ABSTRACT
Let P = (X, < P) be a partial order on a set of n elements X = x1, x2,..., xn. Define the quantum sorting problem QSORTP as: given n distinct numbers x1, x2,..., xn consistent with P, sort them by a quantum decision tree using comparisons of the form "xi: xj". Let Qε(P) be the minimum number of queries used by any quantum decision tree for solving QSORTP with error less than ε (where 0 < ε < 1/10 is fixed). It was proved by Hoyer, Neerbek and Shi (Algorithmica 34 (2002), 429--448) that, when P0 is the empty partial order, Qε(P0) ≥ 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 Qε(P) ≥ c log2 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 Korner's graph entropy that was first noted and developed by Kahn and Kim (JCSS 51(1995), 390--399).
- S. Aaronson. Quantum lower bounds for the collision problem. In Proc. 34th Annual ACM Symposium on Theory of Computing, pages 635--642. ACM, 2002. Google ScholarDigital Library
- 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 352--357. IEEE, 1999. Google ScholarDigital Library
- A. Ambainis. Quantum lower bounds by quantum arguments. In Proc. 32nd Annual ACM Symposium on Theory of Computing, pages 636--643. ACM, 2000. Google ScholarDigital Library
- 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 352--361. IEEE, 1998. Google ScholarDigital Library
- C. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. SIAM J. on Computing, 26:1510--1523, 1997. Google ScholarDigital Library
- H. Buhrman and R. de Wolf. A lower bound for quantum search of an ordered list. Information Processing Letters, 70:205--209, 1999. Google ScholarDigital Library
- C. Csiszar, J. Korner, L. Lovasz, K. Marton, and G. Simonyi. Entropy splitting for antiblocking corners and perfect graphs. Combinatorica, 10:27--40, 1990.Google ScholarCross Ref
- 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/9812057, 1998.Google Scholar
- M. Fredman. How good is the information bound in sorting. Theoretical Computer Science, 1:355--361, 1976.Google ScholarCross Ref
- P. Hoyer, J. Neerbek, and Y. Shi. Quantum complexities of ordered searching, sorting, and element distinctness. Algorithmica, 34:429--448, 2002.Google ScholarCross Ref
- J. Kahn and J. Kim. Entropy and sorting. Journal of Computer and System Sciences, 51:390--399, 1995. Google ScholarDigital Library
- J. Kahn and N. Linial. Balancing poset extensions via Brunn-Minkowski. Combinatorica, 11:363--368, 1991.Google ScholarCross Ref
- J. Kahn and M. Saks. Balancing poset extensions. Order, 1:113--126, 1984.Google ScholarCross Ref
- J. Korner. Coding of an information source having ambiguous alphabet and the entropy of graphs. In Transactions of 6th Prague Conference on Information Theory, etc., pages 411--425. Academia, Prague, 1973.Google Scholar
- J. Korner. Fredman-Komlos bounds and information theory. SIAM Journal on Alg. Disc. Meth., 7:560--570, 1986. Google ScholarDigital Library
- 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 91--100. IEEE, 1990.Google ScholarCross Ref
- J. Radhakhrishnan. Better bounds for threshold formulas. In Proc. 32nd Annual IEEE Symposium on Foundations of Computer Science, pages 314--323. IEEE, 1991. Google ScholarDigital Library
- Y. Shi. Entropy lower bounds of quantum decision tree complexity. Information Processing Letters, 81(1):23--27, 2002.Google ScholarCross Ref
- 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 513--519. IEEE, 2002. Google ScholarDigital Library
- G. Simonyi. Perfect graphs and graph entropy: An updated survey. Chapter 13 in Perfect Graphs, edited by J. Ramirez-Alfonsin and B. Reed, pages 293--328. John Wiley and Sons, 2001.Google Scholar
- R. Stanley. Two poset polytopes. Discrete Computational Geometry, 1:9--23, 1986. Google ScholarDigital Library
Index Terms
- Graph entropy and quantum sorting problems
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