In the claw detection problem we are given two functions f:D ? R and g:D ? R (|D| = n, |R| = k), and we have to determine if there is exist x,y ? D such that f(x) = g(y). We show that the quantum query complexity of this problem is between ?(n^{1/2}k^{1/6}) and O(n^{1/2+?}k^{1/4}) when 2 ? k < n

Ambainis, Andris

Balodis, Kaspars

Iraids, J?nis

English

arXiv

Dagstuhl Research Online Publication Server

A Note About Claw Function with a Small RangeAndris Ambainis #Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, LatviaKaspars Balodis #Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, LatviaJānis Iraids #Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, LatviaAbstractIn the claw detection problem we are given two functions f : D → R and g : D → R (|D| = n,|R| = k), and we have to determine if there is exist x, y ∈ D such that f(x) = g(y). We showthat the quantum query complexity of this problem is between Ω(n1/2k1/6)and O(n1/2+εk1/4)when2 ≤ k < n.2012 ACM Subject Classification Theory of computation → Quantum query complexityKeywords and phrases collision, claw, quantum query complexityDigital Object Identifier 10.4230/LIPIcs.TQC.2021.6Funding Supported by the project “Quantum algorithms: from complexity theory to experiment”funded under ERDF programme 1.1.1.5.1 IntroductionIn this note we study the Claw problem in which given two discrete functions f : D → Rand g : D → R (|D| = n, |R| = k) we have to determine if there is a collision, i.e., inputsx, y ∈ D such that f(x) = g(y). In contrast to the Element-Distinctness problem, wherethe input is a single function f : D → R and we have to determine if f is injective, Claw isnon-trivial even when k < n. This is the setting we focus on.Both Claw and Element-Distinctness have wide applications as useful subroutinesin more complex algorithms [5, 12] and as a means of lower bounding complexity [10, 1].Claw and Element-Distinctness were first tackled by Buhrman et al. in 2000 [8] wherethey gave an O(n3/4)algorithm and Ω(n1/2)lower bound. In 2003 Ambainis, introducing anovel technique of quantum walks, improved the upper bound to O(n2/3)in the query model[4]. It was soon realized that a similar approach works for Claw [9, 13, 15]. MeanwhileAaronson and Shi showed a lower bound Ω(n2/3)that holds if the range k = Ω(n2)[2].Eventually Ambainis showed that the Ω(n2/3)bound holds even if k = n [3]. The same lowerbound has since been reproved using the adversary method [14]. Until now, only the Ω(n1/2)bound based on reduction of searching was known for Claw with k = o(n) [8].We consider quantum query complexity of Claw where the input functions are givenas a list of their values in black box. Let Q(f) denote the bounded error quantum querycomplexity of f . For a short overview of black box model refer to Buhrman and de Wolf’ssurvey [7]. Let [n] denote {1, 2, . . . , n}. Let Clawn→k : [k]2n → {0, 1} be defined asClawn→k(x1, . . . , xn, y1, . . . , yn) ={1, if ∃i, j xi = yj0, otherwise.Our contribution is a quantum algorithm for Clawn→k with quantum query complexityQ(Clawn→k) = O(n1/2+εk1/4)and a lower bound Q(Clawn→k) = Ω(n1/2k1/6). In section 2we describe the algorithm, and in section 3 we give the lower bound.