This paper introduces a formalism that aims to describe the intricacies of quantum computation by establishing a connection with the mathematical foundations of tensor theory and multilinear maps. The focus is on providing a comprehensive representation of quantum states for multiple qubits and the quantum gates that manipulate them. The proposed formalism could contribute to a more intuitive representation of qubit states, and to a clear visualisation of the entanglement property. The main advantages of this formalism are that it preserves the fundamental structure of the Hilbert space to which quantum states belong, and also reduces the computational cost associated with classical prediction of the effect of quantum gates on multi-qubit states. A connection between the ability to generate entanglement and the quantum gate representation is also established.
This paper introduces a formalism that aims to describe the intricacies of quantum computation by establishing a connection with the mathematical foundations of tensor theory and multilinear maps. The focus is on providing a comprehensive representation of quantum states for multiple qubits and the quantum gates that manipulate them. The proposed formalism could contribute to a more intuitive representation of qubit states, and to a clear visualisation of the entanglement property. The main advantages of this formalism are that it preserves the fundamental structure of the Hilbert space to which quantum states belong, and also reduces the computational cost associated with classical prediction of the effect of quantum gates on multi-qubit states. A connection between the ability to generate entanglement and the quantum gate representation is also established.
Quantum computing is fundamentally based on the mathematical structure of complex Hilbert spaces, linear operators, tensor products, and unitary transformations. Among these concepts, tensor products provide the formal mechanism for constructing the state space of composite quantum systems, whereas linear and unitary operators describe quantum transformations, measurements, and computational gates. This article examines the mathematical relationship between linear operators and tensor products and analyzes their role in the formation, manipulation, and characterization of quantum entanglement. Particular attention is given to two-qubit systems, separable and entangled states, Schmidt decomposition, controlled operations, and the role of non-factorizable operators in quantum circuits. The analysis demonstrates that tensor products perform a dual function: they provide the mathematical representation of composite systems and enable the construction of local operations acting on individual subsystems. At the same time, genuinely entangling transformations, such as the controlled-NOT operation, cannot in general be represented as a single tensor product of independent local operators. Recent research further demonstrates that tensor-product and operator-Schmidt decompositions can be used to characterize operator entanglement, reduce circuit complexity, and construct efficient representations of quantum processes. The results establish that linear algebra and tensor-product structures are not merely formal tools but constitute the mathematical foundation connecting quantum entanglement with quantum computation.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
The exponential scaling of Hilbert spaces presents a major computational bottleneck in simulating and controlling large scale multi qubit systems. This paper establishes the tensor product space not just as composite vector space, but fundamentally as a multi linear mapping that describes quantum operations across desperate subsystems. By modelling quantum gates and straight through this lense, we preserve the internal structural integrity of quantum operations while exponentially decreasing the computational overhead needed for simulation and control.
Quantum mechanics has proven its position as the primary theory for accurately describing microscopic domains. Numerous fields in physics utilize quantum mechanics to great effects, with one prominent application being quantum computing. Quantum computing extends the field of quantum mechanics to information theory, striving to perform computational tasks utilizing quantum phenomena with notably greater efficiency compared to classical alternatives. In this thesis, I will go through the basics required in the field of quantum computing. To understand quantum computing demands adapting a new type of matrix mechanical view. This involves considering Hilbert spaces, which are linear vector space having a well defined inner product. Hilbert spaces contain state vectors describing quantum states, as well as operators for altering and measuring these states. The dynamics of quantum systems are governed by three quantum postulates. These specify Hilbert spaces and state vectors as the fundamental components of quantum systems, Hermitian operators as providing the physical manifestation of these systems, and finally, the Schrödinger's equation to describe the time evolution of such systems. These notions are required in producing quantum computing. Quantum computing uses qubits as the computational basis, analogous to classical bits but with the added capability of quantum superposition. Qubits are represented as two-state vectors, and can be physically realized with any quantum-behaving binary-valued system, like the electron spins. Qubits can be combined into higher dimensional states with the use of tensor products, causing the system's Hilbert space to expand as 2^n with n being the number of qubits, allowing for exponential computational speedup. Computational logic can be constructed by acting on the qubits using unitary operators, known as quantum gates. The combinations of qubits and quantum gates can be visualized as quantum circuits, depicted using specialized circuit diagram notations. An essential part of quantum computing, and the phenomena enabling exponential computational speedup, is quantum entanglement. Entanglement arises in interacting qubits, intertwining them so that the individual states become correlated. Entanglement can be characterized by being unable to reduce a combined state into its pure constituents. The amount of entanglement in a system can be measured using von Neumann entropy, analogous to Shannon entropy in classical information theory. Entanglement serves various purposes in quantum computing and is a defining feature distinguishing it from the classical domain. In this thesis, the entanglement properties of a periodic four-qubit circuit with alternating quantum gates were simulated. The simulation was performed computationally, and measured entanglement entropy as a function of the amount of circuit periods. The results demonstrated different characteristics of entanglement depending on the parameters used for the quantum gates, with the entanglement either being periodic, non-occurring or chaotic in nature.
Quantum circuits, fundamental to quantum computation, represent quantum states through a sequence of unitary transformations. The exponential growth in the Hilbert space dimension with increasing circuit size presents a significant challenge to simulation and analysis. This work introduces a novel tensor network approach to formally model and manipulate quantum circuits, specifically designed to mitigate this exponential scaling issue. The core idea is to represent the circuit's state vector as a tensor network, leveraging techniques for efficient compression and expansion. We explore the application of variational tensor networks (VTNs) to approximate the dynamics of quantum circuits, offering a pathway towards more scalable simulations. The effectiveness of this method is demonstrated through theoretical analysis and conceptual design, focusing on key aspects of tensor network construction and manipulation tailored to quantum circuit representations. This approach aims to provide a more scalable representation compared to existing methods.
This presentation provides a comprehensive overview of the foundational principles of quantum computation. It begins by contrasting classical and quantum information, highlighting how quantum bits (qubits) leverage superposition, entanglement, and interference to process information in ways that surpass classical limits. The document explains the four postulates of quantum mechanics: state space (qubits as unit vectors in a Hilbert space), time evolution (via unitary operators), measurement (probabilistic collapse of the state), and composite systems (tensor products of subsystems). Key concepts such as the Bloch sphere representation, quantum gates (e.g., Hadamard, CNOT, Pauli gates), and quantum circuits are discussed, illustrating how quantum operations manipulate qubit states. The presentation also covers quantum evolution, the gate model for computation, and the universality of certain gate sets, emphasizing the potential for quantum parallelism and the transformative impact of quantum computing on fields like cryptography and optimization.