Research Projects

Chief Investigators: 

• Dr Ria Rushin Joseph

• Prof Anil Prabhakar

Chief Investigators: 

• Dr Ria Rushin Joseph

• Prof Jinho Choi

Chief Investigators: 

• Prof Seng Loke 

• Prof Jinho Choi

• Dr Jihong Park

• Dr Sutharshan Rajasekarar

• Dr Ria Rushin Joseph

•  Dr Chathu Ranaweera

Chief Investigators: 

• Dr Ria Rushin Joseph (Deakin SIT), ria.joseph@deakin.edu.au

• Laura Rosales Zarate (Centro de Investigaciones en Optica, CIO Leon Mexico

•  Distinguished Professor Peter D Drummond (Swinburne University)

Project Description:

For short times, Heisenberg equations should give the same result as the Fokker-Planck equation. Here I am going to check the Majorana Hubbard model with the Heisenberg equations. The idea is that if you have a Majorana operator like Xij, then its expectation value evolves in time. At short times, this can be worked out either with the Heisenberg equations or with the Q-function. By assuming an initial Gaussian state, you can use Gaussian factorizations. They ought to agree to O(dt). This can be worked out either with the Heisenberg equations or with the Q-function. The expectation values ought to agree. 

Keywords: 

Heisenberg equations, Gaussian factorisations

Project Status: Draft

References:

• Joseph, R. R., Drummond, P. D., & Rosales-Zárate, L. E. (2021). Interacting-fermion dynamics in Majorana phase space. Physical Review A, 104(6), 062208.

• Ria Rushin Joseph, Laura E. C. Rosales-Zárate, and Peter D. Drummond, “Phase space methods for Majorana fermions,” J. Phys. A 51, 245302 (2018).

Chief Investigators: 

• Dr Ria Rushin Joseph (Deakin SIT), ria.joseph@deakin.edu.au

• Jesse van Rhijn (University of Twente)

• Distinguished Professor Peter D Drummond (Swinburne University)

Project Description:

A path integral or stochastic bridge is the probability of a random path between two states. For state-dependent diffusion, such path integrals correspond to a stochastic process in a variable metric Riemannian manifold. We show that this is most readily analysed through embedding the path as a projection from a higher dimensional Euclidean space. This resolves long-standing ambiguities in previous definitions of curved space path integrals. By the Nash embedding theorem, projections are always possible for closed spaces, and also exist for many open Riemann spaces. We show that the resulting embedded stochastic path integrals are identical to the steady-state of a stochastic partial differential equation with an additional time dimension. This leads to efficient methods for probabilistic sampling of path integrals in many curved spaces, including arbitrary closed surfaces. Suitable projection algorithms are derived and tested, for both the stochastic ordinary and partial differential equation cases. We demonstrate the applicability of these methods to positive and negative curvature Riemann spaces, including hyperspheres and variable curvature ellipsoidal and hyberboloidal examples. This approach to path integration has applications ranging from quantum phase-space and operator representation theory to many other stochastic problems including nanotechnology, cell biology, economics and control theory.

Keywords: 

curved space diffusion, path integral, projection algorithm

Project Status: On-going

References:

• Rushin Joseph, R., van Rhijn, J., & Drummond, P. D. (2021). A midpoint projection algorithm for stochastic differential equations on manifolds. arXiv e-prints, arXiv-2112.

(Accepted at PRE: https://journals.aps.org/pre/accepted/44071Ra8I171202600778bb236f1532d936c0de81)

•  P. D. Drummond, Physical Review E 96, 042123 (2017).

• R. Graham, Physical Review Letters 38, 51 (1977).

Chief Investigators: 

• Dr Ria Rushin Joseph (Deakin SIT), ria.joseph@deakin.edu.au

• Jesse van Rhijn (University of Twente)

• Distinguished Professor Peter D Drummond (Swinburne University)

Project Description:

We derive an algorithm for solving partial stochastic differential equations (SPDEs), by using variable changes and projections. The algorithm given here is particularly useful in cases where the final SPDE has a quadratic derivative term. These arise in a number of applications. In some cases the physical partial stochastic differential equation has these terms initially. In other cases, the equation is used to solve a path integral or a stochastic bridge. In these cases such terms occur from using the SPDE as an algorithmic procedure to sample the path integral. The difficulty with such nonlinear differential terms is that they are not very well-defined, and standard procedures can result in ambiguous results. The technique given here solves the SPDE in a transformed or higher-dimensional space, which is then transformed or projected to the target space, thus defining the nonlinear stochastic derivative terms unambiguously.

Keywords: 

Stochastic Partial differential equations, Fokker Planck equations, Cole-Hopf

Project Status: On-going

References:

• Rushin Joseph, R., van Rhijn, J., & Drummond, P. D. (2021). A midpoint projection algorithm for stochastic differential equations on manifolds. arXiv e-prints, arXiv-2112.

(Accepted at PRE: https://journals.aps.org/pre/accepted/44071Ra8I171202600778bb236f1532d936c0de81)

• S. Carter, P. Drummond, M. Reid, R. M. Shelby, Squeezing of quantum solitons, Physical Review Letters 58 (18) (1987) 1841.