© Andris Ambainis, Kaspars Balodis, and Jānis Iraids;licensed under Creative Commons License CC-BY 4.016th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021).Editor: Min-Hsiu Hsieh; Article No. 6; pp. 6:1–6:5Leibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany6:2 A Note About Claw Function with a Small Range2 Results▶ Theorem 1. For all ε > 0, we have Q(Clawn→k) = O(n1/2+εk1/4).Proof. Let X = (x1, . . . , xn), Y = (y1, . . . , yn) be the inputs of the function. We denotek = nκ .Consider the following algorithm parametrized by α ∈ [0, 1].1. a. Select a random sample A = {a1, . . . , aℓ} ⊆ [n] of size ℓ = 4 · nα · ln n and query thevariables xa1 , . . . , xaℓ .Denote by XA = {xa | a ∈ A} the set containing their values. Do a Grover search foran element y ∈ Y such that y ∈ XA. If found, output 1.b. Select a random sample A′ = {a′1, . . . , a′ℓ} ⊆ Y of size ℓ and query the variablesya′1 , . . . , ya′ℓ .Denote by YA′ = {ya′ | a′ ∈ A′} the set containing their values. Do a Grover searchfor an element x ∈ X such that x ∈ YA′ . If found, output 1.2. Run Claw4b ln n→k algorithm (with the value of b specified below) with the followingoracle:a. To get xi: do a pseudorandom permutation on x1, . . . , xn using seed i and usingGrover’s minimum search return the first value xj such that xj /∈ XA.b. To get yi: do a pseudorandom permutation on y1, . . . , yn using seed i and usingGrover’s minimum search return the first value yj such that yj /∈ XA′ .Let B = {i ∈ [n] | xi /∈ XA}, B′ = {i ∈ [n] | yi /∈ YA′} be the sets containing the indices ofthe variables which have values not seen in the steps 1a and 1b. We denote |B| = b = nβ .Let us calculate the probability that after step 1a there exists an unseen value v which isrepresented in at least n1−α variables, i.e., v /∈ XA ∧ |{i ∈ [n] | xi = v}| ≥ n1−α. Consideran arbitrary value v∗ ∈ [k] such that |{i | xi = v∗}| ≥ n1−α. For i ∈ [ℓ], let Zi be theevent that xai = v∗. ∀i ∈ [ℓ] Pr[Zi] ≥ n1−αn . Let Z =∑i∈[ℓ] Zi. Then E[Z] = ℓ · E[Z1] ≥4 · nα · ln n · n1−αn = 4 ln n. Using Chernoff inequality (see e.g. [11]),Pr[Z = 0] ≤ exp(−12 E[Z])≤ exp(−2 ln n) = 1n2.The probability that there exists such v∗ ∈ [k] is at most nκn2 = o(1). Therefore, withprobability 1 − o(1) after step 1a, every value v ∈ XB is represented in the input less thann1−α times. The same reasoning can be applied to step 1b and the set B′. Therefore, withprobability 1 − o(1) both b and b′ are at most k · n1−α = nκ+1−α.Similarly, we show that with probability 1 − o(1) each x ∈ B appears as the first elementfrom B in at least one of the permutations of the oracle in step 2. Let W xi be the eventthat x ∈ B appears in the i-th permutation as the first element from B. E[W xi ] = 1b .Let W x =∑i∈[4b ln n] Wxi . E[W x] = 4b ln n · 1b = 4 ln n. Pr[Wx = 0] ≤ exp(−2 ln n) = 1n2 .Pr[∃x ∈ B : W x = 0] ≤ nn2 =1n = o(1). The same argument works for B′. Therefore, if thereis a collision, it will be found by the algorithm with probability 1 − o(1).We also show that with probability 1 − o(1), in all permutations the first element fromB appears no further than in position 4nb ln n (and similarly for B′). We denote by Pi,jthe event that in the i-th permutation in the j-th position is an element from B. E[Pi,j ] =bn . We denote Pi =∑j∈[4· nb ·ln n] Pi,j . E[Pi] = 4 · ln n. Pr[Pi = 0] ≤ exp(−2 ln n) =1n2 .Pr[∃i ∈ [4b ln n] : Pi = 0] ≤ 4b ln nn2 ≤4n ln nn2 = o(1). Therefore, the Grover’s minimum searchwill use at most Õ(√nnβ)queries.A. Ambainis, K. Balodis, and J. Iraids 6:3The steps 1a and 1b use Õ(nα) queries to obtain the random sample, and O(√n) queriesto check if there is a colliding element on the other side of the input. The oracle in step 2uses Õ(√nnβ)queries to obtain one value of xi or yi.Therefore the total complexity of the algorithm isÕ(nα + n 12 + Q(Claw4b ln n→k) · n12 −12 β).By using the O(n2/3)algorithm in step 2,Q(Claw4b ln n→k) · n12 −12 β = n 23 β+ 12 − 12 β= n 12 + 16 β≤ n 12 + 16 (κ+1−α)= n4+κ−α6 ,and the total complexity is minimized by setting α = 4+κ7 . However, we can do betterthan that. Notice that the O(n2/3)algorithm might not be the best choice for solvingClaw4b ln n→k in step 2.Let A0 denote the regular O(n2/3)Clawn→k