184 research outputs found
Computational approach to quantum encoder design for purity optimization
In this paper, we address the problem of designing a quantum encoder that
maximizes the minimum output purity of a given decohering channel, where the
minimum is taken over all possible pure inputs. This problem is cast as a
max-min optimization problem with a rank constraint on an appropriately defined
matrix variable. The problem is computationally very hard because it is
non-convex with respect to both the objective function (output purity) and the
rank constraint. Despite this difficulty, we provide a tractable computational
algorithm that produces the exact optimal solution for codespace of dimension
two. Moreover, this algorithm is easily extended to cover the general class of
codespaces, in which case the solution is suboptimal in the sense that the
suboptimized output purity serves as a lower bound of the exact optimal purity.
The algorithm consists of a sequence of semidefinite programmings and can be
performed easily. Two typical quantum error channels are investigated to
illustrate the effectiveness of our method.Comment: 13 pages, 1 figur
Universal Convexification via Risk-Aversion
We develop a framework for convexifying a fairly general class of
optimization problems. Under additional assumptions, we analyze the
suboptimality of the solution to the convexified problem relative to the
original nonconvex problem and prove additive approximation guarantees. We then
develop algorithms based on stochastic gradient methods to solve the resulting
optimization problems and show bounds on convergence rates. %We show a simple
application of this framework to supervised learning, where one can perform
integration explicitly and can use standard (non-stochastic) optimization
algorithms with better convergence guarantees. We then extend this framework to
apply to a general class of discrete-time dynamical systems. In this context,
our convexification approach falls under the well-studied paradigm of
risk-sensitive Markov Decision Processes. We derive the first known model-based
and model-free policy gradient optimization algorithms with guaranteed
convergence to the optimal solution. Finally, we present numerical results
validating our formulation in different applications
Amplitude and Sign Adjustment for Peak-to-Average-Power Reduction
In this letter, we propose a method to reduce the peak-to-mean-envelope-power ratio (PMEPR) of multicarrier signals by modifying the constellation. For-ary phase-shift keying constellations, we minimize the maximum of the multicarrier signal over the sign and amplitude of each subcarrier. In order to find an efficient solution to the aforementioned nonconvex optimization problem, we present a suboptimal solution by first optimizing over the signs, and then optimizing over the amplitudes given the signs. We prove that the minimization of the maximum of a continuous multicarrier signal over the amplitude of each subcarrier can be written as a convex optimization problem with linear matrix inequality constraints. We also generalize the idea to other constellations such as 16-quadrature amplitude modulation. Simulation results show that by an average power increase of 0.21 dB, and not sending information over the sign of each subcarrier, PMEPR can be decreased by 5.1 dB for a system with 128 subcarriers
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