718 research outputs found
Stabilization by Unbounded-Variation Noises
In this paper, we claim the availability of deterministic noises for
stabilization of the origins of dynamical systems, provided that the noises
have unbounded variations. To achieve the result, we first consider the system
representations based on rough path analysis; then, we provide the notion of
asymptotic stability in roughness to analyze the stability for the systems. In
the procedure, we also confirm that the system representations include
stochastic differential equations; we also found that asymptotic stability in
roughness is the same property as uniform almost sure asymptotic stability
provided by Bardi and Cesaroni. After the discussion, we confirm that there is
a case that deterministic noises are capable of making the origin become
asymptotically stable in roughness while stochastic noises do not achieve the
same stabilization results.Comment: 22 pages, 5 figure
Non-Smooth Stochastic Lyapunov Functions With Weak Extension of Viscosity Solutions
This paper proposes a notion of viscosity weak supersolutions to build a
bridge between stochastic Lyapunov stability theory and viscosity solution
theory. Different from ordinary differential equations, stochastic differential
equations can have the origins being stable despite having no smooth stochastic
Lyapunov functions (SLFs). The feature naturally requires that the related
Lyapunov equations are illustrated via viscosity solution theory, which deals
with non-smooth solutions to partial differential equations. This paper claims
that stochastic Lyapunov stability theory needs a weak extension of viscosity
supersolutions, and the proposed viscosity weak supersolutions describe
non-smooth SLFs ensuring a large class of the origins being noisily
(asymptotically) stable and (asymptotically) stable in probability. The
contribution of the non-smooth SLFs are confirmed by a few examples;
especially, they ensure that all the linear-quadratic-Gaussian (LQG) controlled
systems have the origins being noisily asymptotically stable for any additive
noises
Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy
We introduce a simple sub-universal quantum computing model, which we call
the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a
classical reversible circuit sandwiched by two layers of Hadamard gates, and
therefore it is in the second level of the Fourier hierarchy. We show that
output probability distributions of the HC1Q model cannot be classically
efficiently sampled within a multiplicative error unless the polynomial-time
hierarchy collapses to the second level. The proof technique is different from
those used for previous sub-universal models, such as IQP, Boson Sampling, and
DQC1, and therefore the technique itself might be useful for finding other
sub-universal models that are hard to classically simulate. We also study the
classical verification of quantum computing in the second level of the Fourier
hierarchy. To this end, we define a promise problem, which we call the
probability distribution distinguishability with maximum norm (PDD-Max). It is
a promise problem to decide whether output probability distributions of two
quantum circuits are far apart or close. We show that PDD-Max is BQP-complete,
but if the two circuits are restricted to some types in the second level of the
Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a
Merlin-Arthur system with quantum polynomial-time Merlin and classical
probabilistic polynomial-time Arthur.Comment: 30 pages, 4 figure
Completeness of Tableau Calculi for Two-Dimensional Hybrid Logics
Hybrid logic is one of the extensions of modal logic. The many-dimensional
product of hybrid logic is called hybrid product logic (HPL). We construct a
sound and complete tableau calculus for two-dimensional HPL. Also, we made a
tableau calculus for hybrid dependent product logic (HdPL), where one dimension
depends on the other. In addition, we add a special rule to the tableau
calculus for HdPL and show that it is still sound and complete. All of them
lack termination, however.Comment: Version 2. 27 pages. 5 figures. This is a preprin
Stochastic Asymptotic Stabilizers for Deterministic Input-Affine Systems based on Stochastic Control Lyapunov Functions
In this paper, a stochastic asymptotic stabilization method is proposed for
deterministic input-affine control systems, which are randomized by including
Gaussian white noises in control inputs. The sufficient condition is derived
for the diffucion coefficients so that there exist stochastic control Lyapunov
functions for the systems. To illustrate the usefulness of the sufficient
condition, the authors propose the stochastic continuous feedback law, which
makes the origin of the Brockett integrator become globally asymptotically
stable in probability.Comment: A preliminary version of this paper appeared in the Proceedings of
the 48th Annual IEEE Conference on Decision and Control [14
Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk
Quantum walks, whose dynamics is prescribed by alternating unitary coin and
shift operators, possess topological phases akin to those of Floquet
topological insulators, driven by a time-periodic field. While there is ample
theoretical work on topological phases of quantum walks where the coin
operators are spin rotations, in experiments a different coin, the Hadamard
operator is often used instead. This was the case in a recent photonic quantum
walk experiment, where protected edge states were observed between two bulks
whose topological invariants, as calculated by the standard theory, were the
same. This hints at a hidden topological invariant in the Hadamard quantum
walk. We establish a relation between the Hadamard and the spin rotation
operator, which allows us to apply the recently developed theory of topological
phases of quantum walks to the one-dimensional Hadamard quantum walk. The
topological invariants we derive account for the edge state observed in the
experiment, we thus reveal the hidden topological invariant of the
one-dimensional Hadamard quantum walk.Comment: 11 pages, 4 figure
Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups
In this paper we consider what can be computed by a user interacting with a potentially malicious server, when the server performs polynomial-time quantum computation but the user can only perform polynomial-time classical (i.e., non-quantum) computation. Understanding the computational power of this model, which corresponds to polynomial-time quantum computation that can be efficiently verified classically, is a well-known open problem in quantum computing. Our result shows that computing the order of a solvable group, which is one of the most general problems for which quantum computing exhibits an exponential speed-up with respect to classical computing, can be realized in this model
Monotonicity for Multiobjective Accelerated Proximal Gradient Methods
Accelerated proximal gradient methods, which are also called fast iterative
shrinkage-thresholding algorithms (FISTA) are known to be efficient for many
applications. Recently, Tanabe et al. proposed an extension of FISTA for
multiobjective optimization problems. However, similarly to the
single-objective minimization case, the objective functions values may increase
in some iterations, and inexact computations of subproblems can also lead to
divergence. Motivated by this, here we propose a variant of the FISTA for
multiobjective optimization, that imposes some monotonicity of the objective
functions values. In the single-objective case, we retrieve the so-called
MFISTA, proposed by Beck and Teboulle. We also prove that our method has global
convergence with rate , where is the number of iterations, and
show some numerical advantages in requiring monotonicity.Comment: - Added new numerical experiment
- …