206 research outputs found
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
Analysis of circuit imperfections in BosonSampling
BosonSampling is a problem where a quantum computer offers a provable speedup
over classical computers. Its main feature is that it can be solved with
current linear optics technology, without the need for a full quantum computer.
In this work, we investigate whether an experimentally realistic BosonSampler
can really solve BosonSampling without any fault-tolerance mechanism. More
precisely, we study how the unavoidable errors linked to an imperfect
calibration of the optical elements affect the final result of the computation.
We show that the fidelity of each optical element must be at least , where refers to the number of single photons in the scheme. Such
a requirement seems to be achievable with state-of-the-art equipment.Comment: 20 pages, 7 figures, v2: new title, to appear in QI
BosonSampling with Lost Photons
BosonSampling is an intermediate model of quantum computation where
linear-optical networks are used to solve sampling problems expected to be hard
for classical computers. Since these devices are not expected to be universal
for quantum computation, it remains an open question of whether any
error-correction techniques can be applied to them, and thus it is important to
investigate how robust the model is under natural experimental imperfections,
such as losses and imperfect control of parameters. Here we investigate the
complexity of BosonSampling under photon losses---more specifically, the case
where an unknown subset of the photons are randomly lost at the sources. We
show that, if out of photons are lost, then we cannot sample
classically from a distribution that is -close (in total
variation distance) to the ideal distribution, unless a
machine can estimate the permanents of Gaussian
matrices in time. In particular, if is constant, this implies
that simulating lossy BosonSampling is hard for a classical computer, under
exactly the same complexity assumption used for the original lossless case.Comment: 12 pages. v2: extended concluding sectio
The Computational Complexity of Linear Optics
We give new evidence that quantum computers -- moreover, rudimentary quantum
computers built entirely out of linear-optical elements -- cannot be
efficiently simulated by classical computers. In particular, we define a model
of computation in which identical photons are generated, sent through a
linear-optical network, then nonadaptively measured to count the number of
photons in each mode. This model is not known or believed to be universal for
quantum computation, and indeed, we discuss the prospects for realizing the
model using current technology. On the other hand, we prove that the model is
able to solve sampling problems and search problems that are classically
intractable under plausible assumptions. Our first result says that, if there
exists a polynomial-time classical algorithm that samples from the same
probability distribution as a linear-optical network, then P^#P=BPP^NP, and
hence the polynomial hierarchy collapses to the third level. Unfortunately,
this result assumes an extremely accurate simulation. Our main result suggests
that even an approximate or noisy classical simulation would already imply a
collapse of the polynomial hierarchy. For this, we need two unproven
conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is
#P-hard to approximate the permanent of a matrix A of independent N(0,1)
Gaussian entries, with high probability over A; and the "Permanent
Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with
high probability over A. We present evidence for these conjectures, both of
which seem interesting even apart from our application. This paper does not
assume knowledge of quantum optics. Indeed, part of its goal is to develop the
beautiful theory of noninteracting bosons underlying our model, and its
connection to the permanent function, in a self-contained way accessible to
theoretical computer scientists.Comment: 94 pages, 4 figure
Generalized multi-photon quantum interference
Non-classical interference of photons lies at the heart of optical quantum
information processing. This effect is exploited in universal quantum gates as
well as in purpose-built quantum computers that solve the BosonSampling
problem. Although non-classical interference is often associated with perfectly
indistinguishable photons this only represents the degenerate case, hard to
achieve under realistic experimental conditions. Here we exploit tunable
distinguishability to reveal the full spectrum of multi-photon non-classical
interference. This we investigate in theory and experiment by controlling the
delay times of three photons injected into an integrated interferometric
network. We derive the entire coincidence landscape and identify transition
matrix immanants as ideally suited functions to describe the generalized case
of input photons with arbitrary distinguishability. We introduce a compact
description by utilizing a natural basis which decouples the input state from
the interferometric network, thereby providing a useful tool for even larger
photon numbers
The Computational Power of Non-interacting Particles
Shortened abstract: In this thesis, I study two restricted models of quantum
computing related to free identical particles.
Free fermions correspond to a set of two-qubit gates known as matchgates.
Matchgates are classically simulable when acting on nearest neighbors on a
path, but universal for quantum computing when acting on distant qubits or when
SWAP gates are available. I generalize these results in two ways. First, I show
that SWAP is only one in a large family of gates that uplift matchgates to
quantum universality. In fact, I show that the set of all matchgates plus any
nonmatchgate parity-preserving two-qubit gate is universal, and interpret this
fact in terms of local invariants of two-qubit gates. Second, I investigate the
power of matchgates in arbitrary connectivity graphs, showing they are
universal on any connected graph other than a path or a cycle, and classically
simulable on a cycle. I also prove the same dichotomy for the XY interaction.
Free bosons give rise to a model known as BosonSampling. BosonSampling
consists of (i) preparing a Fock state of n photons, (ii) interfering these
photons in an m-mode linear interferometer, and (iii) measuring the output in
the Fock basis. Sampling approximately from the resulting distribution should
be classically hard, under reasonable complexity assumptions. Here I show that
exact BosonSampling remains hard even if the linear-optical circuit has
constant depth. I also report several experiments where three-photon
interference was observed in integrated interferometers of various sizes,
providing some of the first implementations of BosonSampling in this regime.
The experiments also focus on the bosonic bunching behavior and on validation
of BosonSampling devices. This thesis contains descriptions of the numerical
analyses done on the experimental data, omitted from the corresponding
publications.Comment: PhD Thesis, defended at Universidade Federal Fluminense on March
2014. Final version, 208 pages. New results in Chapter 5 correspond to
arXiv:1106.1863, arXiv:1207.2126, and arXiv:1308.1463. New results in Chapter
6 correspond to arXiv:1212.2783, arXiv:1305.3188, arXiv:1311.1622 and
arXiv:1412.678
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