We study a reduced quantum circuit computation paradigm in which the only allowable gates either permute the computational basis states or else apply a ``global Hadamard operation'', \emph{i.e.} apply a Hadamard operation to every qubit simultaneously. In this model, we discuss complexity bounds (lower-bounding the number of global Hadamard operations) for common quantum algorithms~: we illustrate upper bounds for Shor's Algorithm, and prove lower bounds for Grover's Algorithm. We also use our formalism to display a gate that is neither quantum-universal nor classically simulable, on the assumption that Integer Factoring is not in \textbf{BPP}.We study a reduced quantum circuit computation paradigm in which the only allowable gates either permute the computational basis states or else apply a ``global Hadamard operation'', \emph{i.e.} apply a Hadamard operation to every qubit simultaneously. In this model, we discuss complexity bounds (lower-bounding the number of global Hadamard operations) for common quantum algorithms~: we illustrate upper bounds for Shor's Algorithm, and prove lower bounds for Grover's Algorithm. We also use our formalism to display a gate that is neither quantum-universal nor classically simulable, on the assumption that Integer Factoring is not in \textbf{BPP}.

Jan 1, 20060 citations

Quantum Gates and Circuits

Belal Ehsan Baaquie, L. C. Kwek

Springer eBooksJan 1, 20233 citations

Optimal number of parametrized rotations and Hadamard gates in parametrized Clifford circuits with non-repeated parameters

Vivien Vandaele, Simon Perdrix, Christophe Vuillot

We present an efficient algorithm to reduce the number of non-Clifford gates in quantum circuits and the number of parametrized rotations in parametrized quantum circuits. The method consists in finding rotations that can be merged into a single rotation gate. This approach has already been considered before and is used as a pre-processing procedure in many optimization algorithms, notably for optimizing the number of Hadamard gates or the number of $T$ gates in Clifford $+T$ circuits. Our algorithm has a better complexity than similar methods and is particularly efficient for circuits with a low number of internal Hadamard gates. Furthermore, we show that this approach is optimal for parametrized circuits composed of Clifford gates and parametrized rotations with non-repeated parameters. For the same type of parametrized quantum circuits, we also prove that a previous procedure optimizing the number of Hadamard gates and internal Hadamard gates is optimal. This procedure is notably used in our low-complexity algorithm for optimally reducing the number of parametrized rotations.We present an efficient algorithm to reduce the number of non-Clifford gates in quantum circuits and the number of parametrized rotations in parametrized quantum circuits. The method consists in finding rotations that can be merged into a single rotation gate. This approach has already been considered before and is used as a pre-processing procedure in many optimization algorithms, notably for optimizing the number of Hadamard gates or the number of $T$ gates in Clifford$+T$ circuits. Our algorithm has a better complexity than similar methods and is particularly efficient for circuits with a low number of internal Hadamard gates. Furthermore, we show that this approach is optimal for parametrized circuits composed of Clifford gates and parametrized rotations with non-repeated parameters. For the same type of parametrized quantum circuits, we also prove that a previous procedure optimizing the number of Hadamard gates and internal Hadamard gates is optimal. This procedure is notably used in our low-complexity algorithm for optimally reducing the number of parametrized rotations.

arXiv:2407.07846Jul 10, 20240 citations

Threshold Circuits and Quantum Gates

Leporati

Jan 8, 20031 citation

STUDY OF THE QUANTUM PHASE INFLUENCE ON THE QUBIT STATE IN CIRCUITS WITH HADAMARD GATES CONNECTED IN SERIES

T.I. Onyshkevych, Valerii Hlukhov

This paper presents a study of the influence of the quantum phase on the qubit state in circuits with Hadamard gates in series for further minimisation in software or hardware modelling. The study is carried out using the Azure Quantum computing platform to simulate quantum algorithms and run quantum applications on dedicated hardware. The study provides an overview of the qubit as a unit of quantum information, describes in detail the operation of the Hadamard gate, simulates the operation of a circuit of two Hadamard gates in series with initial inputs and , and simulates the operation of a circuit with a sequential shift on and Hadamard gate inputs and . In the course of the study, it was found how the same intermediate state of the qubit can have different outputs after the Hadamard gate, by means of phase. Also, the results of the schemes where the first Hadamard gate is replaced by a sequential shift on . The results obtained can be used to test the Hadamard gate model when creating its digital model. Key words: Hadamard gate, quantum computing, quantum gates, superposition.This paper presents a study of the influence of the quantum phase on the qubit state in circuits with Hadamard gates in series for further minimisation in software or hardware modelling. The study is carried out using the Azure Quantum computing platform to simulate quantum algorithms and run quantum applications on dedicated hardware. The study provides an overview of the qubit as a unit of quantum information, describes in detail the operation of the Hadamard gate, simulates the operation of a circuit of two Hadamard gates in series with initial inputs and , and simulates the operation of a circuit with a sequential shift on and Hadamard gate inputs and . In the course of the study, it was found how the same intermediate state of the qubit can have different outputs after the Hadamard gate, by means of phase. Also, the results of the schemes where the first Hadamard gate is replaced by a sequential shift on . The results obtained can be used to test the Hadamard gate model when creating its digital model. Key words: Hadamard gate, quantum computing, quantum gates, superposition.

Komp'ûternì sistemi ta merežìMay 12, 20250 citations

T

gates in Clifford

+T

circuits. Our algorithm has a better complexity than similar methods and is particularly efficient for circuits with a low number of internal Hadamard gates. Furthermore, we show that this approach is optimal for parametrized circuits composed of Clifford gates and parametrized rotations with non-repeated parameters. For the same type of parametrized quantum circuits, we also prove that a previous procedure optimizing the number of Hadamard gates and internal Hadamard gates is optimal. This procedure is notably used in our low-complexity algorithm for optimally reducing the number of parametrized rotations.