175 research outputs found
Geometric optimization problems in quantum computation and discrete mathematics: Stabilizer states and lattices
This thesis consists of two parts:
Part I deals with properties of stabilizer states and their convex
hull, the stabilizer polytope. Stabilizer states, Pauli measurements
and Clifford unitaries are the three building blocks of the stabilizer
formalism whose computational power is limited by the Gottesman-
Knill theorem. This model is usually enriched by a magic state to get
a universal model for quantum computation, referred to as quantum
computation with magic states (QCM). The first part of this thesis
will investigate the role of stabilizer states within QCM from three
different angles.
The first considered quantity is the stabilizer extent, which provides
a tool to measure the non-stabilizerness or magic of a quantum state.
It assigns a quantity to each state roughly measuring how many stabilizer
states are required to approximate the state. It has been shown
that the extent is multiplicative under taking tensor products when
the considered state is a product state whose components are composed
of maximally three qubits. In Chapter 2, we will prove that
this property does not hold in general, more precisely, that the stabilizer
extent is strictly submultiplicative. We obtain this result as
a consequence of rather general properties of stabilizer states. Informally
our result implies that one should not expect a dictionary to be
multiplicative under taking tensor products whenever the dictionary
size grows subexponentially in the dimension.
In Chapter 3, we consider QCM from a resource theoretic perspective.
The resource theory of magic is based on two types of quantum
channels, completely stabilizer preserving maps and stabilizer operations.
Both classes have the property that they cannot generate additional
magic resources. We will show that these two classes of quantum
channels do not coincide, specifically, that stabilizer operations are a
strict subset of the set of completely stabilizer preserving channels.
This might have the consequence that certain tasks which are usually
realized by stabilizer operations could in principle be performed better
by completely stabilizer preserving maps.
In Chapter 4, the last one of Part I, we consider QCM via the polar
dual stabilizer polytope (also called the Lambda-polytope). This polytope
is a superset of the quantum state space and every quantum state
can be written as a convex combination of its vertices. A way to
classically simulate quantum computing with magic states is based on
simulating Pauli measurements and Clifford unitaries on the vertices
of the Lambda-polytope. The complexity of classical simulation with respect
to the polytope is determined by classically simulating the updates
of vertices under Clifford unitaries and Pauli measurements. However,
a complete description of this polytope as a convex hull of its vertices is
only known in low dimensions (for up to two qubits or one qudit when
odd dimensional systems are considered). We make progress on this
question by characterizing a certain class of operators that live on the
boundary of the Lambda-polytope when the underlying dimension is an odd
prime. This class encompasses for instance Wigner operators, which
have been shown to be vertices of Lambda. We conjecture that this class
contains even more vertices of Lambda. Eventually, we will shortly sketch
why applying Clifford unitaries and Pauli measurements to this class
of operators can be efficiently classically simulated.
Part II of this thesis deals with lattices. Lattices are discrete subgroups
of the Euclidean space. They occur in various different areas of
mathematics, physics and computer science. We will investigate two
types of optimization problems related to lattices.
In Chapter 6 we are concerned with optimization within the space of
lattices. That is, we want to compare the Gaussian potential energy
of different lattices. To make the energy of lattices comparable we
focus on lattices with point density one. In particular, we focus on
even unimodular lattices and show that, up to dimension 24, they are
all critical for the Gaussian potential energy. Furthermore, we find
that all n-dimensional even unimodular lattices with n 24 are local
minima or saddle points. In contrast in dimension 32, there are even
unimodular lattices which are local maxima and others which are not
even critical.
In Chapter 7 we consider flat tori R^n/L, where L is an n-dimensional
lattice. A flat torus comes with a metric and our goal is to approximate
this metric with a Hilbert space metric. To achieve this, we
derive an infinite-dimensional semidefinite optimization program that
computes the least distortion embedding of the metric space R^n/L into
a Hilbert space. This program allows us to make several interesting
statements about the nature of least distortion embeddings of flat tori.
