Graph Entropy and Quantum Sorting Problems

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1 Graph Entropy and Quantum Sorting Problems [Extended Abstract] Andrew Chi-Chih Yao Computer Science Department Princeton University Princeton, NJ ABSTRACT Let P (X, < P ) be a partial order on a set of n elements X {x,x 2,,x n}. Define the quantum sorting problem QSORT P as: given n distinct numbers x,x 2,,x n consistent with P, sort them by a quantum decision tree using comparisons of the form x i : x j. Let Q ɛ(p ) be the minimum number of queries used by any quantum decision tree for solving QSORT P with error less than ɛ (where 0 <ɛ</0 is fixed). It was proved by Høyer, Neerbek and Shi (Algorithmica 34 (2002), ) that, when P 0 is the empty partial order, Q ɛ(p 0) Ω(n log n), i.e., the classical information lower bound holds for quantum decision trees when the input permutations are unrestricted. In this paper we show that the classical information lower bound holds, up to an additive linear term, for quantum decision trees for any partial order P. Precisely, we prove Q ɛ(p ) c log 2 e(p ) c n where c, c > 0 are constants and e(p ) is the number of linear orderings consistent with P. Our proof builds on an interesting connection between sorting and Körner s graph entropy that was first noted and developed by Kahn and Kim (JCSS 5(995), ). Categories and Subject Descriptors F. [Theory of Computation]: Computation by Abstract Devices General Terms Theory Keywords Graph entropy, information lower bound, partial order, quantum algorithms, sorting This research was supported in part by the National Science Foundation grant CCR Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. STOC 04, June 3-5, 2004, Chicago, Illinois, USA. Copyright 2004 ACM /04/ $ INTRODUCTION How much can a computation be helped by the use of quantum algorithms has often been studied in the blackbox model, which is also called the oracle model, or the quantum decision tree model. To determine the value of a function f(x,x 2,,x N ), queries of the form x i? are successively asked and the quantum state gets updated by this information. After a predetermined number of steps, the quantum state is measured to produce the output. This standard model has been extensively studied in recent years, and several lower bound techniques have been developed and applied. For detailed descriptions of the model, we refer the readers to recent literature on this subject (see e.g., [, 2, 3, 4, 5, 0, 9]). Exactly when and how much speed-up can be achieved is still unresolved for many problems in the quantum decision tree model. In this paper we focus on a specific issue which relates to the information lower bound for classical decision trees. To identify an unknown item taken from a pool of M possibilities, it takes at least log 2 M tests in the classical framework if the information obtained from one test is only bit. However, this is no longer true in general when quantum states can be used to collect and process information. Two of the fundamental search problems in which information bounds play a part are the ordered table search, and the sorting problem. In both cases, there is a natural information bound in the classical decision tree model, with a matching upper bound. In the quantum setting, the ordered table search problem has been extensively studied (Farhi et al [8], Ambainis [2], Buhrman and de Wolf [6]), and it was demonstrated in these papers that the information bound Ω(log n) is asymptotically valid for quantum decision trees. For the