•  P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.

• C. W. Gardiner, Handbook of Stochastic Methods, 2nd Edition, Springer-Verlag, Berlin, 1985.

•  P. D. Drummond, M. Hillery, The Quantum Theory of Nonlinear Optics, Cambridge University Press, 2014.

Chief Investigators: 

• Hon. Assoc. Prof. Jacob L. Cybulski (Deakin SIT), jacob.cybulski@deakin.edu.au

• Assist. Prof. Sebastian Zając (PKO Bank / SGH Warsaw School of Economics)

Project Description:

Time series anomalies have been investigated in a variety of different contexts, e.g. in assessing price volatility, when appraising financial risk, in cardiac patient monitoring, machine condition analysis, while conducting video surveillance or traffic monitoring, in observations of seismic activity or astronomical events. All such real-life examples involve streaming data, where the series are infinite and ongoing, data features are evolving over time, anomalies are infrequent, and yet they potentially lead to catastrophic outcomes.

Anomaly detection in streaming data is challenging, especially when new data are voluminous, multivariate, with complex time-state relationships, and generated in real-time, where anomaly detection must happen at the earliest possible time, their impact is to be assessed accurately, business decisions need to be made quickly and the ensuing actions taken instantly.

Traditional statistical and machine learning methods of detection and prediction of anomalies in data streams rely either on the prior analysis of the off-loaded volumes of streaming data or are limited to the real-time analysis of small segments of data immediately preceding the present event. They therefore lack in speed and completeness of data analysis.

This project therefore explores the use of novel quantum computing methods of early anomaly detection in streaming data. Quantum methods, whether pure or hybrid, commonly exceed the performance and effectiveness of classical approaches when dealing with highly complex problems. While targeting the future generation of quantum devices, featuring 1000s of high quality qubits, quantum methods will allow effective real-time analysis of all the past and current states of complex multivariate data streams. The project also aims to determine suitability of different quantum methods for the analysis of distinct types of anomalies in various circumstances.

Keywords: 

Quantum Machine Learning, Anomaly Detection, Data Streaming, Time Series Analysis.

Project Status: Exploratory Investigation, Planning and Scoping (on-going)

References:

• Foorthuis, R. On the Nature and Types of Anomalies: A Review of Deviations in Data. International Journal of Data Science and Analytics, 12(4):297–331, Oct. 2021.

• Shaukat, K., Alam, T. M., Luo, S., Shabbir, S., Hameed, I. A., Li, J., Abbas, S. K., and Javed, U. A review of time-series anomaly detection techniques: A step to future perspectives. In Future of Information and Communication Conference, pages 865–877. Springer, 2021.

• Schuld, Maria, and Francesco Petruccione. Machine Learning with Quantum Computers. 2nd ed. 2021 edition. Springer, 2021.

Chief Investigator: Hon. Assoc. Prof. Jacob L. Cybulski, jacob.cybulski@deakin.edu.au

Student Researcher: Thanh (Tim) Nguyen (Hons Student, SIT)

Project Description:

Quantum neural networks (QNN) are machine learning models which aim to utilise quantum computing algorithms to improve training regimes for classical artificial neural networks. One of the approaches to implementing QNNs is to employ variational quantum algorithms (VQA), which take advantage of optimisation algorithms executed on classical hardware to train parametrised quantum circuits, while using quantum machines to model the landscape of the loss function and efficiently estimate its gradient. As compared with classical neural networks, using VQA allows training of a well-designed QNN to rapidly converge to a solution, while avoiding many problems commonly present in training classical neural networks. 

QNNs however have their own trainability issues, which can manifest in large variational QNN circuits, either due to their depth, the large number of qubits, or poor initialisation of circuit parameters. One such problem is the emergence of large flat areas in the loss function landscape, called barren plateaus, which impede effective circuit optimisation. The methods of preventing the formation of barren plateaus or mitigating their presence in circuit optimisation have been proposed and their variants are vigorously pursued. However, efficacy of each method in the context of the QNN architecture and training data remains an open question. 

This project therefore aims to investigate the effectiveness of different approaches to dealing with barren plateaus in various QNN developmental circumstances. The project is undertaken in two phases. In the first phase, three different approaches are selected to deal with barren plateaus and are then evaluated against different VQA quantum circuit structures (depth, qubits and initialisation) and a random gradient landscape. In the second phase, performance of the selected methods is analysed and compared when used with several QNNs built using standard data sets.

Keywords: 

Quantum Machine Learning, Variational Quantum Algorithms, Quantum Neural Networks, Barren Plateaus, Python Programming, Qiskit.

Project Status: Completed (Feb 2023)

References:

• Nguyen, N. C. T.  and J. L. Cybulski (2023): "Investigation of Barren Plateaus in Quantum Neural Network Development." Presented at 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Waseda University, Tokyo, Japan, August 20-25, 2023.

• Cybulski, J. L.  and T. Nguyen (2023): "Impact of Barren Plateaus Countermeasures on the Quantum Neural Network Capacity to Learn." Quantum Information Processing, To appear.