Hilbert Space, Tensor Product, and Multi-Qubit

Hiu Yung Wong

Sylvia Pulmannová

Puran Singh Mehra

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.

Chinnaraji Annamalai

Classical computing architectures encode information into discrete binary digits residing in isolated regions of phase space. Quantum computing replaces this paradigm by leveraging complex state vectors defined within continuous, multi-dimensional Hilbert spaces. This paper provides a unified mathematical framework examining finite-dimensional Hilbert spaces in quantum information processing. It traces the structural scaling of quantum state spaces from single-qubit systems up through composite tensor product structures and multi-qubit registers. By detailing computational basis representations, state transformations, probability amplitudes, and multipartite entanglement classes, such as Bell, GHZ, and W states, this paper demonstrates how exponential dimensional scaling enables quantum parallelism while presenting computational barriers to classical simulation.