32 research outputs found
Lower Bounds for Quantum Search and Derandomization
We prove lower bounds on the error probability of a quantum algorithm for
searching through an unordered list of N items, as a function of the number T
of queries it makes. In particular, if T=O(sqrt{N}) then the error is lower
bounded by a constant. If we want error <1/2^N then we need T=Omega(N) queries.
We apply this to show that a quantum computer cannot do much better than a
classical computer when amplifying the success probability of an RP-machine. A
classical computer can achieve error <=1/2^k using k applications of the
RP-machine, a quantum computer still needs at least ck applications for this
(when treating the machine as a black-box), where c>0 is a constant independent
of k. Furthermore, we prove a lower bound of Omega(sqrt{log N}/loglog N)
queries for quantum bounded-error search of an ordered list of N items.Comment: 12 pages LaTeX. Submitted to CCC'99 (formerly Structures
Tensor Norms and the Classical Communication Complexity of Nonlocal Quantum Measurement
We initiate the study of quantifying nonlocalness of a bipartite measurement
by the minimum amount of classical communication required to simulate the
measurement. We derive general upper bounds, which are expressed in terms of
certain tensor norms of the measurement operator. As applications, we show that
(a) If the amount of communication is constant, quantum and classical
communication protocols with unlimited amount of shared entanglement or shared
randomness compute the same set of functions; (b) A local hidden variable model
needs only a constant amount of communication to create, within an arbitrarily
small statistical distance, a distribution resulted from local measurements of
an entangled quantum state, as long as the number of measurement outcomes is
constant.Comment: A preliminary version of this paper appears as part of an article in
Proceedings of the the 37th ACM Symposium on Theory of Computing (STOC 2005),
460--467, 200
Communication Complexity Lower Bounds by Polynomials
The quantum version of communication complexity allows the two communicating
parties to exchange qubits and/or to make use of prior entanglement (shared
EPR-pairs). Some lower bound techniques are available for qubit communication
complexity, but except for the inner product function, no bounds are known for
the model with unlimited prior entanglement. We show that the log-rank lower
bound extends to the strongest model (qubit communication + unlimited prior
entanglement). By relating the rank of the communication matrix to properties
of polynomials, we are able to derive some strong bounds for exact protocols.
In particular, we prove both the "log-rank conjecture" and the polynomial
equivalence of quantum and classical communication complexity for various
classes of functions. We also derive some weaker bounds for bounded-error
quantum protocols.Comment: 16 pages LaTeX, no figures. 2nd version: rewritten and some results
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