algorithm. For i > 0, let Ai denote aversion of algorithm from Theorem 1 that in step 2 calls Ai−1. Then we show that for all nand all 0 ≤ κ ≤ 23 ,Q(Ai) = Õ(nTi(κ)),where Ti(κ) =(2i−1)κ+2i+12i+2−1 .The proof is by induction on i. For i = 0, we trivially have that Q(A0) = Õ(n2/3). Forthe inductive step, consider the analysis of our algorithm. Let us set α = Ti(κ). First, noticethat Ti(κ) is non-decreasing in κ and Ti( 23)= 23 for all i. Thus for all κ ≤23 , we haveTi(κ) ≤ 23 , hence α ≤23 andκ1−α+κ ≤23 . Second, since the coefficient of κ is2i−12i+2−1 ≤ 1 thefunction Ti(κ) is above κ for κ ≤ 23 , establishing α − κ ≥ 0. This confirms that α = Ti(κ)is a valid choice of α.It remains to show that the complexity of step 2 does not exceed Õ(nTi(κ)). By theinductive assumption and analysis of the algorithm, the complexity (up to logarithmic factors)of the second step is n to the power of (1 − α + κ) · Ti−1(κ1−α+κ)+ α−κ2 . Finally, we haveto show that(1 − Ti(κ) + κ) · Ti−1(κ1 − Ti(κ) + κ)+ Ti(κ) − κ2 ≤ Ti(κ).By expanding Ti−1(κ) and with a slight rearrangement, we obtain(2i−1 − 1)κ + 2i(1 − Ti(κ) + κ)2i+1 − 1 ≤Ti(κ) + κ2 .We can further rearrange the required inequality by bringing Ti(κ) to right hand side andeverything else to the other. Then we get(2i−1 − 1 + 2i − 2i+1−12 )κ + 2i2i+1 − 1 ≤ Ti(κ)(12 +2i2i+1 − 1).After simplification we obtain (2i−1)κ+2i+12i+2−1 ≤ Ti(κ), which is true.Since limi→∞ 2i−12i+2−1 =14 and limi→∞2i+12i+2−1 =12 , the result follows. ◀TQC 20216:4 A Note About Claw Function with a Small Range3 Lower BoundWe show a Ω(n1/2k1/6)quantum query complexity lower bound for Clawn→k.▶ Theorem 2. For all k ≥ 2, we have Q(Clawn→k) = Ω(n1/2k1/6).Proof. Let pSearchm : (∗ ∪ [k])m → [k] be the partial function defined aspSearchm(x1, x2, . . . , xm) ={xi, if xi ̸= ∗, ∀j ̸= i : xj = ∗undefined, otherwise.Consider the function fn,k = Clawk→k ◦ pSearch⌊n/k⌋. One can straightforwardly reducefn,k(x, y) to Clawn→k+2(x′, y′) by settingx′i ={xi, if xi ̸= ∗k + 1, if xi = ∗andy′i ={yi, if yi ̸= ∗k + 2, if yi = ∗.Now we show that Q(fn,k) = Ω(k2/3√n/k)= Ω(n1/2k1/6). The fact that Q(Clawk→k) =Ω(k2/3)has been established by Zhang [16]. Furthermore, thanks to the work done byBrassard et al. in [6, Theorem 13] we know that for pSearchm a composition theorem holds:Q(h ◦ pSearchm) = Ω(Q(h) · Q(pSearchm)) = Ω(Q(h) ·√m). Therefore,Q(Clawn→k) ≥ Q(Clawk−2→k−2 ◦ pSearch⌊ nk−2 ⌋)= Ω(k2/3√nk)= Ω(n1/2k1/6).◀4 Open ProblemsCan we show that Q(Clawn→n2/3)= Ω(n2/3)? In particular, our algorithm struggles withinstances where there are n2/32 singletons only two (or none) of which are matching andthe remaining variables are evenly distributed with Θ(n1/3)copies each, such that none arematching. Thus our algorithm then either has to waste time sampling all the high-frequencydecoy values or have most variables not sampled by step 2. If this lower bound held, it wouldimply a better lower bound for evaluating constant depth formulas and Boolean matrixproduct verification [10, Theorem 5].References1 Scott Aaronson, Nai-Hui Chia, Han-Hsuan Lin, Chunhao Wang, and Ruizhe Zhang. On theQuantum Complexity of Closest Pair and Related Problems. In Shubhangi Saraf, editor,35th Computational Complexity Conference (CCC 2020), volume 169 of Leibniz InternationalProceedings in Informatics (LIPIcs), pages 16:1–16:43, Dagstuhl, Germany, 2020. SchlossDagstuhl–Leibniz-Zentrum für Informatik. doi:10.4230/LIPIcs.CCC.2020.16.2 Scott Aaronson and Yaoyun Shi. 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A Note About Claw Function with a Small Range

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A Note About Claw Function with a Small Range

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