1,206 research outputs found
Complexity classification of two-qubit commuting hamiltonians
We classify two-qubit commuting Hamiltonians in terms of their computational
complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can
apply to any pair of qubits, starting in a computational basis state. We prove
a dichotomy theorem: either this model is efficiently classically simulable or
it allows one to sample from probability distributions which cannot be sampled
from classically unless the polynomial hierarchy collapses. Furthermore, the
only simulable Hamiltonians are those which fail to generate entanglement. This
shows that generic two-qubit commuting Hamiltonians can be used to perform
computational tasks which are intractable for classical computers under
plausible assumptions. Our proof makes use of new postselection gadgets and Lie
theory.Comment: 34 page
Quantum Sampling Problems, BosonSampling and Quantum Supremacy
There is a large body of evidence for the potential of greater computational
power using information carriers that are quantum mechanical over those
governed by the laws of classical mechanics. But the question of the exact
nature of the power contributed by quantum mechanics remains only partially
answered. Furthermore, there exists doubt over the practicality of achieving a
large enough quantum computation that definitively demonstrates quantum
supremacy. Recently the study of computational problems that produce samples
from probability distributions has added to both our understanding of the power
of quantum algorithms and lowered the requirements for demonstration of fast
quantum algorithms. The proposed quantum sampling problems do not require a
quantum computer capable of universal operations and also permit physically
realistic errors in their operation. This is an encouraging step towards an
experimental demonstration of quantum algorithmic supremacy. In this paper, we
will review sampling problems and the arguments that have been used to deduce
when sampling problems are hard for classical computers to simulate. Two
classes of quantum sampling problems that demonstrate the supremacy of quantum
algorithms are BosonSampling and IQP Sampling. We will present the details of
these classes and recent experimental progress towards demonstrating quantum
supremacy in BosonSampling.Comment: Survey paper first submitted for publication in October 2016. 10
pages, 4 figures, 1 tabl
The Power of Quantum Fourier Sampling
A line of work initiated by Terhal and DiVincenzo and Bremner, Jozsa, and
Shepherd, shows that quantum computers can efficiently sample from probability
distributions that cannot be exactly sampled efficiently on a classical
computer, unless the PH collapses. Aaronson and Arkhipov take this further by
considering a distribution that can be sampled efficiently by linear optical
quantum computation, that under two feasible conjectures, cannot even be
approximately sampled classically within bounded total variation distance,
unless the PH collapses.
In this work we use Quantum Fourier Sampling to construct a class of
distributions that can be sampled by a quantum computer. We then argue that
these distributions cannot be approximately sampled classically, unless the PH
collapses, under variants of the Aaronson and Arkhipov conjectures.
In particular, we show a general class of quantumly sampleable distributions
each of which is based on an "Efficiently Specifiable" polynomial, for which a
classical approximate sampler implies an average-case approximation. This class
of polynomials contains the Permanent but also includes, for example, the
Hamiltonian Cycle polynomial, and many other familiar #P-hard polynomials.
Although our construction, unlike that proposed by Aaronson and Arkhipov,
likely requires a universal quantum computer, we are able to use this
additional power to weaken the conjectures needed to prove approximate sampling
hardness results
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