How Many Functions Can Be Distinguished with K Quantum Queries?

Home , Michael Sipser

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D 1+ N + N + + N . (3) ≤ 1 2 ··· k If the probability of successfully identifying which function the oracle holds is only required to be p for each of the D functions, then

D 1 1+ N + N + + N . (4) ≤ p 1 2 ··· k h i We also give two examples of sets of D functions (and values of k and p) where (3) and (4) are equalities. In these cases the quantum algorithms succeed with fewer queries than the best corresponding classical algorithms. One of these examples shows that van Dam’s algorithm [4] distinguishing all 2N functions with high probability after N/2+O(√N) queries is best possible, answering a question posed in his paper. We also give an example showing that the bound (3) is not always tight. An interesting consequence of (3) is a lower bound on the number of quantum queries needed to sort n items in the comparison model. Here, we have D = n! functions, corresponding to the n! possible orderings, to be distinguished. The domain of these functions is the set of N = n pairs of items. If k = (1 ǫ)n, the bound (3) is violated for ǫ> 0 and n 2 − large, as is easily checked. Hence, for any ǫ> 0 and n suﬃciently large, n items cannot be sorted with (1 ǫ)n quantum queries. −

II. MAIN RESULT

Given an oracle associated with any function F of the form (2), a quantum query is an application of the unitary operator, F , deﬁned by

F x,q,wb = x, q F (x),w (5) | i | · i where x runs from 1 to N, q = b1, and w indexes the work space. A quantum algorithm that ± makes k queries starts with an initial state s and alternately applies F and F -independent | i unitary operators, Vi, producing b

ψF = VkFVk−1 V1F s . (6) | i ··· | i b b Suppose that the oracle holds one of the D functions F1, F2,...,FD, all of the form (2). If the oracle holds Fj, then the ﬁnal state of the algorithm is ψF , but we do not (yet) | j i know what j is. To identify j we divide the Hilbert space into D orthogonal subspaces with corresponding projectors P1,P2,...,PD. We then make simultaneous measurements corresponding to this commuting set of projectors. One and only one of these measurements yields a 1. If the 1 is associated with Pℓ we announce that the oracle holds Fℓ.

2 Following [3], if the oracle holds F , so that the state before the measurement was ψF 2 | i given by (6), we know that for each ℓ, Pℓ ψF is a 2k-degree polynomial in the values k | ik F (1), F (2),...,F (N). More precisely,

2 mℓ 2 Pℓ ψF = Qℓr (F (1),...,F (N)) (7) | i r=1 X where each Qℓr is a k-th degree multilinear polynomial and mℓ is the dimension of the ℓ-th subspace. Note that formula (7) holds for any F whether or not F = Fj for some j. The algorithm succeeds, with probability at least p, if for each j =1,...,D, we have

m 2 j 2 Pj ψF = Qjr (Fj(1),...,Fj(N)) p . (8) | j i ≥ r=1 X

We now prove the following lemma: Let F0 be any one of the functions of form (2). If Q is a polynomial of degree at most k such that

2 Q (F0(1),...,F0(N)) =1 (9)

then 2 2N Q (F (1),...,F (N)) (10) ≥ N N F 1+ 1 + + k X ··· where the sum is over all 2N functions of the form (2). Proof: Without loss of generality we can take F0(1) = F0(2) = = F0(N) = 1. Now ···

Q (F (1),...,F (N)) = a0 + axF (x)+ axyF (x)F (y)+ (11) x x

F (x1)F (x2) F (xg)F (y1) F (yh)=0 (12) ··· ··· XF as long as the sets x1,...,xg and y1,...,yh are not equal and x1,...,xg are distinct, as { } { } are y1,...,yh. This means that

2 N 2 2 2 Q (F (1),...,F (N)) =2 a0 + ax + axy + . (13) | | | | | | ··· F x x

Now (9) with F0(x) 1 means ≡ 2 a0 + ax + axy + =1 . (14) ··· x x

Because of the constraint (14), the minimum value of (13) is achieved when all the coeﬃcients are equal. Since there are 1 + N + + N coeﬃcients, the inequality (10) is established. 1 ··· k Suppose we are given an algorithm that meets condition (8) for j = 1,...,D. Then by the above lemma,

3 m j 2 2N p Qjr (F (1),...,F (N)) N N . (15) =1 ≥ r F 1+ 1 + + k X X ··· Summing over j using (7) yields

2 D2N p Pj ψF . (16) | i ≥ N N j F 1+ 1 + + k X X ··· N For each F , the sum on j gives 1 since ψF = 1. Therefore the lefthand side of (16) is 2 | i

and (4) follows.

III. EXAMPLES

0. If k = N, all 2N functions can be distinguished classically and therefore quantum mechanically. In this case (3) becomes

2N = D 1+ N + N + + N =2N . (17) ≤ 1 2 ··· N

1. For k = 1, if N =2n 1 there are N + 1 functions that can be distinguished [2] so the bound (3) is best possible.− The functions can be written as

a·x fa(x)=( 1) (18) −

with x 1,...,N and a 0, 1,...,N and a x = aixi where a1 an and x1 xn ∈ { } ∈{ } · i ··· ··· are the binary representations of a and x. To see howP these can be distinguished we work in a Hilbert space with basis x, q , x =1,...,N and q = 1, where a quantum query is defined as in (5) and the work{| bitsi} have been suppressed. We± define 1 x = x, +1 x, 1 x =1,...,N (19) | i √2{| i−| − i} and 1 0 = 1, +1 + 1, 1 , (20) | i √2{| i | − i} so x , x =0, 1,...,N, is an orthonormal set. Now by (5), if we define F (0) to be +1, we have{| i}

F x = F (x) x x =0, 1,...,N (21) | i | i and in particular, b

a·x fa x =( 1) x x =0, 1,...,N (22) | i − | i Now let b

4 1 N s = x (23) | i √N +1 x=0 | i X and observe that the N + 1 states fa s are orthogonal for a =0, 1,...,N. | i 2. In [4] an algorithm is presentedb that distinguishes all 2N functions in k calls with probability 1+ N + + N /2N . With this value of p, and D =2N , the bound (4) 1 ··· k becomes an equality. Furthermore, (4) shows that this algorithm is best possible.

3. Nowhere in this paper have we exploited the fact that for an algorithm that succeeds with probability 1, it must be the case that Pℓ ψF = 0 for ℓ = j. With this additional | j i 6 3 3 constraint it can be shown that for N =3, no set of 7=1+ 1 + 2 functions can be

distinguished with 2 quantum queries. Thus for N = 3 and k = 2 the bound (3) is not tight.

♥ This work was supported in part by The Department of Energy under cooperative agreement DE-FC02-94ER40818 and by the National Science Foundation under grant NSF 95–03322 CCR. ♦ [email protected] ; [email protected] ♣ [email protected] ♠ [email protected] [1] L.K. Grover, “A fast quantum mechanical algorithm for database search”, quant-ph/9605043, and Proceedings of the 28th Symposium on Theory of Computing, 1996, 212–219. [2] E. Bernstein and U. Vazirani, “Quantum Complexity Theory”, SIAM Journal on Computing, 26, 1997, 1411–73. [3] R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf, “Quantum Lower Bounds by Polynomials”, quant-ph/9802049, and Proceedings of the 39th Symposium on Foundations of Computer Science, 1998, 352–361. [4] Wim van Dam, “Quantum Oracle Interrogation: Getting all information for almost half the price”, quant-ph/9805006,and Proceedings of the 39th Symposium on Foundations of Computer Science, 1998, 362–367.