Circuit partitioning is a powerful technique in the Noisy Intermediate Scale Quantum era. This method allows the evaluation of many-qubit algorithms with fewer qubits by writing the expectation...Show moreCircuit partitioning is a powerful technique in the Noisy Intermediate Scale Quantum era. This method allows the evaluation of many-qubit algorithms with fewer qubits by writing the expectation value of an observable as a weighted sum of products of inner products which are obtained by evaluating smaller circuits. However the number of terms in this sum generally scales exponentially with the depth of the circuit. The recently introduced Subset Partition Model only evaluates a subset of these terms. Here we explore the application of the Subset Partition Model for Variational Quantum Eigensolvers. Based on our investigation, we conjecture that this model may not be generally applicable to this Variational Quantum Algorithm due to the breaking of the Variational Principle. In addition, optimizing Partitioned Quantum Circuits is challenging due to the Barren Plateau problem that arises for even small circuit sizes. To address this problem, this thesis proposes a novel optimization algorithm that takes advantage of the additional structure on the cost function induced by circuit partitioning. By utilizing this new algorithm, we demonstrate that we can alleviate the barren plateau problem in the optimization of Partitioned Quantum Circuits.Show less
The search is on for near term applicable quantum algorithms. The financial sector is one field where improvements to computationally challenging tasks could be highly beneficial. We have explored...Show moreThe search is on for near term applicable quantum algorithms. The financial sector is one field where improvements to computationally challenging tasks could be highly beneficial. We have explored the use of quantum machine learning for one such task, the generation of synthetic financial data. Building on the current classical state of the art, we have implemented a Wasserstein generative adversarial network with gradient penalty for the generation of synthetic time series. We have expanded this classical framework using a parametrised quantum generator circuit. By using Pauli string expectation values, we can generate multi-dimensional continuous samples. This approach has allowed for the generation of small scale synthetic time series samples, based on the S&P 500 index, that show characteristic signs of the same temporal properties present in real financial data.Show less
Quantum machine learning is currently regarded as one of the most promising candidates for solving problems that appear out of reach using classical computers. Recently, a novel subfield of quantum...Show moreQuantum machine learning is currently regarded as one of the most promising candidates for solving problems that appear out of reach using classical computers. Recently, a novel subfield of quantum learning was opened up by Havlíček et al., who proposed a quantum learning algorithm which is closely related to support vector machines, yet which can be implemented on currently available quantum hardware. In this thesis, contribute to quantum machine learning by presenting new results on the capabilities of this algorithm, placing it in the perspectives of classical learning theory and quantum complexity. As the follow-up research which has since been published mainly focusses on details of experimental implementation, results in this direction are still lacking. Specifically, we compare the hyperplane (explicit) and kernel (implicit) formulations of the classifier algorithm, study its generalisation performance in the framework of statistical learning theory, and pin down the precise requirements for a quantum advantage using this algorithm. To this end, we apply the so-called representer theorem, known from the study of kernel methods in machine learning, to show training set optimality of the implicit formulation under regularised error measures. Furthermore, we show a tight upper bound on the fat shattering dimension of this type of quantum classifier, and discuss the implications for generalisation performance. Lastly, we carry out a complexity theoretic study showing that classical intractability of evaluating quantum kernels implies also the intractability of these quantum classifiers. We argue that despite this fact, we cannot claim that there exist problems which are hard to learn classically, but not quantumly, in the PAC learning sense, and subsequently describe the complexity theoretic requirements of quantum CLF learning to achieve quantum learning supremacy.Show less