sorting problem, Høyer, Neerbek and Shi [0] showed that the information lower bound also remains valid for quantum decision trees, i.e., Ω(n log n) quantumqueries are needed. In this paper, we study a class of problems known as the sorting problems for partial orders. Let P be a partial order on a set of n elements {x,x 2,,x n}. Given n input numbers consistent with P, Fredman [9] showed that there is a classical decision tree using log e(p )+2n binary comparisons x i : x j to determine the linear orderings of these numbers, where e(p ) is the number of linear orderings consistent with P. Subsequent work by Kahn and Saks [3] (see also [, 2]) showed that O(log e(p )) comparisons are sufficient. Thus, the information lower bound log 2 e(p )is asymptotically tight for classical decision trees. For quan- 2

2 tum decision trees, it is not clear whether the information lower bound is valid (except it is known to be true when P is empty as shown in [0]). This is the focus of our inquiry. Define the quantum sorting problem QSORT P as: given n distinct numbers x,x 2,,x n consistent with P, sort them by a quantum decision tree using comparisons of the form x i : x j. Let Q ɛ(p ) be the minimum number of queries used by any quantum decision tree for solving QSORT P with error less than ɛ (where 0 <ɛ</0 is fixed). Our main result is that the classical information lower bound holds, up to additive linear term, for quantum decision trees for any partial order P.Let0<ɛ</0 be any fixed constant. Theorem There exist absolute constants c, c such that Q ɛ(p ) c log e(p ) c n. Our proof relies on the general approach used in [0], and builds on an interesting connection between sorting and Körner s graph entropy in the classical decision tree setting that was first noted and developed by Kahn and Kim []. The main underlying idea in our proof is that, when adapted for partial order P, the complexity measure used in [0] turns out to be almost the same as the entropy considered in []. 2. PRELIMINARIES 2. Review of Lower Bounds in Høyer et al We review the lower bound proof by Høyer, Neerbek and Shi [0] on quantum sorting. We begin with a general framework. Let f : S {0, } m,wheres {0, } N. Consider a quantum decision tree for computing f with probability error bounded by ɛ>0. Let ξ j x > be the quantum state after j oracle steps, when x S is the oracle. Then ξ 0 x > is equal to some fixed initial state (independent of x). After T steps at the end of computation, we must have <ξ T x ξ T y > ɛ if f(x) f(y), where ɛ 2(ɛ( ɛ)) /2. To prove a lower bound, a weight function ω : S S [0, ) is chosen. Define, for each 0 j T, W j x,y S w(x, y) <ξ j x ξ j y >. If one can manage to show that, independent of the quantum algorithm used, there is an upper bound δ to W j W j+, and a lower bound M to W 0 W T, then one obtains a lower bound T M/δ on the quantum complexity. Consider the sorting of n numbers x,x 2,,x n by comparisons of the form x i : x j. In this case we have a function f : S {0, } m where S {0, } N (with N n(n ) and m log 2 (n!)). The set S consists of all the oracles x (x i,j i j) thatrepresentasetofn(n ) bits consistent with some underlying linear orderings of the x i s. Thus, we can identify each oracle x σ S with a unique permutation σ of {, 2,,n}, suchthatx σ i,j ifandonlyif σ(i) <σ(j). In [0], a lower bound to sorting n elements was obtained by an ingenious choice of the weight function ω. Let σ be any permutation of {, 2,,n}. For every k n and d n k, define a new permutation σ (k,d) (k, k+,,k+d) σ. Inotherwords,let(x,x 