13 research outputs found
Critical Transitions In a Model of a Genetic Regulatory System
We consider a model for substrate-depletion oscillations in genetic systems,
based on a stochastic differential equation with a slowly evolving external
signal. We show the existence of critical transitions in the system. We apply
two methods to numerically test the synthetic time series generated by the
system for early indicators of critical transitions: a detrended fluctuation
analysis method, and a novel method based on topological data analysis
(persistence diagrams).Comment: 19 pages, 8 figure
Non-global parameter estimation using local ensemble Kalman filtering
We study parameter estimation for non-global parameters in a low-dimensional
chaotic model using the local ensemble transform Kalman filter (LETKF). By
modifying existing techniques for using observational data to estimate global
parameters, we present a methodology whereby spatially-varying parameters can
be estimated using observations only within a localized region of space. Taking
a low-dimensional nonlinear chaotic conceptual model for atmospheric dynamics
as our numerical testbed, we show that this parameter estimation methodology
accurately estimates parameters which vary in both space and time, as well as
parameters representing physics absent from the model
Zeno-effect Computation: Opportunities and Challenges
Adiabatic quantum computing has demonstrated how quantum Zeno can be used to
construct quantum optimisers. However, much less work has been done to
understand how more general Zeno effects could be used in a similar setting. We
use a construction based on three state systems rather than directly in qubits,
so that a qubit can remain after projecting out one of the states. We find that
our model of computing is able to recover the dynamics of a transverse field
Ising model, several generalisations are possible, but our methods allow for
constraints to be implemented non-perturbatively and does not need tunable
couplers, unlike simple transverse field implementations. We further discuss
how to implement the protocol physically using methods building on STIRAP
protocols for state transfer. We find a substantial challenge, that settings
defined exclusively by measurement or dissipative Zeno effects do not allow for
frustration, and in these settings pathological spectral features arise leading
to unfavorable runtime scaling. We discuss methods to overcome this challenge
for example including gain as well as loss as is often done in optical Ising
machines
Using machine learning to predict catastrophes in dynamical systems
Nonlinear dynamical systems, which include models of the Earth\u27s climate, financial markets and complex ecosystems, often undergo abrupt transitions that lead to radically different behavior. The ability to predict such qualitative and potentially disruptive changes is an important problem with far-reaching implications. Even with robust mathematical models, predicting such critical transitions prior to their occurrence is extremely difficult. In this work, we propose a machine learning method to study the parameter space of a complex system, where the dynamics is coarsely characterized using topological invariants. We show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner, we are able to predict with high accuracy the locations of critical transitions in parameter space. (C) 2011 Elsevier B.V. All rights reserved
Quantum Natural Gradient with Efficient Backtracking Line Search
We consider the Quantum Natural Gradient Descent (QNGD) scheme which was
recently proposed to train variational quantum algorithms. QNGD is Steepest
Gradient Descent (SGD) operating on the complex projective space equipped with
the Fubini-Study metric. Here we present an adaptive implementation of QNGD
based on Armijo's rule, which is an efficient backtracking line search that
enjoys a proven convergence. The proposed algorithm is tested using noisy
simulators on three different models with various initializations. Our results
show that Adaptive QNGD dynamically adapts the step size and consistently
outperforms the original QNGD, which requires knowledge of optimal step size to
{perform competitively}. In addition, we show that the additional complexity
involved in performing the line search in Adaptive QNGD is minimal, ensuring
the gains provided by the proposed adaptive strategy dominates any increase in
complexity. Additionally, our benchmarking demonstrates that a simple SGD
algorithm (implemented in the Euclidean space) equipped with the adaptive
scheme above, can yield performances similar to the QNGD scheme with optimal
step size.
Our results are yet another confirmation of the importance of differential
geometry in variational quantum computations. As a matter of fact, we foresee
advanced mathematics to play a prominent role in the NISQ era in guiding the
design of faster and more efficient algorithms.Comment: 14 page
Understanding domain-wall encoding theoretically and experimentally
We analyze the method of encoding pairwise interactions of higher-than-binary
discrete variables (these models are sometimes referred to as discrete
quadratic models) into binary variables based on domain walls on one
dimensional Ising chains. We discuss how this is relevant to quantum annealing,
but also many gate model algorithms such as VQE and QAOA. We theoretically show
that for problems of practical interest for quantum computing and assuming only
quadratic interactions are available between the binary variables, it is not
possible to have a more efficient general encoding in terms of number of binary
variables per discrete variable. We furthermore use a D-Wave Advantage 1.1 flux
qubit quantum annealing computer to show that the dynamics effectively freeze
later for a domain-wall encoding compared to a traditional one-hot encoding.
This second result could help explain the dramatic performance improvement of
domain wall over one hot which has been seen in a recent experiment on D-Wave
hardware. This is an important result because usually problem encoding and the
underlying physics are considered separately, our work suggests that
considering them together may be a more useful paradigm. We argue that this
experimental result is also likely to carry over to a number of other settings,
we discuss how this has implications for gate-model and quantum-inspired
algorithms.Comment: 15 pages, 16 figures, typo in metadata fixed in v2, referee requested
changes in v3, accepted in Royal Society Philosophical Transactions A,
current version matches author accepted manuscrip
Grover Speedup from Many Forms of the Zeno Effect
It has previously been established that adiabatic quantum computation,
operating based on a continuous Zeno effect due to dynamical phases between
eigenstates, is able to realise an optimal Grover-like quantum speedup. In
other words is able to solve an unstructured search problem with the same
scaling as Grover's original algorithm. A natural question is
whether other manifestations of the Zeno effect can also support an optimal
speedup in a physically realistic model (through direct analog application
rather than indirectly by supporting a universal gateset). In this paper we
show that they can support such a speedup, whether due to measurement,
decoherence, or even decay of the excited state into a computationally useless
state. Our results also suggest a wide variety of methods to realise speedup
which do not rely on Zeno behaviour. We group these algorithms into three
families to facilitate a structured understanding of how speedups can be
obtained: one based on phase kicks, containing adiabatic computation and
continuous-time quantum walks; one based on dephasing and measurement; and
finally one based on destruction of the amplitude within the excited state, for
which we are not aware of any previous results. These results suggest that
there may be exciting opportunities for new paradigms of analog quantum
computing based on these effects