22 research outputs found
Phase Unwrapping and One-Dimensional Sign Problems
Sign problems in path integrals arise when different field configurations
contribute with different signs or phases. Phase unwrapping describes a family
of signal processing techniques in which phase differences between elements of
a time series are integrated to construct non-compact unwrapped phase
differences. By combining phase unwrapping with a cumulant expansion, path
integrals with sign problems arising from phase fluctuations can be
systematically approximated as linear combinations of path integrals without
sign problems. This work explores phase unwrapping in zero-plus-one-dimensional
complex scalar field theory. Results with improved signal-to-noise ratios for
the spectrum of scalar field theory can be obtained from unwrapped phases, but
the size of cumulant expansion truncation errors is found to be undesirably
sensitive to the parameters of the phase unwrapping algorithm employed. It is
argued that this numerical sensitivity arises from discretization artifacts
that become large when phases fluctuate close to singularities of a complex
logarithm in the definition of the unwrapped phase.Comment: 42 pages, 16 figures. Journal versio
Unwrapping phase fluctuations in one dimension
Correlation functions in one-dimensional complex scalar field theory provide
a toy model for phase fluctuations, sign problems, and signal-to-noise problems
in lattice field theory. Phase unwrapping techniques from signal processing are
applied to lattice field theory in order to map compact random phases to
noncompact random variables that can be numerically sampled without sign or
signal-to-noise problems. A cumulant expansion can be used to reconstruct
average correlation functions from moments of unwrapped phases, but points
where the field magnitude fluctuates close to zero lead to ambiguities in the
definition of the unwrapped phase and significant noise at higher orders in the
cumulant expansion. Phase unwrapping algorithms that average fluctuations over
physical length scales improve, but do not completely resolve, these issues in
one dimension. Similar issues are seen in other applications of phase
unwrapping, where they are found to be more tractable in higher dimensions.Comment: 14 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1806.0183
Algorithms, Bounds, and Strategies for Entangled XOR Games
We study the complexity of computing the commuting-operator value
of entangled XOR games with any number of players. We introduce necessary and
sufficient criteria for an XOR game to have , and use these
criteria to derive the following results:
1. An algorithm for symmetric games that decides in polynomial time whether
or , a task that was not previously known to be
decidable, together with a simple tensor-product strategy that achieves value 1
in the former case. The only previous candidate algorithm for this problem was
the Navascu\'{e}s-Pironio-Ac\'{i}n (also known as noncommutative Sum of Squares
or ncSoS) hierarchy, but no convergence bounds were known.
2. A family of games with three players and with , where it
takes doubly exponential time for the ncSoS algorithm to witness this (in
contrast with our algorithm which runs in polynomial time).
3. A family of games achieving a bias difference
arbitrarily close to the maximum possible value of (and as a consequence,
achieving an unbounded bias ratio), answering an open question of Bri\"{e}t and
Vidick.
4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games:
that is, we show that there exists a constant depending
only on the number of players, such that a random -XOR game over an
alphabet of size has with high probability when the number
of clauses is above .
5. A lower bound of on the number of levels
in the ncSoS hierarchy required to detect unsatisfiability for most random
3-XOR games. This is in contrast with the classical case where the -th level
of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all
possible solutions.Comment: 55 page
Path integral contour deformations for noisy observables
Monte Carlo studies of many quantum systems face exponentially severe
signal-to-noise problems. We show that noise arising from complex phase
fluctuations of observables can be reduced without introducing bias using path
integral contour deformation techniques. A numerical study of contour
deformations for correlation functions in Abelian gauge theory and complex
scalar field theory demonstrates that variance can be reduced by orders of
magnitude without modifying Monte Carlo sampling.Comment: 7 pages, 2 figure
Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics
Sampling from known probability distributions is a ubiquitous task in
computational science, underlying calculations in domains from linguistics to
biology and physics. Generative machine-learning (ML) models have emerged as a
promising tool in this space, building on the success of this approach in
applications such as image, text, and audio generation. Often, however,
generative tasks in scientific domains have unique structures and features --
such as complex symmetries and the requirement of exactness guarantees -- that
present both challenges and opportunities for ML. This Perspective outlines the
advances in ML-based sampling motivated by lattice quantum field theory, in
particular for the theory of quantum chromodynamics. Enabling calculations of
the structure and interactions of matter from our most fundamental
understanding of particle physics, lattice quantum chromodynamics is one of the
main consumers of open-science supercomputing worldwide. The design of ML
algorithms for this application faces profound challenges, including the
necessity of scaling custom ML architectures to the largest supercomputers, but
also promises immense benefits, and is spurring a wave of development in
ML-based sampling more broadly. In lattice field theory, if this approach can
realize its early promise it will be a transformative step towards
first-principles physics calculations in particle, nuclear and condensed matter
physics that are intractable with traditional approaches.Comment: 11 pages, 5 figure
Signal-to-noise improvement through neural network contour deformations for 3D lattice gauge theory
Complex contour deformations of the path integral have been demonstrated to
significantly improve the signal-to-noise ratio of observables in previous
studies of two-dimensional gauge theories with open boundary conditions. In
this work, new developments based on gauge fixing and a neural network
definition of the deformation are introduced, which enable an effective
application to theories in higher dimensions and with generic boundary
conditions. Improvements of the signal-to-noise ratio by up to three orders of
magnitude for Wilson loop measurements are shown in lattice gauge
theory in three spacetime dimensions.Comment: 9 pages, 3 figures. Proceedings for the 40th Lattice conference at
Fermilab from July 31 to August 4, 202