H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms, Proc. IEEE FOCS, pp. 358--368, 1999. cs.CC/9904019.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....gaps be tween the measures, including a total f for whi C(f) is superquadratic in QC (f) and a symmetric partial f for which QC (f) O (1) yet Q2 (f) f2 (n logn) 1. Background Most of what is known about the power of quantum computing can be cast in the query or decision tree model [1, 2, 3, 5, 6, 9, 10, 11, 18, 22, 23]. Here one counts only the number of queries to the input, not the number of computational steps. The appeal of this model lies in its extreme simplicityin contrast to (say) the Turing machine model, one feels the query model ought to be completely understandable. In spite of this, open ....

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms, in Proc. IEEE FOCS'99, pp. 358-368, 1999. cs. CC/9904019.

.... gaps between the mea sures, including a total f for which (f) is superquadratic in Q (f) and a symmetric partial f for which QC (f) O (1) yet Q2 (f) f (n logn) 1 Background Most of what is known about the power of quantum computing can be cast in the query or decision tree model [1, 2, 3, 5, 6, 9, 10, 11, 17, 21, 22]. Here one counts only the number of queries to the input, not the number of computational steps. The appeal of this model lies in its extreme simplicity in contrast to (say) the Turing machine model, one feels the query model ought to be completely understandable. In spite of this, open ....

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms, in Proc. 1EEE FOCS'99, pp. 358-368, 1999. cs. CC/9904019.

....The initial state b0 is a basis state. At any point in time t, i t T, the state bt is obtained by applying Ut to bt . At time T, we measure the state bT. We define the quantum decision tree following D.Deutsch and R. Jozsa [4] and Harry Buhrman,Richard Cleve,Roland de Wolf and Christof Zalka [3]. For input length N, the initial state b0 is independent of the input X XoX. XN . We allow arbitrary unitary transformations independent of X. In addition, we allow A to make quantum queries. This is the transformation U taking the basis state li, b, z to li, b Xi, zl, where: i is a ....

Harry Buhrman, Pdchard Cleve, Ronald de Wolf, Christof Zalka. Bounds for SmallError and Zero-Error Quantum Algorithms. Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS '99), pages 358-368.

....lower bound for binary search. It may be possible to simplify other lower bounds proven previously by di erent methods. In some cases, it is quite easy to reprove the result by our method (like Theorem 5.4) but there are two cases in which we could not do that. The rst is the bound of [7] on the number of queries needed to achieve very small probability of error in database search problem. The second is the lower bound on the ordered search[1] It seems unlikely that our technique can be useful in the rst case but there is a chance that some variant of our idea may work for ....

H. Buhrman, R. Cleve, R. de Wolf, C. Zalka. Bounds for small-error and zero-error quantum algorithms. Proceedings of FOCS'99, pages 358-368. Also cs.CC/9904019.

....lower bound for binary search. It may be possible to simplify other lower bounds proven previously by different methods. In some cases, it is quite easy to reprove the result by our method (like Theorem 5) but there are two cases in which we could not do that. The first is the bound of [6] on the number of queries needed to achieve very small probability of error in database search problem. The second is the lower bound on the ordered search[1] It seems unlikely that our technique can be useful in the first case but there is a chance that some variant of our idea may work for ....

H. Buhrman, R. Cleve, R. de Wolf, C. Zalka. Bounds for small-error and zero-error quantum algorithms. Proceedings of FOCS'99, pages 358-368. Also cs.CC/9904019.

No context found.

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zeroerror quantum algorithms. In Proceedings of 40th IEEE FOCS, pages 358-368, 1999. cs.CC/9904019. 19

.... the two main examples being Shor s algorithm for factoring [Sho97] and Grover s algorithm for searching [Gro96] Whereas the first so far has remained a somewhat isolated result, the second has been applied as a building block in quite a few other quantum algorithms [BHT97, BHMT00, BH97, BCW98, BCWZ99] One of the earliest applications of Grover s algorithm was the algorithm of Brassard, Hyer, and Tapp for finding a collision in a 2 to 1 function f . A collision is a pair of distinct elements x, y such that f(x) f(y) Suppose thesizeoff s domain is N . For a classical randomized algorithm, #( ....

