56,063 research outputs found
Multiparty Quantum Coin Flipping
We investigate coin-flipping protocols for multiple parties in a quantum
broadcast setting:
(1) We propose and motivate a definition for quantum broadcast. Our model of
quantum broadcast channel is new.
(2) We discovered that quantum broadcast is essentially a combination of
pairwise quantum channels and a classical broadcast channel. This is a somewhat
surprising conclusion, but helps us in both our lower and upper bounds.
(3) We provide tight upper and lower bounds on the optimal bias epsilon of a
coin which can be flipped by k parties of which exactly g parties are honest:
for any 1 <= g <= k, epsilon = 1/2 - Theta(g/k).
Thus, as long as a constant fraction of the players are honest, they can
prevent the coin from being fixed with at least a constant probability. This
result stands in sharp contrast with the classical setting, where no
non-trivial coin-flipping is possible when g <= k/2.Comment: v2: bounds now tight via new protocol; to appear at IEEE Conference
on Computational Complexity 200
Randomized and Quantum Algorithms Yield a Speed-Up for Initial-Value Problems
Quantum algorithms and complexity have recently been studied not only for
discrete, but also for some numerical problems. Most attention has been paid so
far to the integration problem, for which a speed-up is shown by quantum
computers with respect to deterministic and randomized algorithms on a
classical computer. In this paper we deal with the randomized and quantum
complexity of initial-value problems. For this nonlinear problem, we show that
both randomized and quantum algorithms yield a speed-up over deterministic
algorithms. Upper bounds on the complexity in the randomized and quantum
settings are shown by constructing algorithms with a suitable cost, where the
construction is based on integral information. Lower bounds result from the
respective bounds for the integration problem.Comment: LaTeX v. 2.09, 13 page
Compound transfer matrices: Constructive and destructive interference
Scattering from a compound barrier, one composed of a number of distinct
non-overlapping sub-barriers, has a number of interesting and subtle
mathematical features. If one is scattering classical particles, where the wave
aspects of the particle can be ignored, the transmission probability of the
compound barrier is simply given by the product of the transmission
probabilities of the individual sub-barriers. In contrast if one is scattering
waves (whether we are dealing with either purely classical waves or quantum
Schrodinger wavefunctions) each sub-barrier contributes phase information (as
well as a transmission probability), and these phases can lead to either
constructive or destructive interference, with the transmission probability
oscillating between nontrivial upper and lower bounds. In this article we shall
study these upper and lower bounds in some detail, and also derive bounds on
the closely related process of quantum excitation (particle production) via
parametric resonance.Comment: V1: 28 pages. V2: 21 pages. Presentation significantly streamlined
and shortened. This version accepted for publication in the Journal of
Mathematical Physic
A Stronger LP Bound for Formula Size Lower Bounds via Clique Constraints
We introduce a new technique proving formula size lower bounds based on the linear programming bound originally introduced by Karchmer, Kushilevitz and Nisan (1995) and the theory of stable set polytope. We apply it to majority functions and prove their formula size lower bounds improved from the classical result of Khrapchenko (1971). Moreover, we introduce a notion of unbalanced recursive ternary majority functions motivated by a decomposition theory of monotone self-dual functions and give integrally matching upper and lower bounds of their formula size. We also show monotone formula size lower bounds of balanced recursive ternary majority functions improved from the quantum adversary bound of Laplante, Lee and Szegedy (2006)
Exponential Lower Bound for 2-Query Locally Decodable Codes via a Quantum Argument
A locally decodable code encodes n-bit strings x in m-bit codewords C(x), in
such a way that one can recover any bit x_i from a corrupted codeword by
querying only a few bits of that word. We use a quantum argument to prove that
LDCs with 2 classical queries need exponential length: m=2^{Omega(n)}.
Previously this was known only for linear codes (Goldreich et al. 02). Our
proof shows that a 2-query LDC can be decoded with only 1 quantum query, and
then proves an exponential lower bound for such 1-query locally
quantum-decodable codes. We also show that q quantum queries allow more
succinct LDCs than the best known LDCs with q classical queries. Finally, we
give new classical lower bounds and quantum upper bounds for the setting of
private information retrieval. In particular, we exhibit a quantum 2-server PIR
scheme with O(n^{3/10}) qubits of communication, improving upon the O(n^{1/3})
bits of communication of the best known classical 2-server PIR.Comment: 16 pages Latex. 2nd version: title changed, large parts rewritten,
some results added or improve
Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks
Many computational problems are subject to a quantum speed-up: one might find that a problem having an Opn3q-time or Opn2q-time classic algorithm can be solved by a known Opn1.5q-time or Opnq-time quantum algorithm. The question naturally arises: how much quantum speed-up is possible? The area of fine-grained complexity allows us to prove optimal lower-bounds on the complexity of various computational problems, based on the conjectured hardness of certain natural, well-studied problems. This theory has recently been extended to the quantum setting, in two independent papers by Buhrman, Patro and Speelman [7], and by Aaronson, Chia, Lin, Wang, and Zhang [1]. In this paper, we further extend the theory of fine-grained complexity to the quantum setting. A fundamental conjecture in the classical setting states that the 3SUM problem cannot be solved by (classical) algorithms in time Opn2´εq, for any ε ą 0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear Opn1´εq-time quantum algorithms for the 3SUM problem. Based on the Quantum-3SUM-Conjecture, we show new lower-bounds on the time complexity of quantum algorithms for several computational problems. Most of our lower-bounds are optimal, in that they match known upper-bounds, and hence they imply tight limits on the quantum speedup that is possible for these problems. These results are proven by adapting to the quantum setting known classical fine-grained reductions from the 3SUM problem. This adaptation is not trivial, however, since the original classical reductions require pre-processing the input in various ways, e.g. by sorting it according to some order, and this pre-processing (provably) cannot be done in sublinear quantum time. We overcome this bottleneck by combining a quantum walk with a classical dynamic data-structure having a certain “history-independence” property. This type of construction has been used in the past to prove upper bounds, and here we use it for the first time as part of a reduction. This general proof strategy allows us to prove tight lower bounds on several computational-geometry problems, on Convolution-3SUM and on the 0-Edge-Weight-Triangle problem, conditional on the Quantum-3SUM-Conjecture. We believe this proof strategy will be useful in proving tight (conditional) lower-bounds, and limits on quantum speed-ups, for many other problems
Dense Quantum Coding and a Lower Bound for 1-way Quantum Automata
We consider the possibility of encoding m classical bits into much fewer n
quantum bits so that an arbitrary bit from the original m bits can be recovered
with a good probability, and we show that non-trivial quantum encodings exist
that have no classical counterparts. On the other hand, we show that quantum
encodings cannot be much more succint as compared to classical encodings, and
we provide a lower bound on such quantum encodings. Finally, using this lower
bound, we prove an exponential lower bound on the size of 1-way quantum finite
automata for a family of languages accepted by linear sized deterministic
finite automata.Comment: 12 pages, 3 figures. Defines random access codes, gives upper and
lower bounds for the number of bits required for such (possibly quantum)
codes. Derives the size lower bound for quantum finite automata of the
earlier version of the paper using these result
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