18,921 research outputs found
Exponential Separation of Quantum and Classical Non-Interactive Multi-Party Communication Complexity
We give the first exponential separation between quantum and classical
multi-party communication complexity in the (non-interactive) one-way and
simultaneous message passing settings.
For every k, we demonstrate a relational communication problem between k
parties that can be solved exactly by a quantum simultaneous message passing
protocol of cost O(log n) and requires protocols of cost n^{c/k^2}, where c>0
is a constant, in the classical non-interactive one-way message passing model
with shared randomness and bounded error.
Thus our separation of corresponding communication classes is superpolynomial
as long as k=o(\sqrt{\log n / \log\log n}) and exponential for k=O(1)
Exponential Separation of Quantum and Classical One-Way Communication Complexity for a Boolean Function
We give an exponential separation between one-way quantum and classical
communication complexity for a Boolean function. Earlier such a separation was
known only for a relation. A very similar result was obtained earlier but
independently by Kerenidis and Raz [KR06]. Our version of the result gives an
example in the bounded storage model of cryptography, where the key is secure
if the adversary has a certain amount of classical storage, but is completely
insecure if he has a similar amount of quantum storage.Comment: 8 pages, no figure
Quantum Versus Randomized Communication Complexity, with Efficient Players
We study a new type of separations between quantum and classical communication complexity, separations that are obtained using quantum protocols where all parties are efficient, in the sense that they can be implemented by small quantum circuits, with oracle access to their inputs. Our main result qualitatively matches the strongest known separation between quantum and classical communication complexity [Dmitry Gavinsky, 2016] and is obtained using a quantum protocol where all parties are efficient. More precisely, we give an explicit partial Boolean function f over inputs of length N, such that:
(1) f can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where at the beginning of the protocol Alice and Bob also have polylog(N) entangled EPR pairs).
(2) Any classical randomized protocol for f, with any number of rounds, has communication complexity at least ??(N^{1/4}).
(3) All parties in the quantum protocol of Item (1) (Alice, Bob and the referee) can be implemented by quantum circuits of size polylog(N) (where Alice and Bob have oracle access to their inputs).
Items (1), (2) qualitatively match the strongest known separation between quantum and classical communication complexity, proved by Gavinsky [Dmitry Gavinsky, 2016]. Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of ?(N^{1/2}) in the classical case, the referee in his quantum protocol is inefficient).
Exponential separations of quantum and classical communication complexity have been studied in numerous previous works, but to the best of our knowledge the efficiency of the parties in the quantum protocol has not been addressed, and in most previous separations the quantum parties seem to be inefficient. The only separations that we know of that have efficient quantum parties are the recent separations that are based on lifting [Arkadev Chattopadhyay et al., 2019; Arkadev Chattopadhyay et al., 2019]. However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of two-way quantum and classical communication complexity but they do not give the stronger separations of simultaneous-message quantum communication complexity vs. two-way classical communication complexity (or even one-way quantum communication complexity vs. two-way classical communication complexity).
Our proof technique is completely new, in the context of communication complexity, and is based on techniques from [Ran Raz and Avishay Tal, 2019]. Our function f is based on a lift of the forrelation problem, using xor as a gadget
Exponential Separation of Quantum and Classical Online Space Complexity
Although quantum algorithms realizing an exponential time speed-up over the
best known classical algorithms exist, no quantum algorithm is known performing
computation using less space resources than classical algorithms. In this
paper, we study, for the first time explicitly, space-bounded quantum
algorithms for computational problems where the input is given not as a whole,
but bit by bit. We show that there exist such problems that a quantum computer
can solve using exponentially less work space than a classical computer. More
precisely, we introduce a very natural and simple model of a space-bounded
quantum online machine and prove an exponential separation of classical and
quantum online space complexity, in the bounded-error setting and for a total
language. The language we consider is inspired by a communication problem (the
set intersection function) that Buhrman, Cleve and Wigderson used to show an
almost quadratic separation of quantum and classical bounded-error
communication complexity. We prove that, in the framework of online space
complexity, the separation becomes exponential.Comment: 13 pages. v3: minor change
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