2,,x n) (σ(),σ(2),,σ(n)) be the assignment of values to x i s corresponding to σ. Then the assignment (x,x 2,,x n) corresponding to σ (k,d) is obtained from (x,x 2,,x n)by replacing the entry i with i + for k i<k+ d, andthe entry k + d with i. Thus, the following is true for τ σ (k,d) : τ (k) if i k + d σ (i) τ (i +) ifk i<k+ d τ (i) otherwise Define the weight function ω(σ, τ ) d if τ σ(k,d) for some k and d; ω(σ, τ ) 0 otherwise. With this choice, they were able to show the bounds below, for any quantum decision tree. Lemma [0] For each 0 j<t, W j W j+ 2πn!. Lemma 2[0] W 0 n!(nh n n), W T ɛ W 0, where H n /j. It follows from Lemmas and 2 that T Ω(n log n) for the sorting problem (with n unrestricted input numbers). 2.2 Review of Polytopes and Graph Entropy We first review some concepts and results from Stanley [2]. Let P (X, < P ) be a partial order on a set X {x,x 2,,x n}. A point y (y,y 2,,y n) R n is said to be consistent with P,ify i y j whenever x i P x j. For any y consistent with P, and for each i n, let d i(y) y i if x i is a minimal element in the partial order P ;otherwise,letd i(y) be the minimum of y i y j for any j satisfying x j < P x i. The order polytope O(P )isthesetofallthepointsy (y,y 2,,y n) [0, ] n consistent with P.Thechain polytope C(P )isthesetofpointsz (z,z 2,,z n) [0, ] n satisfying z i +z i2 + +z ik ifx i < P x i2 < P < P x ik is a chain in P.Defineatransfer map φ : O(P ) C(P )by the formula φ(y) (d (y),d 2(y),,d n(y)). Let (P ) be the set of all permutations of {, 2,,n}, and recall e(p ) (P ). Foreachσ (P ), let O σ(p )be the subset of points in O(P ) consistent with the permutation σ. Lemma 3 [2] (a) For any σ (P ), φ is a linear, measurepreserving bijection when its domain is restricted to O σ(p ). (b) φ is a continuous, piecewise-linear, measure-preserving bijection from O(P )ontoc(p). We now review a special entropy in connection with partial orders, which is based on the concept graph entropy first introduced by Körner [4]. Graph entropy has an extensive literature, including applications to complexity theory (e.g. Körner [5], Newman, Ragde and Wigderson [6], Radhakhrishnan [7]). We refer the readers to Csiszár et al [7], or Simonyi s survey [20] for additional information. We restrict our discussions here to those needed for this paper. For any z (z,z 2,,z n)withz i > 0, define ψ(z) log 2 n n log 2 z i. () In connection with sorting problems (in the classical setting), Kahn and Kim [] define the following entropy notion associated with P : H(P )max{ψ(z) z C(P ) }. (2) This can be described alternatively as the graph entropy of the comparability graph of P. It is known [7] that the maximum is finite and achieved at a unique point z in the polytope. Lemma 5 [] log 2 e(p ) nh(p ) O(log 2 e(p )). 3

3 3. PROOF OF THEOREM We adopt the general approach in [0] described in the previous section. Let the set of oracles S consist of those labelled by permutations in (P ). Define the weight function ω in the same way, except of course ω is defined only on S S. Consider any quantum decision tree B for QSORT P with error bounded by ɛ. Define ɛ,w j as in the previous section. Let A P σ,τ (P ) ω(σ, τ ). Lemma 6 For each 0 j<t, W j W j+ 2πe(P ). Lemma 7 W 0 W T ( ɛ )A P. The proof of Lemma 6 is exactly the same as the proof of Lemma as given in [0]. Lemma 7 follows easily from the requirement on the initial and final quantum states produced by the quantum decision tree B. Proposition There exist absolute constants λ, λ > 0 such that e(p ) AP λ log e(p ) λ n. It follows