4,197 research outputs found
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of -query quantum algorithms in terms of the
unit ball of a space of degree- polynomials. Based on this, we obtain a
refined notion of approximate polynomial degree that equals the quantum query
complexity, answering a question of Aaronson et al. (CCC'16). Our proof is
based on a fundamental result of Christensen and Sinclair (J. Funct. Anal.,
1987) that generalizes the well-known Stinespring representation for quantum
channels to multilinear forms. Using our characterization, we show that many
polynomials of degree four are far from those coming from two-query quantum
algorithms. We also give a simple and short proof of one of the results of
Aaronson et al. showing an equivalence between one-query quantum algorithms and
bounded quadratic polynomials.Comment: 24 pages, 3 figures. v2: 27 pages, minor changes in response to
referee comment
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC\u2716). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms.
Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms.
We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of t-query quantum algorithms in terms of the unit ball of a
space of degree-2t polynomials. Based on this, we obtain a refined notion of approximate polynomial
degree that equals the quantum query complexity, answering a question of Aaronson et
al. (CCC’16). Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct.
Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels
to multilinear forms. Using our characterization, we show that many polynomials of degree
four are far from those coming from two-query quantum algorithms. We also give a simple and
short proof of one of the results of Aaronson et al. showing an equivalence between one-query
quantum algorithms and bounded quadratic polynomials
Quantum query algorithms are completely bounded forms
We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC’16). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms. Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials
On Statistical Query Sampling and NMR Quantum Computing
We introduce a ``Statistical Query Sampling'' model, in which the goal of an
algorithm is to produce an element in a hidden set with
reasonable probability. The algorithm gains information about through
oracle calls (statistical queries), where the algorithm submits a query
function and receives an approximation to . We
show how this model is related to NMR quantum computing, in which only
statistical properties of an ensemble of quantum systems can be measured, and
in particular to the question of whether one can translate standard quantum
algorithms to the NMR setting without putting all of their classical
post-processing into the quantum system. Using Fourier analysis techniques
developed in the related context of {em statistical query learning}, we prove a
number of lower bounds (both information-theoretic and cryptographic) on the
ability of algorithms to produces an , even when the set is fairly
simple. These lower bounds point out a difficulty in efficiently applying NMR
quantum computing to algorithms such as Shor's and Simon's algorithm that
involve significant classical post-processing. We also explicitly relate the
notion of statistical query sampling to that of statistical query learning.
An extended abstract appeared in the 18th Aunnual IEEE Conference of
Computational Complexity (CCC 2003), 2003.
Keywords: statistical query, NMR quantum computing, lower boundComment: 17 pages, no figures. Appeared in 18th Aunnual IEEE Conference of
Computational Complexity (CCC 2003
Failure of the trilinear operator space Grothendieck theorem
We give a counterexample to a trilinear version of the operator space
Grothendieck theorem. In particular, we show that for trilinear forms on
, the ratio of the symmetrized completely bounded norm and the
jointly completely bounded norm is in general unbounded, answering a question
of Pisier. The proof is based on a non-commutative version of the generalized
von Neumann inequality from additive combinatorics.Comment: Reformatted for Discrete Analysi
Fixed-point quantum search with an optimal number of queries
Grover's quantum search and its generalization, quantum amplitude
amplification, provide quadratic advantage over classical algorithms for a
diverse set of tasks, but are tricky to use without knowing beforehand what
fraction of the initial state is comprised of the target states. In
contrast, fixed-point search algorithms need only a reliable lower bound on
this fraction, but, as a consequence, lose the very quadratic advantage that
makes Grover's algorithm so appealing. Here we provide the first version of
amplitude amplification that achieves fixed-point behavior without sacrificing
the quantum speedup. Our result incorporates an adjustable bound on the failure
probability, and, for a given number of oracle queries, guarantees that this
bound is satisfied over the broadest possible range of .Comment: 4 pages plus references, 2 figure
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