G = −J s J x can be replaced by a more general operator which is also unitary and it can be in one of the two following equivalent forms with V ∈ U(N).These symmetries survive in the case of joined searches as follows.For merged algorithms the unitary symmetry is U(N merg ), i.e.,
The notion of number partition is associated to the joining of quantum searches as follows: given a number of search algorithms m with database dimensions N 1 , N 2 , ..., N m , we can join them either by merging or by concatenating in various ways therefore number m is partitioned as λ ⊢ m, where every part λ i of λ denotes the number of algorithms joined in a similar way, i.e., either by merging or by concatenation.Thus every possible joining scheme corresponds to a Ferrers diagram and vice versa, i.e., π → T π , ρ → T ρ , we find by way of example that T π ≤ T ρ .
The linear combination of φ µ 's functions is also a Schur-concave function, and this in particular is valid for the search complexity T π associated with a partition π, i.e., π → T π , or explicitly For n = 16 and the partition π = (6, 4, 4, 2), there are four functions of variables x = (x 1 , ..., x 16 ), each one of them and their linear combination is a Schur-concave function.
Proof.(Lemma 4) Applying Jensen inequality [14] for the convex function x → √ x yields In the relation above the equality is reached iff 0 , where {x} denotes the fractional part of the real number x.Notice that the special case where all the numbers appearing in the integral part are all integers, never occurs due to the involvement of π .
Proof.(Lemma 6) Let N 1 , ..., N l be the sizes of databases, then the complexity equals Due to AM-GM inequality, we take that The equality holds iff N 1 = ... = N l ≡ N, and therefore the minimum is Remark 2.
(i) For comparison reasons we find that the complexity of l-concatenation algorithm is (ii) The total tableau complexity for a joining scheme described by its corresponding Young diagram λ is computed as follows: let a Young diagram λ = (i 1 , i 2 , ..., i r ) then the total search algorithm consists of r groups of concatenated sub-algorithms where each group contains i 1 , i 2 , ..., i r merged algorithms.Via previous lemma and remark, the tableau complexity equalsT , where i 0 = 0 and i

Main Proposition
Next, we consider the ratio of the extreme values of complexities T conc /T merg ("all concatenated" over "all merged"), and regarding the sequence of the dimensions {N i } n i=1 , two cases are arising for its asymptotic behaviour.Moreover, for arbitrary positive integers (database sizes) N i , i = 1, 2, ..., n we prove that for the continuous analogues (continuous functions) for the complexities T conc , T merg , it holds that In more details, if {N i } n i=1 is an unbounded sequence, specifically N i 's are consecutive terms of the geometric sequence 2 i (which is the most natural and reasonable choice for database sizes), we conclude that i=1 is bounded), it results that the ratio T conc /T merg is asymptotically linear with respect to the number n of the databases.This fact leads to an interesting observation: although the qualitative difference between a bounded and an unbounded sequence of database sizes is essential (notice that N i = 2 i increases exponentially fast), however, the quantitative change that entails to the ratio of complexities, is only quadratic (quadratic reduction) with respect to the database population.

Proposition 2. For arbitrary positive integers (database sizes) N
(a) consecutive terms of the unbounded sequence (b) terms of a bounded sequence of positive integers with p = sup{N i } n i=1 , q = inf{N i } n i=1 , then : Proof.Applying the Cauchy-Schwarz inequality we obtain that: (a) Carrying out trivial calculations, we take : In this first case, we have that N i = 2 i , so and . Therefore: The RHS of the above asymptotically equals to ( √ 2 + 1) 2 , so , then for all i = 1, 2, ..., n, is valid that 2 ≤ q ≤ N i ≤ p, so Moreover π

Geometry of Complexity Reduction
All concave functions fulfil a very intuitive geometric condition with their graph, namely that the center of mass of a set of points lying on the graph is lying not above the graph itself.Quantifying this geometric property leads to the Jensen inequality [14], which in fact is the reason for achieving complexity reduction in various forms of joining schemes.This is demonstrated below by means of a numerical example.Example 2. Numerical example (see Figure 3).Let the Young diagram of shape (5,4,1) and let the following Young tableau (strictly increasing row and column-wise, no repetitions)   the center of mass vector c 2 = (616.0,19.556) and its crossing point with the graph q 2 = (616.0,√ 616.0) = (616.0,24.8193)
Equal Complexity Tableaux and Shapes.Motivated by the geometric explanation of the complexity measure for various schemes of joining quantum search algorithms as has been studied in previous section, we proceed to address the problem of determining shapes and tableaux describing ways of joining searches.We study joining of database spaces of equal dimension N. The complexity is differentiated from one scheme to the other due to the difference of the associated Young diagram shapes, so we call it shape complexity.Joined quantum searches, all of which have equal Hilbert space dimension N and share the same shape complexity, are displayed as a pattern of bold typed integer partitions from 3 to 7 within the Young lattice.The pattern of equal complexities is independent from N. c 1 y + c 2 y + c 3 y ≤ q 1 y + q 2 y + q 3 y Figure 4 displays the contour of equal complexity families of joined quantum algorithms having unequal database sizes.A constant complexity difference (vertical segments) is chosen between tableau complexity (lower broken line) describing concatenation of groups of merged quantum searches and its upper bound (upper full line) describing the same group jointed by concatenation only.