immediately from Lemmas 6, 7 and Proposition that T ( ɛ )A P 2πe(P ) ɛ 2π (λ log e(p ) λ n), which gives Theorem. We prove Proposition in two steps. Proposition 2 e(p ) AP Ω(nE z C(P )(ψ(z))). Proposition 3 There exists an absolute constant µ>0 such that E z C(P ) (ψ(z)) H(P ) µ. Clearly, Proposition follows from Lemma 5, Propositions 2 and 3. We now prove Proposition 2. Note that each permutation σ (P ) gives rise naturally to a point (σ(),σ(2),,σ(n)) in R n, and we shall use the notation d i(σ) with this understanding. Lemma 8 For any σ (P ), log 2 d i(σ) log 2 (n +)+E y Oσ(P )(log 2 d i(y)). Proof of Lemma 8 First consider the case when x i is not a minimal element under partial order P.Letjbesuch that x j < P x i, d i(σ) σ(i) σ(j). Then the expected value of d i(y) for a random y O σ(p ) is the expected difference between the σ(i)-th and the σ(j)-th smallest elements among n randomly chosen real numbers in the interval [0, ]. By standard results from order statistics, we have E y Oσ(P )(d i(y)) di(σ). (3) n + It is easy to verify that the above formula is valid in the other case (when x i is a minimal element under P ). Now using the convexity of logarithm and Equation (3), we obtain E y Oσ(P )(log 2 d i(y)) log 2 (E y Oσ(P )(d i(y)) log 2 ( n + di(σ)). This proves Lemma 8. Q.E.D. It follows from Lemma 8 that log e(p ) 2 d i(σ) σ (P ) log 2 n + E y Oσ(P )(log e(p ) 2 d i(y)) σ (P ) log 2 n + E y O(P ) (log 2 d i(y)). (4) From the definition of A P and ω, wehave A P ( d i(σ) ) σ (P ) σ (P ) Ω(log 2 d i(σ)). (5) It follows from (4) and (5) that e(p ) AP Ω(n log 2 n + E y O(P ) (log 2 d i(y))) Ω(n log 2 n + E y O(P ) ( log 2 d i(y))). By Lemma 3, φ P is a - onto measure-preserving mapping from O(P )toc(p). This leads to e(p ) AP Ω(n log 2 n + E z(z,,z n) C(P )( log 2 z i)) Ω(nE z C(P ) (ψ(z))). This proves Proposition 2. To finish the proof of Proposition, we now prove Proposition 3. First we derive two lemmas. Let Q n be the set of all y (y,,y n) R n such that y i 0and yi. Lemma 9 Let µ 00. Take a random y (y,y 2,,y n) uniformly chosen from Q n. Then, Pr{ log 2 n log y 2 n + µn} e µn/4. i Proof of Lemma 9 The proof is technical, and will be delayed to the Appendix. Let F n R n be the set of all w (w,w 2,,w n)such that w i 0 for all i and wi n. For any w (w,w 2,,w n) F n, let Y denote the function on F n defined by Y (w) n log 2 w i. Lemma 0 Let µ 200,andletD F n be a polytope of volume no less than. Then E w D(Y ) µ, for all sufficiently large n. Proof of Lemma 0 We assume that n is sufficiently large for all asymptotic inequalities (such as n 2 e n <e n/6 ) valid. Let Fn be the set of all w F n satisfying Y (w) µ/2. Let Q n be the set of all y Q n satisfying k n log 2 y k n log 2 n + µn/2. Note that F n is Q n scaled up by a factor of n on all sides, and Fn is Q n scaled up by a factor of n. The probability of a random w F n falling into Fn is exactly equal to the probability for a random y Q n falling into Q n. By Lemma 9, we conclude that Vol(Fn ) e µn/8 Vol(F n) e µn/8 n n n! e µn/9. (6) 4

