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75
Improved quantum communication complexity bounds for disjointness and equality
- In Proc. Intl. Symp. on Theoretical Aspects of Computer Science (STACS
, 2002
"... Abstract. We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and non-deterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound being n/2. We show this by improving a general bo ..."
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Cited by 30 (5 self)
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Abstract. We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and non-deterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound being n/2. We show this by improving a general bound for non-deterministic protocols of de Wolf. We also give an O ( √ n·c log ∗ n)-qubit bounded-error protocol for disjointness, modifying and improving the earlier O ( √ n log n) protocol of Buhrman, Cleve, and Wigderson, and prove an Ω ( √ n) lower bound for a class of protocols that includes the BCW-protocol as well as our new protocol. 1
Quantum Search Algorithms
, 2005
"... We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1 ..."
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Cited by 27 (1 self)
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We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1
On the power of Ambainis’s lower bounds
- Theoretical Computer Science, 339(2-3):241– 256, 2005. Earlier version in ICALP’04. 569 Copyright © by SIAM. Unauthorized
"... The polynomial method and Ambainis’s lower bound method are two main quantum lower bound techniques. Recently Ambainis showed that the polynomial method is not tight. The present paper aims at studying the limitation of Ambainis’s lower bounds. We first give a generalization of the three known Ambai ..."
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Cited by 27 (0 self)
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The polynomial method and Ambainis’s lower bound method are two main quantum lower bound techniques. Recently Ambainis showed that the polynomial method is not tight. The present paper aims at studying the limitation of Ambainis’s lower bounds. We first give a generalization of the three known Ambainis’s lower bound theorems. Then it is shown that all these four Ambainis’s lower bounds have an upper bound, which is in terms of certificate complexity. This implies that for some problems such as TRIANGLE, k-CLIQUE, and BIPARTITE/GRAPH MATCHING whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis’s techniques. Another consequence is that all the Ambainis’s lower bounds are not tight. Finally, we show that for total functions, this upper bound for Ambainis’s lower bounds can be further improved. This also implies limitation of Ambainis’s method on some specific problems such as AND-OR TREE, whose precise quantum complexity is still unknown. 1
Quantum information processing in continuous time
, 2004
"... Quantum mechanical computers can solve certain problems asymptotically faster than any classical computing device. Several fast quantum algorithms are known, but the nature of quantum speedup is not well understood, and inventing new quantum algorithms seems to be difficult. In this thesis, we explo ..."
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Cited by 20 (6 self)
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Quantum mechanical computers can solve certain problems asymptotically faster than any classical computing device. Several fast quantum algorithms are known, but the nature of quantum speedup is not well understood, and inventing new quantum algorithms seems to be difficult. In this thesis, we explore two approaches to designing quantum algorithms based on continuous-time Hamiltonian dynamics. In quantum computation by adiabatic evolution, the computer is prepared in the known ground state of a simple Hamiltonian, which is slowly modified so that its ground state encodes the solution to a problem. We argue that this approach should be inherently robust against low-temperature thermal noise and certain control errors, and we support this claim using simulations. We then show that any adiabatic algorithm can be implemented in a different way, using only a sequence of measurements of the Hamiltonian. We illustrate how this approach can achieve quadratic speedup for the unstructured search problem. We also demonstrate two examples of quantum speedup by quantum walk, a quantum mechanical analog of random walk. First, we consider the problem of searching a region
On the power of Ambainis lower bounds
, 2005
"... The polynomial method and the Ambainis lower bound (or Alb, for short) method are two main quantum lower bound techniques. While recently Ambainis showed that the polynomial method is not tight, the present paper aims at studying the power and limitation of Alb’s. We first use known Alb’s to derive( ..."
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Cited by 17 (4 self)
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The polynomial method and the Ambainis lower bound (or Alb, for short) method are two main quantum lower bound techniques. While recently Ambainis showed that the polynomial method is not tight, the present paper aims at studying the power and limitation of Alb’s. We first use known Alb’s to derive(n1.5) lower bounds for BIPARTITENESS, BIPARTITENESS MATCHING and GRAPH MATCHING, in which the lower bound for BIPARTITENESS improves the previous(n) one.We then show that all the three known Ambainis lower bounds have a limitation N min{C0(f), C1(f)}, where C0(f) and C1(f) are the 0- and 1-certificate complexities, respectively. This implies that for many problems such asTRIANGLE, k-CLIQUE, BIPARTITENESS andBIPARTITE/GRAPHMATCHING which draw wide interest and whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis techniques. Another consequence is that all theAmbainis lower bounds are not tight. For total functions, this upper bound for Alb’s can be further improved to min{√C0(f)C1(f), N · CI(f)}, where CI(f) is the size of max intersection of a 0- and a 1-certificate set. Again this implies that Alb’s cannot improve the best known lower bound for some specific problems such as AND-OR TREE, whose precise quantum query complexity is still open. Finally, we generalize the three known Alb’s and give a new Alb style lower bound method, which may be easier to use for some problems.
Quantum identification of Boolean oracles.
- In Proceedings of the 21st Annual Symposium on Theoretical Aspects of Computer Science (STACS 2004),
, 2004
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Average-case quantum query complexity
- Journal of Physics A: Mathematical and General
"... We compare classical and quantum query complexities of total Boolean functions. It has been shown that for worst-case complexity, the gap between quantum and classical can be at most polynomial [BBC + 98]. We give (non-uniform) distributions where the gap for average-case complexity of the OR-functi ..."
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Cited by 11 (4 self)
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We compare classical and quantum query complexities of total Boolean functions. It has been shown that for worst-case complexity, the gap between quantum and classical can be at most polynomial [BBC + 98]. We give (non-uniform) distributions where the gap for average-case complexity of the OR-function can be exponential or even larger. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than the classical complexity. 1
QUANTUM SEARCH WITH VARIABLE TIMES
, 2008
"... Since Grover’s seminal work, quantum search has been studied in great detail. In the usual search problem, we have a collection of n items x1,..., xn and we would like to find i: xi = 1. We consider a new variant of this problem in which evaluating xi for different i may take a different number of ..."
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Cited by 10 (2 self)
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Since Grover’s seminal work, quantum search has been studied in great detail. In the usual search problem, we have a collection of n items x1,..., xn and we would like to find i: xi = 1. We consider a new variant of this problem in which evaluating xi for different i may take a different number of time steps. Let ti be the number of time steps required to evaluate xi. If the numbers ti are known in advance, we give an algorithm that solves the problem in O ( p t 2 1 + t2 2 +... + t2 n) steps. This is optimal, as we also show a matching lower bound. The case, when ti are not known in advance, can be solved with a polylogarithmic overhead. We also give an application of our new search algorithm to computing read-once functions.
