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International Journal of Statistics and Applied Mathematics

High arrangement group quantum circuits Hilbert spaces for in sequence special consideration

Kuchi Padmavathi, Yesuratnam Anitha Kumari

Published May 1, 2018• 0 citations
OpenAlex

Abstract

A quantum circuit is a model for Hilbert space computation in which a computation is a sequence of quantum spaces, which are reversible transformations on a quantum mechanical space of an n-bit register. This analogous structure is referred to as an n-space register. We define and can build efficient depth-universal and almost size-universal quantum circuits Such circuits can be viewed as general purpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blow-up in the universal circuits constructed here. The architecture scalability afforded by recent proposals of a large-scale photonic-based quantum space computer allows us to move on to a discussion of High performance Hilbert quantum circuits.

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A quantum circuit is a model for Hilbert space computation in which a computation is a sequence of quantum spaces, which are reversible transformations on a quantum mechanical space of an n-bit register. This analogous structure is referred to as an n-space register. We define and can build efficient depth-universal and almost size-universal quantum circuits Such circuits can be viewed as general purpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blow-up in the universal circuits constructed here. The architecture scalability afforded by recent proposals of a large-scale photonic-based quantum space computer allows us to move on to a discussion of High performance Hilbert quantum circuits.
We define and construct efficient depth-universal and almost-size-universal quantum circuits. Such circuits can be viewed as general-purpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blow-up in the universal circuits constructed here. We prove that this construction is nearly optimal.
Universal circuits can be viewed as general-purpose simulators for central classes of circuits and can be used to capture the computational power of the circuit class being simulated. We define and construct quantum universal circuits which are efficient and has very little overhead in simulation. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blow-up in the universal circuits constructed here which is nearly optimal.
The quantum Fourier transform and quantum wavelet transform have been cornerstones of quantum information processing. However, for non-stationary signals and anomaly detection, the Hilbert transform can be a more powerful tool, yet no prior work has provided efficient quantum implementations for the discrete Hilbert transform. This letter presents a novel construction for a quantum Hilbert transform in polylogarithmic size and logarithmic depth for a signal of length NNN, exponentially fewer operations than classical algorithms for the same mapping. We generalize this algorithm to create any ddd-dimensional Hilbert transform in depth O(dlog⁡N)O(d\log N)O(dlogN). Simulations demonstrate effectiveness for tasks such as power systems control and image processing, with exact agreement with classical results.
Quantum computing is a process that incorporates interacting physical systems that represent quantum bits and quantum gates. We present the quantum bit (qubit), the quantum register and the quantum gates. The qubit is described as a vector in a two-dimensional Hilbert space and the quantum register, which comprises a number of qubits, as a vector in a multidimensional Hilbert space. Quantum gates are Hilbert space operators that rotate the qubit or the quantum register vectors. Quantum computations are modeled and described using a quantum circuit model. We also present a quantum computer simulator based on the circuit model of quantum computation. In this model quantum computations and quantum algorithms are represented by circuits, which comprise quantum gates and quantum registers. The well-known Deutsch's algorithm is described and the corresponding quantum circuit is presented. Possible applications of quantum computers are be presented and discussed.
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.