865 research outputs found
Upper and lower bounds for first order expressibility
AbstractWe study first order expressibility as a measure of complexity. We introduce the new class Var&Sz[v(n),z(n)] of languages expressible by a uniform sequence of sentences with v(n) variables and size O[z(n)]. When v(n) is constant our uniformity condition is syntactical and thus the following characterizations of P and PSPACE come entirely from logic. NSPACE|log n|⊆⋃k=1,2,…Var&Sz|k, log(n)|⊆DSPACE|log2(n)|,P=⋃k=1,2,…Var&Sz|k, nk|,PSPACE=⋃k=1,2,…Var&Sz|k, 2nk|. The above means, for example, that the properties expressible with constantly many variables in polynomial size sentences are just the polynomial time recognizable properties. These results hold for languages with an ordering relation, e.g., for graphs the vertices are numbered. We introduce an “alternating pebbling game” to prove lower bounds on the number of variables and size needed to express properties without the ordering. We show, for example, that k variables are needed to express Clique(k), suggesting that this problem requires DTIME[nk]
Lower bound of the expressibility of ansatzes for Variational Quantum Algorithms
The expressibility of an ansatz used in a variational quantum algorithm is
defined as the uniformity with which it can explore the space of unitary
matrices. The expressibility of a particular ansatz has a well-defined upper
bound. In this work, we show that the expressibiliity also has a well-defined
lower bound in the hypothesis space. We provide an analytical expression for
the lower bound of the covering number, which is directly related to
expressibility. We also perform numerical simulations to to support our claim.
To numerically calculate the bond length of a diatomic molecule, we take
hydrogen () as a prototype system and calculate the error in the energy
for the equilibrium energy point for different ansatzes. We study the variation
of energy error with circuit depths and show that in each ansatz template, a
plateau exists for a range of circuit depths, which we call the set of
acceptable points, and the corresponding expressibility is known as the best
expressive region. We report that the width of this best expressive region in
the hypothesis space is inversely proportional to the average error. Our
analysis reveals that alongside trainability, the lower bound of expressibility
also plays a crucial role in selecting variational quantum ansatzes
Explicit linear kernels via dynamic programming
Several algorithmic meta-theorems on kernelization have appeared in the last
years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of
bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding
a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed
topological minor. Typically, these results guarantee the existence of linear
or polynomial kernels on sparse graph classes for problems satisfying some
generic conditions but, mainly due to their generality, it is not clear how to
derive from them constructive kernels with explicit constants. In this paper we
make a step toward a fully constructive meta-kernelization theory on sparse
graphs. Our approach is based on a more explicit protrusion replacement
machinery that, instead of expressibility in CMSO logic, uses dynamic
programming, which allows us to find an explicit upper bound on the size of the
derived kernels. We demonstrate the usefulness of our techniques by providing
the first explicit linear kernels for -Dominating Set and -Scattered Set
on apex-minor-free graphs, and for Planar-\mathcal{F}-Deletion on graphs
excluding a fixed (topological) minor in the case where all the graphs in
\mathcal{F} are connected.Comment: 32 page
Quantum Deep Dreaming: A Novel Approach for Quantum Circuit Design
One of the challenges currently facing the quantum computing community is the
design of quantum circuits which can efficiently run on near-term quantum
computers, known as the quantum compiling problem. Algorithms such as the
Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization
Algorithm (QAOA), and Quantum Architecture Search (QAS) have been shown to
generate or find optimal near-term quantum circuits. However, these methods are
computationally expensive and yield little insight into the circuit design
process. In this paper, we propose Quantum Deep Dreaming (QDD), an algorithm
that generates optimal quantum circuit architectures for specified objectives,
such as ground state preparation, while providing insight into the circuit
design process. In QDD, we first train a neural network to predict some
property of a quantum circuit (such as VQE energy). Then, we employ the Deep
Dreaming technique on the trained network to iteratively update an initial
circuit to achieve a target property value (such as ground state VQE energy).
Importantly, this iterative updating allows us to analyze the intermediate
circuits of the dreaming process and gain insights into the circuit features
that the network is modifying during dreaming. We demonstrate that QDD
successfully generates, or 'dreams', circuits of six qubits close to ground
state energy (Transverse Field Ising Model VQE energy) and that dreaming
analysis yields circuit design insights. QDD is designed to optimize circuits
with any target property and can be applied to circuit design problems both
within and outside of quantum chemistry. Hence, QDD lays the foundation for the
future discovery of optimized quantum circuits and for increased
interpretability of automated quantum algorithm design.Comment: Undergraduate Thesis. Defended 19th April 2022, McMaster University.
Supervised by Dr. Alan Aspuru-Guzik. 40 Pages, 9 Figures, 1 Appendi
Tarski's influence on computer science
The influence of Alfred Tarski on computer science was indirect but
significant in a number of directions and was in certain respects fundamental.
Here surveyed is the work of Tarski on the decision procedure for algebra and
geometry, the method of elimination of quantifiers, the semantics of formal
languages, modeltheoretic preservation theorems, and algebraic logic; various
connections of each with computer science are taken up
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