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

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

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