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arXiv:2411.09010 · Advanced Quantum Technologies

Use of Electron Paramagnetic Resonance (EPR) Technique to Build Quantum Computers: n ‐Qubit ( n = 1–4) Toffoli Gates

Serena La Manna, Sushil K. Misra

Published Aug 19, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1002/qute.202500115

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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.
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.
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Modern experimental techniques can generate magnetic fields of the form H(t) = H0 z-hat + H1 [x-hat cos(ωt) + y-hat sin(ωt)], at frequencies within an order of magnitude of the nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) frequencies, ωn0 and ωe0, respectively, when acting on atoms or molecules. We derive simple closed-form expressions for the exact nuclear- and electronic-spin wave functions that enable controlled transitions between entangled states, allowing an atom or molecule to function as a quantum computer. These solutions also enable precise NMR or EPR measurements of nuclear moments in atoms and molecules. We present examples relevant to measurements of the nuclear moments of 14N, 7Li, and 133Cs. Because existing hyperfine measurements of the lowest three nuclear moments of 133Cs are mutually inconsistent, the proposed NMR/EPR experiments provide a route to measuring all seven of its nuclear moments with high precision.
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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.