875 research outputs found
Quantum communication complexity of linear regression
Dequantized algorithms show that quantum computers do not have exponential
speedups for many linear algebra problems in terms of time and query
complexity. In this work, we show that quantum computers can have exponential
speedups in terms of communication complexity for some fundamental linear
algebra problems. We mainly focus on solving linear regression and Hamiltonian
simulation. In the quantum case, the task is to prepare the quantum state of
the result. To allow for a fair comparison, in the classical case the task is
to sample from the result. We investigate these two problems in two-party and
multiparty models, propose near-optimal quantum protocols and prove
quantum/classical lower bounds. In this process, we propose an efficient
quantum protocol for quantum singular value transformation, which is a powerful
technique for designing quantum algorithms. As a result, for many linear
algebra problems where quantum computers lose exponential speedups in terms of
time and query complexity, it is possible to have exponential speedups in terms
of communication complexity.Comment: 28 page
The Structure of Promises in Quantum Speedups
It has long been known that in the usual black-box model, one cannot get
super-polynomial quantum speedups without some promise on the inputs. In this
paper, we examine certain types of symmetric promises, and show that they also
cannot give rise to super-polynomial quantum speedups. We conclude that
exponential quantum speedups only occur given "structured" promises on the
input.
Specifically, we show that there is a polynomial relationship of degree
between and for any function defined on permutations
(elements of in which each alphabet element occurs
exactly once). We generalize this result to all functions defined on orbits
of the symmetric group action (which acts on an element of by permuting its entries). We also show that when is constant, any
function defined on a "symmetric set" - one invariant under -
satisfies .Comment: 15 page
The need for structure in quantum speedups
Is there a general theorem that tells us when we can hope for exponential speedups from quantum algorithms, and when we cannot? In this paper, we make two advances toward such a theorem, in the black-box model where most quantum algorithms operate.
First, we show that for any problem that is invariant under permuting inputs and outputs (like the collision or the element distinctness problems), the quantum query complexity is at least the 9th root of the classical randomized query complexity. This resolves a conjecture of Watrous from 2002.
Second, inspired by recent work of O'Donnell et al. and Dinur et al., we conjecture that every bounded low-degree polynomial has a "highly influential" variable. Assuming this conjecture, we show that every T-query quantum algorithm can be simulated on most inputs by a poly(T)-query classical algorithm, and that one essentially cannot hope to prove P!=BQP relative to a random oracle
Quantum machine learning: a classical perspective
Recently, increased computational power and data availability, as well as
algorithmic advances, have led machine learning techniques to impressive
results in regression, classification, data-generation and reinforcement
learning tasks. Despite these successes, the proximity to the physical limits
of chip fabrication alongside the increasing size of datasets are motivating a
growing number of researchers to explore the possibility of harnessing the
power of quantum computation to speed-up classical machine learning algorithms.
Here we review the literature in quantum machine learning and discuss
perspectives for a mixed readership of classical machine learning and quantum
computation experts. Particular emphasis will be placed on clarifying the
limitations of quantum algorithms, how they compare with their best classical
counterparts and why quantum resources are expected to provide advantages for
learning problems. Learning in the presence of noise and certain
computationally hard problems in machine learning are identified as promising
directions for the field. Practical questions, like how to upload classical
data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde
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