....a graph Finally we consider a related search problem, which is to find a triangle in a graph, provided one exists. Consider an undirected graph G = V,E)on V = n nodes with m = E edges. There are N = # n 2 # potential edges in E, which we can query in a black box fashion (see also [BCWZ99, Section 7] The triangle finding problem is: Triangle finding problem Given undirected graph G = V,E) find distinct vertices a, b, c # V such that (a, b) a, c) b, c) # E. Since there are # n 3 # n 3 triples a, b, c, and we can decide whether a given triple is a triangle ....

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms. In Proceedings of 40th FOCS, pages 358--368, 1999. cs.CC/9904019.

....two modes of computation: exact and bounded error. An intermediate type of protocols are zero error or Las Vegas protocols. These never output an incorrect answer, but may claim ignorance with probability at most 1 2. Some quantum14 classical separations for zero error protocols may be found in [18,34]. ffl One way communication. Suppose the communication is one way: Alice just sends qubits to Bob. Klauck [34] showed for all total functions that quantum communication is not significantly better than classical communication in this case. ffl Rounds. It is well known in classical communication ....

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms. In Proceedings of 40th FOCS, pages 358--368, 1999. cs.CC/9904019.

....transformations ( quantum gates ) on few qubits at a time, see [30] 2 The subscript 2 in R 2 (f) refers to the 2 sided error of the algorithm: it may err on 0 inputs as well as on 1 inputs. We will not discuss zero error (Las Vegas) or one sided error randomized decision trees here. See [38,31,22,23,20,8] for some results concerning such trees. 6 We formalize a query to an input x 2 f0; 1g n as a unitary transformation O which maps ji; b; zi to ji; b Phi x i ; zi. Here ji; b; zi is some m qubit basis state, where i takes dlog ne bits, b is one bit, z denotes the (m Gamma dlog ne Gamma ....

.... p n for all f . Combining Theorems 17 and 18 with Theorems 12 and 13 we obtain the polynomial relations between classical and quantum complexities of [3] Corollary 4 D(f) 2 O(QE (f) 4 ) and D(f) 2 O(Q 2 (f) 6 ) Some other quantum lower bounds via degree lower bounds may be found in [3,1,29,14,8]. The biggest gap that is known between D(f) and QE (f) is only a factor of 2: D(PARITY n ) n and QE (PARITY n ) dn=2e. The biggest gap we know between D(f) and Q 2 (f) is quadratic: D(OR n ) n and Q 2 (OR n ) 2 Theta( p n) by Grover s quantum search algorithm [19] Also, R 2 (OR n ) 2 ....

[Article contains additional citation context not shown here]

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms. In Proceedings of 40th FOCS, pages 358--368, 1999. cs.CC/9904019.

.... main examples being Shor s algorithm for factoring [Sho97] and Grover s algorithm for searching [Gro96] Whereas the rst so far has remained a somewhat isolated although seminal result, the second has been applied as a building block in quite a few other quantum algorithms [BH97, BHT97, BCW98, BCWZ99, NW99, BHMT00] One of the earliest applications of Grover s algorithm was the algorithm of Brassard, H yer, and Tapp [BHT97] for nding a collision in a 2 to 1 function f . A collision is a pair of distinct elements x; y such that f(x) f(y) Suppose the size of f s domain is N . For a ....

....triangle in a graph Finally we consider a related search problem, which is to nd a triangle in a graph, provided one exists. Consider an undirected graph G = V; E) on jV j = n nodes with m = jEj edges. There are N = n 2 edge slots in E, which we can query in a black box fashion (see also [BCWZ99, Section 7] The triangle nding problem is: 11 Triangle nding problem Given undirected graph G = V; E) nd distinct vertices a; b; c 2 V such that (a; b) a; c) b; c) 2 E. Since there are n 3 3 triples a; b; c, and we can decide whether a given triple is a triangle using 3 ....

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms. In Proceedings of 40th FOCS, pages 358-368, 1999. cs.CC/9904019.

.... main examples being Shor s algorithm for factoring [Sho97] and Grover s algorithm for searching [Gro96] Whereas the first so far has remained a somewhat isolated although seminal result, the second has been applied as a building block in quite a few other quantum algorithms [BH97, BHT97, BCW98, BCWZ99, NW99, BHMT00] Research partially supported by the EU fifth framework program QAIP, IST 1999 11234. CWI, P.O. Box 94079, Amsterdam, The Netherlands. email: buhrman cwi.nl. z Universit e Paris Sud, LRI, 91405 Orsay, France. email: durr lri.fr. x NSA, Suite 6111, Fort George G. Meade, MD ....

