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arXiv:1606.09290

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

Fırat Diker

Published Jun 28, 2016• 12 citations
arXivHTMLPDF

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. Also, the practical feasibility is discussed regarding photon sources and various applications of $cNOT$ gates.

DOI
10.48550/arxiv.1606.09290

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Deterministic construction of arbitrary WWW states with quadratically increasing number of two-qubit gates
We propose a quantum circuit composed of cNOTcNOTcNOT gates and four single-qubit gates to generate a WWW state of three qubits. This circuit was then enhanced by integrating two-qubit gates to create a WWW state of four and five qubits. After a couple of enhancements, we show that an arbitrary WWW 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 nnn-qubit WWW-state generation can be achieved with quadratically increasing number of two-qubit gates. Also, the practical feasibility is discussed regarding photon sources and various applications of cNOTcNOTcNOT gates.
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.
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.
We propose an optical scheme to build an entangled network composed of W state based on polarization encoded qubits (photons). This new setup consists of 2 cNOT gates, 4 V gates, 2 Hadamard gates and basic optical tools such as polarizing beamsplitters (PBSs) and path couplers (PCs). V gate is a specially-designed tool acting as a two-qubit gate which is composed of a cNOT gate, 3 PBSs and a PC. By using this gate, one benefits from the temporarily generated optical degree of freedom, which is the spatial mode of a photon in the proposed scheme. Using an extra degree of freedom allows us to perform more capable processing for W-state creation protocols. We use four photons as input, which means we do not need entanglement as a resource. Also, we show that the proposed scheme can be implemented by operating the quantum optical gates which can be realized via current photonics technology. Additionally, we provide discussions of quantum contextuality and non-locality for W states.
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.
Optimal construction of quantum operations is a fundamental problem in the realization of quantum computation. We here introduce a newly discovered quantum gate, B, that can implement any arbitrary two-qubit quantum operation with minimal number of both two- and single-qubit gates. We show this by giving an analytic circuit that implements a generic nonlocal two-qubit operation from just two applications of the B gate. Realization of the B gate is illustrated with an example of charge-coupled superconducting qubits for which the B gate is seen to be generated in shorter time than the CNOT gate.