4 Let J n be the set of all w (w,w 2,,w n) F n with 0 w 2 µn. With the help of equation (6) we obtain log 2 w dw log 2 w dw + log 2 w dw J n Fn J n n n (n )! 2 µn 0 log 2 w dw + Vol(Fn ) log 2 (2 µn ) e n (µn +)2 µn + e µn/9 µn e µn/0. This implies Y (w) dw Fn n log 2 w dw log 2 w i dw e µn/0. (7) Using (7) we obtain Y (w)dw Y (w)dw + D D Fn Vol(D Fn )µ/2 As Vol(D), this leads to D Vol(D)µ/2 e µn/0. E w D(Y ) D Y (w)dw Vol(D) µ 2 e µn/0 Vol(D) µ. Y (w)dw Y (w) dw This completes the proof of Lemma 0. Q.E.D. We now use Lemma 0 to prove Proposition 3. By the definition of H(P ), Proposition 3 can be written as E z C(P ) (ψ(z)) ψ(a) µ, where a is the unique point z C(P ) for ψ(z) to achieve its maximum. As seen in [7, ], C(P ) is contained in the positive quadrant (all z i 0) of R n bounded by z i a i n. Make the change of variables z i a iw i for i n. Then z w gives a one-to-one linear mapping of C(P )in the z i space onto a polytope C (P )inthew i space, with the transformation dz ( ai)dw between the infinitesimal volume elements. Note that C (P ) (which includes all points in [0, ] n ) is a polytope of volume in the positive quadrant of R n bounded by wi n, i.e., C (P ) F n. By Lemma 0, we have E w C (P )(Y ) µ. (8) Now note that ψ(z) log 2 n + log n 2 (a iw i) log 2 n + log n 2 (a i)+ log n 2 (w i) ψ(a)+y (w). Thus, E z C(P ) (ψ(z)) ψ(a)+e w C (P )(Y ). (9) It follows from (8) and (9) that E z C(P ) (ψ(z)) ψ(a) µ. This proves Proposition DISCUSSIONS We suggest several open problems for future investigations. Firstly, can one strengthen Theorem to Q ɛ(p ) Ω(log e(p ))? We conjecture that in fact the quantity A P in Lemma 7 provides such a lower bound, even though the estimation techniques used in the present paper are not strong enough to prove it. Secondly, to what extent do various classical sorting lower bounds remain valid? For example, suppose we allow any ternary polynomial tests p(x,x 2,,x n) : 0, what is the number of quantum queries needed to sort n numbers? Lastly, there are other information lower bounds whose validity in quantum case is non-obvious. For example, see Shi [8]. It would be very interesting to study systematically all the classical information lower bounds for decision trees. 5. REFERENCES [] S. Aaronson. Quantum lower bounds for the collision problem. In Proc. 34th Annual ACM Symposium on Theory of Computing, pages ACM, [2] A. Ambainis. A better lower bound for quantum algorithms searching an ordered list. In Proc. 40th Annual IEEE Symposium on Foundations of Computer Science, pages IEEE, 999. [3] A. Ambainis. Quantum lower bounds by quantum arguments. In Proc. 32nd Annual ACM Symposium on Theory of Computing, pages ACM, [4] R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf. Quantum lower bounds by polynomials. In Proc. 39th Annual IEEE Symposium on Foundations of Computer Science, pages IEEE, 998. [5] C. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. SIAM J. on Computing, 26:50 523, 997. [6] H. Buhrman and R. de Wolf. A lower bound for quantum search of an ordered list. Information Processing Letters, 70: , 999. [7] C. Csiszár, J. Körner, L. Lovász, K. Marton, and G. Simonyi. Entropy splitting for antiblocking corners and perfect graphs. Combinatorica, 0:27 40, 990. [8] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. A limit on the speed of quantum computation for insertion into an ordered list. arxiv.org e-print archive, quant-ph/982057,

5 [9] M. Fredman. How good is the information bound in sorting. Theoretical Computer Science, :355 36, 976. [0] P. Høyer, J. Neerbek, and Y. Shi. Quantum complexities of ordered searching, sorting, and element distinctness. Algorithmica, 34: , [] J. Kahn and J. Kim. Entropy and sorting. Journal of Computer and System Sciences, 5: , 995. [2] J. Kahn and N. Linial. Balancing poset extensions via Brunn-Minkowski. Combinatorica, : , 99. [3] J. Kahn and M. Saks. Balancing poset extensions. Order, :3 26, 984. [4] J. Körner. Coding of an information source having ambiguous alphabet and the entropy of graphs. In Transactions of 