....graph Finally we consider a related search problem, which is to find a triangle in a graph, provided one exists. Consider an undirected graph G = V; E) on jV j = n nodes with m = jEj edges. There are N = Gamma n 2 Delta edge slots in E, which we can query in a black box fashion (see also [BCWZ99, Section 7] The triangle finding problem is: Triangle finding problem Given undirected graph G = V; E) find distinct vertices a; b; c 2 V such that (a; b) a; c) b; c) 2 E. Since there are Gamma n 3 Delta n 3 triples a; b; c, and we can decide whether a given triple is a ....

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms. In Proceedings of 40th FOCS, pages 358--368, 1999. cs.CC/9904019.

....measure the m qubit register jOEi, then we will see the basis state jii with probability jff i j 2 . Since P i jff i j 2 = 1, we thus have a valid probability distribution 1 We will not discuss zero error (or Las Vegas) randomized decision trees here. See [SW86, Nis91, HNW93, HW91, Haj91, BCWZ99] for some results concerning such trees. 4 over the classical m bit strings. After the measurement, jOEi has collapsed to the specific observed basis state jii and all other information in the state will be lost. Apart from measuring jOEi, we can also apply a unitary transformation to it. ....

.... 17 and 18 with Theorems 12 and 13 we obtain the polynomial relations between classical and quantum complexities of [BBC 98] Corollary 4 D(f) 2 O(QE (f) 4 ) and D(f) 2 O(Q 2 (f) 6 ) Some other quantum lower bounds via degree lower bounds may be found in [BBC 98, Amb99, NW99, FGGS99, BCWZ99] The biggest gap known between D(f) and QE (f) is only a factor of 2: D(PARITY) n and QE (PARITY) dn=2e. The biggest gap we know between D(f) and Q 2 (f) is quadratic: D(OR) n and Q 2 (OR) 2 Theta( p n) Gro96] Also, R 2 (OR) 2 Theta(n) deg(OR) n, g deg(OR) 2 Theta( p n) Open ....

[Article contains additional citation context not shown here]

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms. In Proceedings of 40th FOCS, pages 358--368, 1999. cs.CC/9904019.

....max ( ndeg(f) 2 ; ndeg(f ) 2 ) deg(f) 2 Q q (f) D q (f) O(NQ q (f)NQ q (f) O(ndeg(f)ndeg(f ) Here the part is open. This conjecture would imply D q (f) 2 O(Q 0 (f) 2 ) Q 0 (f) is zero error quantum query complexity; the quadratic relation would be close to optimal [BCWZ99] and would also imply D q (f) 2 O(deg(f) 2 ) also close to optimal [NS94] The currently best known relations have a fourth power instead of a square. Similarly 3 , for communication complexity we have (see [KN97, Section 2.11] maxfN c (f) N c (f)g D c (f) O(N c (f)N c (f) An ....

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms. In Proceedings of 40th FOCS, pages 358--368, 1999. cs.CC/9904019. 3 The similarities between the settings of query complexity and communication complexity are striking. See also [BW99a]. 9

....case of d level AND OR trees. For example, let g be a 2 level AND of ORs on n variables with fan out p n and f(x; y) g(xy) It is not hard to see that g has (2 p n Gamma1) p n monomials and hence Q (f) n=2. In contrast, the zero error quantum complexity of f is O(n 3=4 log n) [8]. 5 Bounded Error Protocols Here we generalize the above approach to bounded error quantum protocols. Define the approximate rank of f , g rank(f ) as the minimum rank among all matrices M that approximate M f entry wise up to 1=3. Let the approximate decomposition number e m(f) be the ....

H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms. In Proceedings of 40th FOCS, 1999. To appear. Also cs.CC/9904019.

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H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for small-error and zero-error quantum algorithms, Proc. IEEE FOCS, pp. 358--368, 1999. cs.CC/9904019.

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H. Buhrman, R. Cleve, R. Wolf, C. Zalka. Bounds for small-error and zero-error quantum algorithms,

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