Quantum Computing and Quantum ML:

Variational Quantum Circuit for Quantum ML:

Quantum Machine Learning (QML) is an emerging research area advocating the use of quantum computing for advancement in machine learning. Since the discovery of the capability of Parametrized Variational Quantum Circuits (VQC) to replace Artificial Neural Networks, they have been widely adopted to different tasks in Quantum Machine Learning. However, despite their potential to outperform neural networks, VQCs are limited to small scale applications given the challenges in scalability of quantum circuits. To address this shortcoming, we propose an algorithm that compresses the quantum state within the circuit using a tensor ring representation. We also show that the tensor networks have deep connections with the quantum hidden Markov models, which allow for efficient representation of tensor networks.


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Quantum Causality:

One of the most important problems in quantum causality is linked to this prominent aphorism that states correlation does not mean causation. We proposed an approach to determine the presence of latent confounders (common causes) in quantum systems. Further, if there is no confounder, we also provide an approach to determine the cause-effect relation between the quantum systems. The approach of determining latent confounders can be applied to classical data, and is shown to outperform the algorithms for determining confounders in classical data. The key reason for the quantum approach outperforming classical approach is that the search over quantum density matrices achieve better optima than probability distributions.


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Boolean functions and Projection Operators:

There is a fundamental correspondence between Boolean functions and projection operators in Hilbert space. I have used it to provide a common mathematical framework for the design of both additive and non-additive quantum error correcting codes and to construct new families of error correcting codes. As shown in the figure alongside, the problem of finding new quantum codes is converted to a problem of finding a boolean function satisfying certain properties; thus converting the problem of finding quantum codes to a classical problem. The correspondence is widely applicable and I have also used it to construct and analyze new families of non-coherent space-time codes for applications in wireless communication.

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Quantum Fault Tolerance:

I am interested in understanding how to reliably determine error thresholds that are both necessary and sufficient to support fault tolerant quantum computing. The standard approach to quantum fault tolerance is to calculate error thresholds on basic gates in the limit of arbitrarily many concatenation levels. I want to be able to take the number of qubits and the target implementation accuracy as a constraint and then provide a framework for engineering the constrained quantum system to the required tolerance. I have developed a different approach which is based on complete analytical solutions to the dynamics of quantum error correction. I believe it will result in better integration of hardware and software since the full error manifold clarifies what is easier to do in physics and what is easier to do in error control algorithms. As an example, the image on the right shows the manifold of initial error probabilities that are consistent with a given implementation inaccuracy when the 7-qubit code is concatenated a specified number of times. My research program was funded by DARPA under the highly competitive QuEST Program.

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