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Unbounded Error Quantum Query Complexity
, 2007
"... This work studies the quantum query complexity of Boolean functions in a scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded error quantum query complexity is exa ..."
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This work studies the quantum query complexity of Boolean functions in a scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, we show that the “blackbox ” approach to convert quantum query algorithms into communication protocols by BuhrmanCleveWigderson [STOC’98] is optimal even in the unbounded error setting. We also study a setting related to the unbounded error model, called the weakly unbounded error setting, where the cost of a query algorithm is given by q + log(1/2(p − 1/2)), where q is the number of queries made and p> 1/2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight Θ(log n) separation between quantum and classical query complexity in the weakly unbounded error setting for a partial Boolean function. We also show the asymptotic equivalence between them for some wellstudied total Boolean functions. 1
Unboundederror classical and quantum communication complexity
 Proc. 18th ISAAC, Lecture Notes in Comput. Sci. 4835
, 2007
"... Abstract. Since the seminal work of Paturi and Simon [25, FOCS’84 & JCSS’86], the unboundederror classical communication complexity of a Boolean function has been studied based on the arrangement of points and hyperplanes. Recently, [13, ICALP’07] found that the unboundederror quantum communica ..."
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Abstract. Since the seminal work of Paturi and Simon [25, FOCS’84 & JCSS’86], the unboundederror classical communication complexity of a Boolean function has been studied based on the arrangement of points and hyperplanes. Recently, [13, ICALP’07] found that the unboundederror quantum communication complexity in the oneway communication model can also be investigated using the arrangement, and showed that it is exactly (without a difference of even one qubit) half of the classical oneway communication complexity. In this paper, we extend the arrangement argument to the twoway and simultaneous message passing (SMP) models. As a result, we show similarly tight bounds of the unboundederror twoway/oneway/SMP quantum/classical communication complexities, implying that all of them are equivalent up to a multiplicative constant of four. Moreover, the arrangement argument is used to show that the gap between weakly unboundederror quantum and classical communication complexities is at most a factor of three. 1
The oneway communication complexity of group membership
, 2009
"... This paper studies the oneway communication complexity of the subgroup membership problem, a classical problem closely related to basic questions in quantum computing. Here Alice receives, as input, a subgroup H of a finite group G; Bob receives an element x ∈ G. Alice is permitted to send a single ..."
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This paper studies the oneway communication complexity of the subgroup membership problem, a classical problem closely related to basic questions in quantum computing. Here Alice receives, as input, a subgroup H of a finite group G; Bob receives an element x ∈ G. Alice is permitted to send a single message to Bob, after which he must decide if his input x is an element of H. We prove the following upper bounds on the classical communication complexity of this problem in the boundederror setting: 1. The problem can be solved with O(logG) communication, provided the subgroup H is normal. 2. The problem can be solved with O(dmax · logG) communication, where dmax is the maximum of the dimensions of the irreducible complex representations of G. 3. For any prime p not dividing G, the problem can be solved with O(dmax · log p) communication, where dmax is the maximum of the dimensions of the irreducible Fprepresentations of G.
The oneway communication complexity . . .
, 2009
"... This paper studies the oneway communication complexity of the subgroup membership problem, a classical problem closely related to basic questions in quantum computing. Here Alice receives, as input, a subgroup H of a finite group G; Bob receives an element x ∈ G. Alice is permitted to send a single ..."
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This paper studies the oneway communication complexity of the subgroup membership problem, a classical problem closely related to basic questions in quantum computing. Here Alice receives, as input, a subgroup H of a finite group G; Bob receives an element x ∈ G. Alice is permitted to send a single message to Bob, after which he must decide if his input x is an element of H. We prove the following upper bounds on the classical communication complexity of this problem in the boundederror setting: 1. The problem can be solved with O(logG) communication, provided the subgroup H is normal. 2. The problem can be solved with O(dmax · logG) communication, where dmax is the maximum of the dimensions of the irreducible complex representations of G. 3. For any prime p not dividing G, the problem can be solved with O(dmax · log p) communication, where dmax is the maximum of the dimensions of the irreducible Fprepresentations of G.