Oracle Algebra and Representations
Definition 2. Let the set ∆ = {1, 2, ...N}, a subset I ⊂ ∆, and the oracle function f, be the characteristic function of I with k elements, defined as f (i) = 1,for i ∈ I, and f (i) = 0,for i / ∈ I.
Let also (a) the Hilbert space l 2 (D), the vector and its orthogonal vector Next, we introduce the following Σ 0 , Σ 1 , Σ 2 , Σ 3 , as the generators of A f : For the oracle function f (i) = 1, 1 ≤ i ≤ k < N, and zero otherwise, the representation above reads , and therefore, for an arbitrary element A ∈ A f , it holds that Therefore, the generators of A f are:  x In order to compute the generators π 12 Σ (1,1,1) a , a = 0, 1, 2, 3, we proceed in an analogous manner to the previous case.For simplicity, we also introduce the following shorthand notation, to denote direct sums of vectors u  , for e.g., π 12 Σ (1,1,1) 1 we obtain that:

Discussion
An important follow up of this work concerns the fact that the collective quantum search can be cast in the language of cooperative game theory, and so wider problems of search complexity reduction can be addressed.In fact, cooperative game theory is an area where multi-agent entities choose to collaborate in various schemes in order to take advantage from the collaboration in lowering some computational load which would enable them to achieve a desirable shared objective, see e.g., [16], for a wealth of principles and examples.For this connection, particularly useful would be the special joining schemes determined by the partitions π = π * , π th and π max , as tools for studying coalition formation of merging teams of searches aiming to trade collectivity for less search complexity.This appears to be a favorite context for implementing and applying the idea of merging.In particular quantum search by merging as outlined here could also be applied in applications where quantum simulation of quantum searching is carried out by multi-particle Hamiltonian models (see e.g., [17] and references therein).These prospects will be taken up elsewhere.

1 , 2 , 1 √
..., l with H r = span{|i } Nr i=1 their database Hilbert spaces, U r (f r ) = −J sr J xr , where the reflection operators J sr = 1 − 2 |s r s r | , and J xr = 1 − 2 |x r x r | , are defined wrt some vectors |x r and |s r , with |s r = Nr Nr i=1 |i , and |x r = Nr i=1 f r (i) |i ∈ H r the target vectors; here f r : Z Nr → Z 2 their respective oracle functions.We further denote the merged space by

Figure 1
Figure1.Plots for T conc /T merg , for non decreasing and bounded above sequence of database sizes (blue curve), and an unbounded one (black curve).Here the bounded sequence N j =

2. 4 . 1 .G
Generalized Azimuthal Symmetry Let the partition τ = (N 1 , N 2 , • • • , N l ) of N of length l = l(τ ), and let the one parameter subgroup U a (1) = e iφaπa(Σ 3 ) , generated by π a (Σ 3 ) ∈ End(H a ).Let the group G = U(N) and the subgroup K = N l a=N 1 U a (1).Consider a concatenation of l searches for a given partition τ ⊢ N, with search operator U and search step implemented by the transformation ρ → U (τ ) G ρU (τ ) † G .Let further the unitary operator V 3 below.

Figure 2 .
Figure 2. Young tableaux for m = 6 and the corresponding complexities T G .
all databases are of equal size N, then for any diagram λ the tableau complexity equals T (N,...,N ) λ = r π 4 √ N .

Figure 3 .
Figure 3. Jensen's inequality for the numerical example.Round dots represent points lying on the graph, and square dots represent center of mass points.

Figure 4 .
Figure 4. Display of the contour of equal complexity families of joined quantum algorithms.
databases ∆ N 1 , ∆ N 2 , ∆ N 3 respectively, as well as direct sums for other operators and the corresponding Σ's.