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This book serves as an introductory yet thorough guide to tensor networks and their applications in quantum computation and quantum information, designed for advanced undergraduate and graduate-level readers. In Part I, foundational topics…
Surface and color codes are two forms of topological quantum error correction in two spatial dimensions with complementary properties. Surface codes have lower-depth error detection circuits and well-developed decoders to interpret and…
Scalable modern-time fault-tolerant quantum computation and quantum communication in a network employ a large number of physical qubits. For example, IBM is reported to have made a 127-qubit quantum computer. Unlike classical computation,…
Hybrid quantum-classical models represent a crucial step toward leveraging near-term quantum devices for sequential data processing. We present Quantum Recurrent Neural Networks (QRNNs) and Quantum Convolutional Neural Networks (QCNNs) as…
Clifford circuits -- i.e. circuits composed of only CNOT, Hadamard, and $\pi/4$ phase gates -- play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and…
We propose to represent both $n$--qubits and quantum gates acting on them as elements in the complex Clifford algebra defined on a complex vector space of dimension $2n.$ In this framework, the Dirac formalism can be realized in…
Characterising quantum processes is a key task in and constitutes a challenge for the development of quantum technologies, especially at the noisy intermediate scale of today's devices. One method for characterising processes is randomised…
Current quantum computing hardware is restricted by the availability of only few, noisy qubits which limits the investigation of larger, more complex molecules in quantum chemistry calculations on quantum computers in the near-term. In this…
We investigate the emergence of quantum complexity and chaos in doped Clifford circuits acting on qudits of odd prime dimension $d$. Using doped Clifford Weingarten calculus and a replica tensor network formalism, we derive exact results…
Efficient simulation of quantum computers relies on understanding and exploiting the properties of quantum states. This is the case for methods such as tensor networks, based on entanglement, and the tableau formalism, which represents…
Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the…
Particle physics underpins our understanding of the world at a fundamental level by describing the interplay of matter and forces through gauge theories. Yet, despite their unmatched success, the intrinsic quantum mechanical nature of gauge…
Quantum neural networks (QNNs), as currently formulated, are near-term quantum machine learning architectures that leverage parameterized quantum circuits with the aim of improving upon the performance of their classical counterparts. In…
Matchgates are a restricted set of two-qubit gates known to be classically simulable under particular conditions. Specifically, if a circuit consists only of nearest-neighbour matchgates, an efficient classical simulation is possible if…
We introduce a class of 3D color codes, which we call stacked codes, together with a fault-tolerant transformation that will map logical qubits encoded in two-dimensional (2D) color codes into stacked codes and back. The stacked code allows…
Learning an unknown quantum process is a central task for validation of the functioning of near-term devices. The task is generally hard, requiring exponentially many measurements if no prior assumptions are made on the process. However, an…
We show that the Clifford gates and stabilizer circuits in the quantum computing literature, which admit efficient classical simulation, are equivalent to affine signatures under a unitary condition. The latter is a known class of tractable…
Matchgates and Clifford circuits are two types of quantum circuits which can be efficiently simulated classically, though the underlying reasons are quite different. Matchgates are essentially the single particle basis transformations in…
We suggest Clifford algebra as a useful simplifying language for present quantum dynamics. Clifford algebras arise from representations of the permutation groups as they arise from representations of the rotation groups. Aggregates using…
Fault-tolerant quantum computation allows quantum computations to be carried out while resisting unwanted noise. Several error-correcting codes have been developed to achieve this task, but none alone are capable of universal quantum…