190 research outputs found
Formal Verification of Full-Wave Rectifier: A Case Study
We present a case study of formal verification of full-wave rectifier for
analog and mixed signal designs. We have used the Checkmate tool from CMU [1],
which is a public domain formal verification tool for hybrid systems. Due to
the restriction imposed by Checkmate it necessitates to make the changes in the
Checkmate implementation to implement the complex and non-linear system.
Full-wave rectifier has been implemented by using the Checkmate custom blocks
and the Simulink blocks from MATLAB from Math works. After establishing the
required changes in the Checkmate implementation we are able to efficiently
verify the safety properties of the full-wave rectifier.Comment: The IEEE 8th International Conference on ASIC (IEEE ASICON 2009),
October 20-23 2009, Changsha, Chin
An Operational Definition of Topological Order
The unrivaled robustness of topologically ordered states of matter against
perturbations has immediate applications in quantum computing and quantum
metrology, yet their very existence poses a challenge to our understanding of
phase transitions. In particular, topological phase transitions cannot be
characterized in terms of local order parameters, as it is the case with
conventional symmetry-breaking phase transitions. Currently, topological order
is mostly discussed in the context of nonlocal topological invariants or
indirect signatures like the topological entanglement entropy. However, a
comprehensive understanding of what actually constitutes topological order is
still lacking. Here we show that one can interpret topological order as the
ability of a system to perform topological error correction. We find that this
operational approach corresponding to a measurable observable does not only lay
the conceptual foundations for previous classifications of topological order,
but it can also be applied to hitherto inaccessible problems, such as the
question of topological order for mixed quantum states arising in open quantum
systems. We demonstrate the existence of topological order in open systems and
their phase transitions to topologically trivial states, including topological
criticality. Our results demonstrate the viability of topological order in
nonequilibrium quantum systems and thus substantially broaden the scope of
possible technological applications. We therefore expect our work to be a
starting point for many future theoretical and experimental investigations,
such as the application of our approach to fracton or Floquet topological
order, or the direct experimental realization of the error correction protocol
presented in our work for the development of future quantum technological
devices.Comment: 7 pages, 5 figure
Novel approaches to topological order involving open boundaries in closed and open quantum systems
The fundamental understanding of phases and their transitions has been a central theme in condensed matter
physics. Until recently, it was largely believed that the Landau symmetry breaking principle was effective in
distinguishing different phases of matter, with broken symmetries signaling the phase transition. But with the
discovery of topological phases which are beyond the Landau symmetry breaking principle, the identification and
classification of quantum phases at absolute zero has opened up new unexplored avenues thus leading to
exciting theoretical discoveries further propelling technological advancement. Topological phases of matter are
characterized by the notion of topological order and in this work we aim to explore and understand topological
phases by introducing novel signatures which characterize topological order. The robustness of these phases
to external perturbation makes them an ideal candidate to store and manipulate quantum information thus
making them an unique and interesting prospect for realizing quantum computers.
There have been several signatures to characterize intrinsic topological order, for instance the invariance of the
topologically ordered state under local operators, the dependence of ground state degeneracy on the underlying
manifold and its robustness to external perturbation, topological entanglement entropy related to the quantum
dimension of the supers-selection sectors, the inability to construct the topologically ordered state from a product
state via constant depth unitary transformations. With toric code as the toy model, we analyze the robustness of
topological order on a manifold supporting open boundaries by computing some of the above signatures which
effectively detect a topological to trivial phase transition. We then probe the existence of a quantum criticality
between distinct topological phases obtained by varying the underlying manifold. In these scenarios, most of the
above signatures turn out to be ineffective in detecting the distinct phases leading to the introduction of an non-
local order parameter whose construction is facilitated by the phenomenon of anyon condensation.
The signatures for quantitatively and qualitatively characterizing intrinsic topological order being highly
scenario dependent and also with its definition for mixed states being elusive we introduce an operational
definition based on concepts of topological error correction. We define a state to be topologically ordered
if the errors in the state can be corrected by an error correction circuit of finite depth. To concretize
the notion of topological to trivial phase transition in an open setting we turn to nonequilibrium
phenomenon, for example: Directed Percolation, with the change in percolation rate driving a dynamical phase
transition between absorbing and active states with the former being topologically ordered while the latter
being topologically trivial. Additionally, we explore the notion of topological phase transitions between
distinct topological phases obtained by varying underlying topology in an open setting, analogous to the
closed setting discussed earlier. To summarize, we have introduced various mixed states which exhibit
topological order and also an operational definition to quantify topological order applicable across
multitude of scenarios.
We extend the above operational definition to quantify and detect quantum phase transitions in the case of
Symmetry Protected Topological (SPT) phases. To further validate the above notion, we consider the perturbed
variants of the Su-Schrieffer–Heeger (SSH) models and detect quantum phase transitions to a high
accuracy by employing the techniques from the framework of tensor networks. It is significant to note the
distinction of the error correction algorithms applied earlier in the case of intrinsic topological order
were independent of symmetry constraints while in the current scenario we impose additional symmetry
constraints to accurately detect the phase transition. In addition, we also devise error correction
strategies with respect to topologically trivial states to detect quantum phase transitions which do
not involve topological phases. This gives rise to a very fundamental question on whether error correction
statistics with a well defined error correction algorithm, not necessarily optimal, are capable of detecting
a equivalence classes of phases and thereby acting as a reliable probe to effectively detect topological/quantum
phase transitions?
From theoretical and numerical end of the spectrum we shift gears to explore possible experimental platforms
with an aim to realize some of the quantum many-body phenomenon discussed earlier. While there have been
several innovative experimental avenues to realize the above, one such promising candidate has been ultracold
polar molecules setups that offer additional degrees of freedom due to the ro-vibrational degrees of freedom.
Based on the chemical reaction between atoms and molecules which results in a quantum Zeno-based
blockade, we devise several optimal strategies to efficiently detect molecules using atom as probe, we further
extend the above technique to entangle the internal states of molecules and atoms. In addition, we also present
optimal strategies for dissipative state engineering using the atom-molecule interactions
Topological phase transitions induced by varying topology and boundaries in the toric code
One of the important characteristics of topological phases of matter is the topology of the underlying manifold on which they are defined. In this paper, we present the sensitivity of such phases of matter to the underlying topology, by studying the phase transitions induced due to the change in the boundary conditions. We claim that these phase transitions are accompanied by broken symmetries in the excitation space and to gain further insight we analyze various signatures like the ground state degeneracy, topological entanglement entropy while introducing the open-loop operator whose expectation value effectively captures the phase transition. Further, we extend the analysis to an open quantum setup by defining effective collapse operators, the dynamics of which cool the system to distinct steady states both of which are topologically ordered. We show that the phase transition between such steady states is effectively captured by the expectation value of the open-loop operator
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