6th Prague Conference on Information Theory, etc., pages Academia, Prague, 973. [5] J. Körner. Fredman-Komlós bounds and information theory. SIAM Journal on Alg. Disc. Meth., 7: , 986. [6] I. Newman, P. Ragde, and A. Wigderson. Perfect hashing, graph entropy and circuit complexity. In Proc. 5th Annual IEEE Symposium on Structure in Complexity Theory, pages IEEE, 990. [7] J. Radhakhrishnan. Better bounds for threshold formulas. In Proc. 32nd Annual IEEE Symposium on Foundations of Computer Science, pages IEEE, 99. [8] Y. Shi. Entropy lower bounds of quantum decision tree complexity. Information Processing Letters, 8():23 27, [9] Y. Shi. Quantum lower bounds for the collision and the element distinctness problems. In Proc. 43rd Annual IEEE Symposium on Foundations of Computer Science, pages IEEE, [20] G. Simonyi. Perfect graphs and graph entropy: An updated survey. Chapter 3 in Perfect Graphs, edited by J. Ramirez-Alfonsin and B. Reed, pages John Wiley and Sons, 200. [2] R. Stanley. Two poset polytopes. Discrete Computational Geometry, :9 23, 986. APPENDIX Proof of Lemma 9 Pick n random real numbers x j uniformly and independently from the interval [0, ], and sort them into ascending order x i x i2 x in.lety (y,y 2,,y n)where y x i,andy j x ij x ij,2 j n. It is easily verified that the generated y is a random point uniformly distributed over Q n. Consider the random variable Y n(x,x 2,,x n) ln y j. (0) j n To prove Lemma 9, we need to show that Pr{Y n n ln n +(µ ln 2)n} e µn/4. () Let us generate x,x 2,,x n sequentially. When we have generated k random numbers x,x 2,,x k,lety k denote the random variable defined as in equation (0) above, except with n replaced by k. Let Y 0 be the constant random variable 0. Lemma A Let k n, andletx,x 2,,x k be distinct. Then Y k (x,x 2,,x k ) Y k (x,x 2,,x k )+ln(2/δ), where δ 0 is the minimum distance between x k and any numbersintheset{0,x,x 2,,x k }. ProofofLemmaA The numbers x,x 2,,x k divide the interval [0, ] into k sub-intervals. Suppose that x k falls into a sub-interval I of length u, and splits it into two parts of length λu and ( λ)u. There are two cases. If I is not the rightmost sub-interval, then Y k (x,x 2,,x k ) Y k (x,x 2,,x k ) ln(λu) ln(( λ)u)+lnu ln( 2 δ ). If I is the rightmost interval, then Y k (x,x 2,,x k ) Y k (x,x 2,,x k ) ln(λu) ln( δ ). This proves Lemma A. Q.E.D. Let us regard Y 0 0,Y,Y 2,,Y n as a sequence of random variables where Y k depends only on x,x 2,...,x k. It follows immediately from Lemma A that, Pr{Y k Y k t} 4ke t (2) for all t 0. Consider the probability distribution ρ k over [0, ]: { 0 t ln(4k) ρ k (t) 4ke t t>ln(4k). Let A k be a real-valued random variable defined on some probability space such that ρ k is the density function for the distribution of the value of A k, i.e., Pr{A k t} t ρ k (τ)dτ. (3) It is easily verified from equations (2) and (3) that Pr{Y k Y k t} Pr{A k t}. That is, Y k Y k is stochastically dominated by A k. Now note that Y n k n (Y k Y k ). This means Pr{Y n T } Pr{A T }, (4) where A k n A k. Lemma A2 Let T n ln n +(µ ln 2)n, then Pr{A T } e µn/4. Proof Let T T ln(4 n n!). Then Pr{A T } dt ρ k (t k ) ( t(t,,tn) k t k T k n (4k)) k n e s(s,,sn) s k 0, k s k T T e v v n (n )! dv. t(t,,tn) t k ln(4k), k t k T k s k ds e k t k dt 6

6 By assumption T T ln(4 n n!) ((µ ln 2) 2)n, wehave for all v T, v n (n )! ev/2. It follows that Pr{A T } T 2e T /2. e v/2 dv This implies Pr{A T } e µn/4, andproveslemmaa2.q.e.d. Equation () follows from Lemma A2 and equation (4). This completes the proof of Lemma 9. 7