In particular, we give a simple proof for a lower bound which gives
a constant factor improvement over the previously best lower bound
on the minimal distortion of an embedding of an n-dimensional flat
torus. Furthermore, we show that there is always an optimal embedding
into a finite-dimensional Hilbert space. Finally, we construct
optimal least distortion embeddings for the standard torus R^n/Z^n and
all 2-dimensional flat tori
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning
The paper introduces the application of information geometry to describe the
ground states of Ising models by utilizing parity-check matrices of cyclic and
quasi-cyclic codes on toric and spherical topologies. The approach establishes
a connection between machine learning and error-correcting coding. This
proposed approach has implications for the development of new embedding methods
based on trapping sets. Statistical physics and number geometry applied for
optimize error-correcting codes, leading to these embedding and sparse
factorization methods. The paper establishes a direct connection between DNN
architecture and error-correcting coding by demonstrating how state-of-the-art
architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range
arena can be equivalent to of block and convolutional LDPC codes (Cage-graph,
Repeat Accumulate). QC codes correspond to certain types of chemical elements,
with the carbon element being represented by the mixed automorphism
Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and
the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix
are elaborated upon in detail. The Quantum Approximate Optimization Algorithm
(QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous
to the back-propagation loss function landscape in training DNNs. This
similarity creates a comparable problem with TS pseudo-codeword, resembling the
belief propagation method. Additionally, the layer depth in QAOA correlates to
the number of decoding belief propagation iterations in the Wiberg decoding
tree. Overall, this work has the potential to advance multiple fields, from
Information Theory, DNN architecture design (sparse and structured prior graph
topology), efficient hardware design for Quantum and Classical DPU/TPU (graph,
quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text
overlap with arXiv:2109.08184 by other author
Operational Research: methods and applications
This is the final version. Available on open access from Taylor & Francis via the DOI in this recordThroughout its history, Operational Research has evolved to include methods, models and algorithms that have been applied to a wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first summarises the up-to-date knowledge and provides an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion and used as a point of reference by a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes
On finding dense sub-lattices as low energy states of a quantum Hamiltonian
Lattice-based cryptography has emerged as one of the most prominent
candidates for post-quantum cryptography, projected to be secure against the
imminent threat of large-scale fault-tolerant quantum computers. The Shortest
Vector Problem (SVP) is to find the shortest non-zero vector in a given
lattice. It is fundamental to lattice-based cryptography and believed to be
hard even for quantum computers. We study a natural generalization of the SVP
known as the -Densest Sub-lattice Problem (-DSP): to find the densest
-dimensional sub-lattice of a given lattice. We formulate -DSP as finding
the first excited state of a Z-basis Hamiltonian, making -DSP amenable to
investigation via an array of quantum algorithms, including Grover search,
quantum Gibbs sampling, adiabatic, and Variational Quantum Algorithms. The
complexity of the algorithms depends on the basis through which the input
lattice is presented. We present a classical polynomial-time algorithm that
takes an arbitrary input basis and preprocesses it into inputs suited to
quantum algorithms. With preprocessing, we prove that qubits suffice
for solving -DSP for dimensional input lattices. We empirically
demonstrate the performance of a Quantum Approximate Optimization Algorithm
-DSP solver for low dimensions, highlighting the influence of a good
preprocessed input basis. We then discuss the hardness of -DSP in relation
to the SVP, to see if there is reason to build post-quantum cryptography on
-DSP. We devise a quantum algorithm that solves -DSP with run-time
exponent . Therefore, for fixed , -DSP is no more than
polynomially harder than the SVP
Operational research:methods and applications
Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
The Fifteenth Marcel Grossmann Meeting
The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity
Advances in Bosonic Quantum Error Correction with Gottesman-Kitaev-Preskill Codes: Theory, Engineering and Applications
Encoding quantum information into a set of harmonic oscillators is considered
a hardware efficient approach to mitigate noise for reliable quantum
information processing. Various codes have been proposed to encode a qubit into
an oscillator -- including cat codes, binomial codes and
Gottesman-Kitaev-Preskill (GKP) codes. These bosonic codes are among the first
to reach a break-even point for quantum error correction. Furthermore, GKP
states not only enable close-to-optimal quantum communication rates in bosonic
channels, but also allow for error correction of an oscillator into many
oscillators. This review focuses on the basic working mechanism, performance
characterization, and the many applications of GKP codes, with emphasis on
recent experimental progress in superconducting circuit architectures and
theoretical progress in multimode GKP qubit codes and
oscillators-to-oscillators (O2O) codes. We begin with a preliminary
continuous-variable formalism needed for bosonic codes. We then proceed to the
quantum engineering involved to physically realize GKP states. We take a deep
dive into GKP stabilization and preparation in superconducting architectures
and examine proposals for realizing GKP states in the optical domain (along
with a concise review of GKP realization in trapped-ion platforms). Finally, we
present multimode GKP qubits and GKP-O2O codes, examine code performance and
discuss applications of GKP codes in quantum information processing tasks such
as computing, communication, and sensing.Comment: 77+5 pages, 31 figures. Minor bugs fixed in v2. comments are welcome
Operational Research: Methods and Applications
Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes
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