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arXiv:2510.02685

Construction of the Complete Set of Maximally Entangled Basis Vectors for N-Qubit Systems

Chi‐Chuan Hwang

Published Oct 3, 2025• 0 citations
arXivHTMLPDF

Abstract

In this study, we first use a three-qubit system as an example to demonstrate the construction of quantum circuits for the eight maximally entangled basis vectors, subsequently extending the approach to N-qubit systems. We employ a random-number approach to generate maximally entangled basis vectors and their corresponding circuits, while also detailing the required number of single-qubit and CNOT gates. This approach not only provides a solid theoretical foundation but also establishes a practical technique for technological applications, bypassing the difficulty of storing large-scale encoding data.

DOI
10.48550/arxiv.2510.02685

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In this study, we first use a three-qubit system as an example to demonstrate the construction of quantum circuits for the eight maximally entangled basis vectors, subsequently extending the approach to N-qubit systems. We employ a random-number approach to generate maximally entangled basis vectors and their corresponding circuits, while also detailing the required number of single-qubit and CNOT gates. This approach not only provides a solid theoretical foundation but also establishes a practical technique for technological applications, bypassing the difficulty of storing large-scale encoding data.
Study of a quantum circuit consisting of four qubits that produces an output state consisting of eight entangled states through the CNOT –gates and the three Hadamard gates. targeting the gates to |q0q1q2q3> to show the other entangled states, We used quantum gates that give the same arrangement as the circuit gates and for the same inputs, taking into account the output of a quantum circuit that contains entangled states. As a general case, we used an X-gate and a SWAP-gate, For inputs|0000> .We obtained a general change to the output state while keeping the eight states maximally entangled.
Abstract This laboratory work presents a practical introduction to the preparation and analysis of entangled quantum states (Bell pairs) using the Qiskit framework. The material is based on educational resources developed by the International Quantum Center at CERN (Switzerland) and is designed to provide hands-on experience with multi-qubit quantum circuits and entanglement generation. In Laboratory Work No. 3, participants construct and simulate a quantum circuit that prepares an entangled two-qubit state. As a representative example of multi-qubit operations in Qiskit, the circuit generates a Bell state by applying a Hadamard (H) gate to the first qubit q[0]q[0]q[0], followed by a controlled-NOT (CNOT) gate with q[0]q[0]q[0] as the control qubit and q[1]q[1]q[1] as the target qubit. This sequence creates a maximally entangled state, demonstrating the fundamental non-classical correlations characteristic of quantum systems. The laboratory exercise includes measurement operations applied to both qubits, illustrating how entanglement manifests in correlated classical outcomes. When the measurement operator is applied to the first qubit q[0]q[0]q[0], the corresponding classical bit records a definite value, and subsequent measurement of the second qubit q[1]q[1]q[1] yields a correlated result consistent with the structure of the Bell state. The probabilistic yet correlated outcomes provide direct evidence of quantum entanglement. The guide emphasizes the conceptual understanding of entanglement, circuit implementation in Qiskit, and the hybrid quantum-classical workflow underlying quantum computation. By bridging theoretical quantum mechanics with executable quantum circuits, this laboratory work enables students and researchers to experimentally explore one of the central phenomena of quantum information science using cloud-based quantum computing resources.
Abstract There are various gate sets that can be used to describe a quantum computation. A particularly popular gate set in the literature on quantum computing consists of arbitrary single-qubit gates and two-qubit CNOT gates. A CNOT gate is however not always the natural multi-qubit interaction that can be implemented on a given physical quantum computer, necessitating a compilation step that transforms these CNOT gates to the native gate set. An especially interesting case where compilation is necessary is for ion trap quantum computers, where the natural entangling operation can act on more than two qubits and can even act globally on all qubits at once. This calls for an entirely different approach to constructing efficient circuits. In this paper we study the problem of converting a given circuit that uses two-qubit gates to one that uses global gates. Our three main contributions are as follows. First, we find an efficient algorithm for transforming an arbitrary circuit consisting of Clifford gates and arbitrary phase gates into a circuit consisting of single-qubit gates and a number of global interactions proportional to the number of non-Clifford phases present in the original circuit. Second, we find a general strategy to transform a global gate that targets all qubits into one that targets only a subset of the qubits. This approach scales linearly with the number of qubits that are not targeted, in contrast to the exponential scaling reported in (Maslov and Nam 2018 New J. Phys. 20 033018). Third, we improve on the number of global gates required to synthesise an arbitrary n-qubit Clifford circuit from the 12n − 18 reported in (Maslov and Nam 2018 New J. Phys. 20 033018) to 6n − 8.
Abstract In the circuit model of quantum computation, an entangling two-qubit gate and a set of single-qubit gates are used as universal gate set or basis gates for doing quantum computation. CNOT and iSWAP gates, the perfect entanglers that can create maximally entangled two-qubit states in one application, are broadly used entangling two-qubit basis gates in quantum computers. In this paper, we analyze the potentiality of B gate, an unexplored non-perfect entangler of the form, exp i π ( σ ˆ x ⨂ σ ˆ x ) 8 + i π ( σ ˆ y ⨂ σ ˆ y ) 16 , as an entangling two-qubit basis gate in quantum computers by studying its ability to generate other two-qubit gates. We derive a necessary condition for a two-qubit gate to be generated by n applications of B gate. Using this condition, we show that the gates that can be generated by two and three applications of B gate are contained in the 50% and 92.97% of the volume of Weyl chamber of two-qubit gates, respectively. We prove that two applications of B gate can generate both SWAP and SWAP
In this research, a quantum system consists of three qubits is considered; it has eight states (|000˃, |001˃, |010˃, |011˃, |100˃, |101˃, |110˃ and |111˃).Each of these states is passed through a quantum circuit made of a sequence of quantum gates.The circuit applies X-gate on the first qubit from the left, then using CNOT-gate on the first and second qubits and using the Hadamard-gate on the first qubit.Finally, Hadamard-gate is applied on the third qubit.When the state |000˃ is passed as input through the proposed quantum circuit then the output includes four entanglement states; for the other input states the output will be another non-repeatable entanglement states.So, for using all states in quantum circuit the result will be eight equations each one consists of four entanglement states; which called ML-Quadripartite.