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

T.I. Onyshkevych, Valerii Hlukhov

Published May 12, 2025• 0 citations

Abstract

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.

DOI: 10.23939/csn2025.01.247

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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.

Abstract The Hadamard gate (H gate) is a fundamental single-qubit quantum gate that transforms a qubit from a basis state ∣0⟩|0\rangle∣0⟩ or ∣1⟩|1\rangle∣1⟩ into an equal superposition of ∣0⟩|0\rangle∣0⟩ and ∣1⟩|1\rangle∣1⟩. By enabling a qubit to simultaneously occupy multiple states, the Hadamard gate is essential for exploiting quantum parallelism, which underlies the computational advantage of quantum computers. The H gate is widely used in quantum algorithms, including quantum Fourier transform, Grover’s search algorithm, and quantum teleportation, as it provides a mechanism for creating superposition and preparing initial states for interference. Mastery of the Hadamard gate is critical for designing quantum circuits that leverage the unique properties of qubits and achieve complex quantum computation.

We studied the outputs that a quantum circuit consisting of four qubits can produce by using quantum gates, which operate on the states of the qubit as a state vector that can be expressed through unitary matrices. We made the change by exploiting the superposition property of the Hadamard gate, and with its presence in the quantum circuit, we used CNOT gates, which produced entangled states . The study explained the output of a quantum circuit consisting of four qubits and gave a methodology for changing this output by installing one input type for the quantum circuit (we used the input ). It showed the importance of measurement in the quantum circuit and its effect on the and the . Starting from, produce all 16 states of the four qubits, which can be expressed as the computational basis state of a vector state using only an x-gate.

A quantum gate or quantum logic gate is an elementary quantum circuit working on a small number of qubits. It means that quantum gates can grasp two primary feature of quantum mechanics that are entirely out of reach for classical gates : superposition and entanglement. In simpler words quantum gates are reversible. In classical computing sets of logic gates are connected to construct digital circuits. Similarly, quantum logic gates operates on input states that are generally in superposition states to compute the output. In this paper the authors will discuss in detail what is single and multiple qubit gates and scope and challenges in quantum gates.

A quantum gate is the quantum analog of the classical logic gate of digital computers. It is a basic structural unit of a quantum circuit which operates on a small number of qubits to deliver a predefined output. A quantum circuit is a graphical representation of the sequence of the quantum gates used and the measurements performed in a quantum computation. Various quantum logic gates, namely, the Hadamard, T phase, NOT, Pauli-X, controlled NOT (CNOT), controlled-Z, SWAP and controlled phase gates as well as higher-level Toffoli and Fredkin gates are described. Conventions of reading quantum circuit diagrams are explained. Operation of a quantum full adder consisting of registers and gates is discussed, elaborating the role of the NOT gate, the sum calculation with CNOTgates and the carry C-out calculation with Toffoli gates. A primary tool used for performing arithmetic operations on a quantum computer is the quantum Fourier transform (QFT). The QFT adder and multiplier circuits are presented.

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}.