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Laser & Optoelectronics Progress

量子Toffoli门的核磁共振物理实现

彭永刚 Peng Yonggang

Published Jan 1, 2023• 2 citations
Publisher

Abstract

The three -qubit Toffoli gate is equivalent to the combination of two two -qubit controlled NOT gates and three two -qubit controlled phase -shift gates.The controlled NOT and phase -shift gates respectively comprise a combination of nuclear -magnetic -resonance pulse sequences and a free development operator of two nuclear spins with time.According to the order in which the controlled NOT and phase -shift gates act on the nuclear spin system, the nuclear -magnetic - resonance -based physical realization of a quantum Toffoli gate is achieved.The time -dependent Schrödinger equation is numerically solved by the Suzuki formula to confirm the correctness and feasibility of the realization of the quantum Toffoli gate based on nuclear magnetic resonance.

DOI
10.3788/lop220821

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The three -qubit Toffoli gate is equivalent to the combination of two two -qubit controlled NOT gates and three two -qubit controlled phase -shift gates.The controlled NOT and phase -shift gates respectively comprise a combination of nuclear -magnetic -resonance pulse sequences and a free development operator of two nuclear spins with time.According to the order in which the controlled NOT and phase -shift gates act on the nuclear spin system, the nuclear -magnetic - resonance -based physical realization of a quantum Toffoli gate is achieved.The time -dependent Schrödinger equation is numerically solved by the Suzuki formula to confirm the correctness and feasibility of the realization of the quantum Toffoli gate based on nuclear magnetic resonance.
We present the experimental realization of this gate with a solution of chlorostyrene molecules. Our method does not depend heavily on the two-qubit controlled operation, which used to serve as the basic quantum operation in quantum computing. At present, we use transition operator that can be applied to all qubits in one operation. It appears that no experimental realization has yet been reported up to now regarding the implementation of quantum Toffoli gate using transition pulse on three-qubit nuclear magnetic resonance quantum computers. In addition, our method is experimentally convenient to be extended to more qubits.
Logic gates such as the controlled-NOT (c-NOT) and Toffoli gates play a key role in quantum information processing (QIP) and quantum computing. A natural extension of such gates would necessarily operate on one quantum bit (qubit) conditional on the state of the remaining qubits in the system. We show that such selective gates, termed $(\mathrm{controlled}{)}^{n}\ensuremath{-}\mathrm{NOT}$ gates, or ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates, are convenient in nuclear magnetic resonance (NMR) implementations of QIP and are straightforward to implement. NMR pulse sequences for these gates can be built using classical methods as well as insights from geometric algebra. These methods yield equivalent NMR pulse sequences for the generation of ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates for any number of control spins. In this work, a catalog of ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates for systems of as many as 16 spins is provided along with an experimental implementation of a ${\mathrm{c}}^{3}\ensuremath{-}\mathrm{N}\mathrm{O}\mathrm{T}$ gate on a four spin system, 13C{}^{13}\mathrm{C}13C alanine.
A quantum circuit network is a set of circuits that implements a certain computation task. Being at the centre of the quantum circuit network, the multi-qubit controlled phase shift is one of the most important quantum gates. In this paper, we apply the method of modular structuring in classical computer architecture to the quantum computer and give a recursive realization of the multi-qubit phase gate. This realization of the controlled phase shift gate is convenient in realizing certain quantum algorithms. We have experimentally implemented this modularized multi-qubit controlled phase gate in a three-qubit nuclear magnetic resonance (NMR) quantum system. The network is demonstrated experimentally using line selective pulses in the NMR technique. The procedure has the advantage of being simple and easy to implement.
We propose a scheme for realizing quantum gates [controlled-NOT (CNOT) and Toffoli] using a two-level quantum system by means of a pulsed bias. We show how to realize a CNOT gate using a two-qubit, two-level coupled system and a Toffoli gate using a three-qubit, two-level coupled system. For the CNOT gate, we show how to reduce the $4\ifmmode\times\else\texttimes\fi{}4$ Hamiltonian of the coupled system to a $2\ifmmode\times\else\texttimes\fi{}2$ Hamiltonian in order to solve for parameter values of the system. When these values are substituted into the coupled system equations for the two-qubit system, the simulation results confirm the CNOT gate operation. We further demonstrate our proposed scheme for a CNOT gate operation using a multilevel system of two coupled superconducting quantum interference devices as a particular example. We next use the same approach to implement a Toffoli gate using a three-qubit system.
In quantum information processing, realization of 3-qubit quantum operation is challenging especially when multiple logic operations are required to build non-trivial quantum circuits. Quantum half-adder is a 3-qubit circuit and in this work, a simple realization scheme for quantum half-adder operation is proposed. The 3-qubit quantum circuit is composed of a Toffoli operation and a controlled-NOT operation in succession. These two operations are shown to be achievable only by two transition pulses in nuclear magnetic resonance based quantum computing at room temperature. In our proposed scheme, three weakly coupled spin 1/2 nuclei of a molecule serve as three qubits. The time-independent internal Hamiltonian of the system allows us to obtain the desired multi-qubit operation. The simple structure of the spin 1/2 systems of our scheme make it easy to be implemented in experiments. Simulation on fidelity estimation reveals that more than 96% fidelity is achievable in worst case scenario.
Abstract It is shown theoretically how to use the EPR (electron paramagnetic resonance) technique, using electron spins as qubits, coupled with each other by the exchange interaction, to set the configuration of n qubits ( n = 1–4) at resonance, in conjunction with pulses, to construct the NOT (one qubit), CNOT (two qubits), CCNOT (three qubits), and CCCNOT (four qubits) Toffoli gates, which can be exploited to build a quantum computer. This is unique to EPR, wherein exchange‐coupled electron spins are used. This is different from NMR (Nuclear Magnetic Resonance), which uses nuclear spins as qubits, that do not couple with each other by the exchange interaction.