3,587 research outputs found
08381 Abstracts Collection -- Computational Complexity of Discrete Problems
From the 14th of September to the 19th of September, the Dagstuhl Seminar
08381 ``Computational Complexity of Discrete Problems\u27\u27 was held in Schloss Dagstuhl - Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work as well as open problems were discussed.
Abstracts of the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this report. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Dagstuhl News January - December 2011
"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic
The Computational Lens: from Quantum Physics to Neuroscience
Two transformative waves of computing have redefined the way we approach
science. The first wave came with the birth of the digital computer, which
enabled scientists to numerically simulate their models and analyze massive
datasets. This technological breakthrough led to the emergence of many
sub-disciplines bearing the prefix "computational" in their names. Currently,
we are in the midst of the second wave, marked by the remarkable advancements
in artificial intelligence. From predicting protein structures to classifying
galaxies, the scope of its applications is vast, and there can only be more
awaiting us on the horizon.
While these two waves influence scientific methodology at the instrumental
level, in this dissertation, I will present the computational lens in science,
aiming at the conceptual level. Specifically, the central thesis posits that
computation serves as a convenient and mechanistic language for understanding
and analyzing information processing systems, offering the advantages of
composability and modularity.
This dissertation begins with an illustration of the blueprint of the
computational lens, supported by a review of relevant previous work.
Subsequently, I will present my own works in quantum physics and neuroscience
as concrete examples. In the concluding chapter, I will contemplate the
potential of applying the computational lens across various scientific fields,
in a way that can provide significant domain insights, and discuss potential
future directions.Comment: PhD thesis, Harvard University, Cambridge, Massachusetts, USA. 2023.
Some chapters report joint wor
Benchmarking treewidth as a practical component of tensor-network--based quantum simulation
Tensor networks are powerful factorization techniques which reduce resource
requirements for numerically simulating principal quantum many-body systems and
algorithms. The computational complexity of a tensor network simulation depends
on the tensor ranks and the order in which they are contracted. Unfortunately,
computing optimal contraction sequences (orderings) in general is known to be a
computationally difficult (NP-complete) task. In 2005, Markov and Shi showed
that optimal contraction sequences correspond to optimal (minimum width) tree
decompositions of a tensor network's line graph, relating the contraction
sequence problem to a rich literature in structural graph theory. While
treewidth-based methods have largely been ignored in favor of dataset-specific
algorithms in the prior tensor networks literature, we demonstrate their
practical relevance for problems arising from two distinct methods used in
quantum simulation: multi-scale entanglement renormalization ansatz (MERA)
datasets and quantum circuits generated by the quantum approximate optimization
algorithm (QAOA). We exhibit multiple regimes where treewidth-based algorithms
outperform domain-specific algorithms, while demonstrating that the optimal
choice of algorithm has a complex dependence on the network density, expected
contraction complexity, and user run time requirements. We further provide an
open source software framework designed with an emphasis on accessibility and
extendability, enabling replicable experimental evaluations and future
exploration of competing methods by practitioners.Comment: Open source code availabl
Modular quantum signal processing in many variables
Despite significant advances in quantum algorithms, quantum programs in
practice are often expressed at the circuit level, forgoing helpful structural
abstractions common to their classical counterparts. Consequently, as many
quantum algorithms have been unified with the advent of quantum signal
processing (QSP) and quantum singular value transformation (QSVT), an
opportunity has appeared to cast these algorithms as modules that can be
combined to constitute complex programs. Complicating this, however, is that
while QSP/QSVT are often described by the polynomial transforms they apply to
the singular values of large linear operators, and the algebraic manipulation
of polynomials is simple, the QSP/QSVT protocols realizing analogous
manipulations of their embedded polynomials are non-obvious. Here we provide a
theory of modular multi-input-output QSP-based superoperators, the basic unit
of which we call a gadget, and show they can be snapped together with LEGO-like
ease at the level of the functions they apply. To demonstrate this ease, we
also provide a Python package for assembling gadgets and compiling them to
circuits. Viewed alternately, gadgets both enable the efficient block encoding
of large families of useful multivariable functions, and substantiate a
functional-programming approach to quantum algorithm design in recasting QSP
and QSVT as monadic types.Comment: 15 pages + 9 figures + 4 tables + 45 pages supplement. For codebase,
see https://github.com/ichuang/pyqsp/tree/bet
A Full Characterization of Quantum Advice
We prove the following surprising result: given any quantum state rho on n
qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of
two-qubit interactions), such that any ground state of H can be used to
simulate rho on all quantum circuits of fixed polynomial size. In terms of
complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which
supersedes the previous result of Aaronson that BQP/qpoly is contained in
PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in
power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools --
including a result of Alon et al. on learning of real-valued concept classes, a
result of Aaronson on the learnability of quantum states, and a result of
Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new
ones. The main new tool is a so-called majority-certificates lemma, which is
closely related to boosting in machine learning, and which seems likely to find
independent applications. In its simplest version, this lemma says the
following. Given any set S of Boolean functions on n variables, any function f
in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm
in S, such that each fi is the unique function in S compatible with O(log|S|)
input/output constraints.Comment: We fixed two significant issues: 1. The definition of YQP machines
needed to be changed to preserve our results. The revised definition is more
natural and has the same intuitive interpretation. 2. We needed properties of
Local Hamiltonian reductions going beyond those proved in previous works
(whose results we'd misstated). We now prove the needed properties. See p. 6
for more on both point
Quantum computing for finance
Quantum computers are expected to surpass the computational capabilities of
classical computers and have a transformative impact on numerous industry
sectors. We present a comprehensive summary of the state of the art of quantum
computing for financial applications, with particular emphasis on stochastic
modeling, optimization, and machine learning. This Review is aimed at
physicists, so it outlines the classical techniques used by the financial
industry and discusses the potential advantages and limitations of quantum
techniques. Finally, we look at the challenges that physicists could help
tackle
- …