15 research outputs found
Adversary lower bounds for nonadaptive quantum algorithms
International audienceWe present general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Our results are based on the adversary method of Ambainis
The Power of Adaptivity in Quantum Query Algorithms
Motivated by limitations on the depth of near-term quantum devices, we study
the depth-computation trade-off in the query model, where the depth corresponds
to the number of adaptive query rounds and the computation per layer
corresponds to the number of parallel queries per round. We achieve the
strongest known separation between quantum algorithms with versus
rounds of adaptivity. We do so by using the -fold Forrelation problem
introduced by Aaronson and Ambainis (SICOMP'18). For , this problem can
be solved using an round quantum algorithm with only one query per round,
yet we show that any round quantum algorithm needs an exponential (in the
number of qubits) number of parallel queries per round.
Our results are proven following the Fourier analytic machinery developed in
recent works on quantum-classical separations. The key new component in our
result are bounds on the Fourier weights of quantum query algorithms with
bounded number of rounds of adaptivity. These may be of independent interest as
they distinguish the polynomials that arise from such algorithms from arbitrary
bounded polynomials of the same degree.Comment: 35 pages, 9 figure
An Algorithmic Argument for Nonadaptive Query Complexity Lower Bounds on Advised Quantum Computation
This paper employs a powerful argument, called an algorithmic argument, to
prove lower bounds of the quantum query complexity of a multiple-block ordered
search problem in which, given a block number i, we are to find a location of a
target keyword in an ordered list of the i-th block. Apart from much studied
polynomial and adversary methods for quantum query complexity lower bounds, our
argument shows that the multiple-block ordered search needs a large number of
nonadaptive oracle queries on a black-box model of quantum computation that is
also supplemented with advice. Our argument is also applied to the notions of
computational complexity theory: quantum truth-table reducibility and quantum
truth-table autoreducibility.Comment: 16 pages. An extended abstract will appear in the Proceedings of the
29th International Symposium on Mathematical Foundations of Computer Science,
Lecture Notes in Computer Science, Springer-Verlag, Prague, August 22-27,
200
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
Randomized vs. Deterministic Separation in Time-Space Tradeoffs of Multi-Output Functions
We prove the first polynomial separation between randomized and deterministic
time-space tradeoffs of multi-output functions. In particular, we present a
total function that on the input of elements in , outputs
elements, such that: (1) There exists a randomized oblivious algorithm with
space , time and one-way access to randomness, that
computes the function with probability ; (2) Any deterministic
oblivious branching program with space and time that computes the
function must satisfy . This implies that
logspace randomized algorithms for multi-output functions cannot be black-box
derandomized without an overhead in time.
Since previously all the polynomial time-space tradeoffs of multi-output
functions are proved via the Borodin-Cook method, which is a probabilistic
method that inherently gives the same lower bound for randomized and
deterministic branching programs, our lower bound proof is intrinsically
different from previous works. We also examine other natural candidates for
proving such separations, and show that any polynomial separation for these
problems would resolve the long-standing open problem of proving
time lower bound for decision problems with
space.Comment: 15 page
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