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AIP Advances

Deterministic construction of arbitrary W states with quadratically increasing number of two-qubit gates

Fırat Diker

Published Jul 1, 2025• 2 citations
Publisher

Abstract

We propose a quantum circuit composed of cNOT gates and four single-qubit gates to generate a W state of three qubits. This circuit was then enhanced by integrating two-qubit gates to create a W state of four and five qubits. After a couple of enhancements, we show that an arbitrary W state can be generated depending only on the degree of enhancement. The generalized formula for the number of two-qubit gates required is given, showing that an n-qubit W-state generation can be achieved with quadratically increasing number of two-qubit gates. The practical feasibility is discussed regarding photon sources and various applications of cNOT gates. In addition, we discuss possible future directions for quantum-state networks, concerning quantum correlations of W states.

DOI
10.1063/5.0241266

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Deterministic construction of arbitrary WWW states with quadratically increasing number of two-qubit gates
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We address the problem of constructing quantum circuits to implement an arbitrary two-qubit quantum computation. We pursue circuits without ancilla qubits and as small a number of elementary quantum gates as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 171717 gates: 151515 one-qubit rotations and 222 controlled-NOT (CNOT) gates. We also constructively prove a worst-case upper bound of 232323 elementary gates, of which at most four (CNOT gates) entail multiqubit interactions. Our analysis shows that synthesis algorithms suggested in previous work, although more general, entail larger quantum circuits than ours in the special case of two qubits. One such algorithm has a worst case of 616161 gates, of which 181818 may be CNOT gates.
We present a deterministic framework for preparing an arbitrary three-qubit pure state. To leverage entanglement structure in the state-preparation task, we classify three-qubit pure states into five types with respect to a 1∣21|21∣2 bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.
In order to demonstrate non-trivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to nderstand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 CNOT gates and 15 elementary one qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, y-rotation, z-rotation, and phase gates.