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Scientific Insights

Ion Trap Quantum Computers Part 2: The Universal 2-Qubit-CNOT Gate

Bernd Baumann

Published Dec 26, 2025• 0 citations
Publisher

Abstract

The Cirac-Zoller mechanism is generally used for the implementation of CNOT gates in ion trap quantum computers. We present here a highly detailed quantum physical description of this approach. To keep the presentation simple, the qubits are modeled as rigid rotors and the radiation field of each of the lasers is treated as a classical field. In combination with the corresponding first part (Ion Trap Quantum Computer Part 1: 1-Qubit Gates), this results in the complete quantum theory of a universal set of quantum gates. This presentation is intended for readers with basic knowledge of quantum physics and assumes familiarity with Part 1.

DOI
10.64897/si.2025v1i2.008

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The Cirac-Zoller mechanism is generally used for the implementation of CNOT gates in ion trap quantum computers. We present here a highly detailed quantum physical description of this approach. To keep the presentation simple, the qubits are modeled as rigid rotors and the radiation field of each of the lasers is treated as a classical field. In combination with the corresponding first part (Ion Trap Quantum Computer Part 1: 1-Qubit Gates), this results in the complete quantum theory of a universal set of quantum gates. This presentation is intended for readers with basic knowledge of quantum physics and assumes familiarity with Part 1.
Ions offer a promising approach to physically realize qubits. The present text extensively elucidates the physics of 1-qubit gates for use in quantum computers based on ion qubits. To keep the discussion as simple as possible, ions are treated mechanically as rigid rotors. The discussed system is excellently suited to practice important elementary methods of quantum physics using a practical application example. For didactic reasons, contrary to the usual approach, a typographical distinction is made between states and operators on the one hand and column vectors and matrices on the other. The article is aimed at readers with basic knowledge of quantum physics, i.e., those familiar with terms like Schrödinger equation, wave function, bra and ket, Pauli matrices, and the like. Rudimentary knowledge of quantum computers is also assumed (qubit, gate, ...).
We investigate the exact effect on ion trap quantum computation after field quantization. First an exact expression of failure probability from field quantization after many CNOT operations in Cirac-Zoller scheme is given. It is proportional to operation number and the amplitude of ∣1⟩x∣0⟩y|1\rangle_x |0\rangle_y∣1⟩x​∣0⟩y​ or ∣1⟩x∣1⟩y|1\rangle_x |1\rangle_y∣1⟩x​∣1⟩y​ in initial state, and inverse proportional to mean number of photons and amplitude of ∣0⟩x∣0⟩y|0\rangle_x |0\rangle_y∣0⟩x​∣0⟩y​ or ∣0⟩x∣1⟩y|0\rangle_x |1\rangle_y∣0⟩x​∣1⟩y​ in initial state. Then we calculate the failure probability when the limitation to mean number of photons in sideband transition is considered. When the initial state is ∣1⟩x∣0⟩y|1\rangle_x |0\rangle_y∣1⟩x​∣0⟩y​ or ∣1⟩x∣1⟩y|1\rangle_x |1\rangle_y∣1⟩x​∣1⟩y​, after about 10210^2102 times of CNOT operations, failure probability is no less than 10−210^{-2}10−2, while 10−210^{-2}10−2 is the known maximum threshold in fault-tolerant quantum computation. Then when the initial state is ∣1⟩x∣0⟩y|1\rangle_x |0\rangle_y∣1⟩x​∣0⟩y​ or ∣1⟩x∣1⟩y|1\rangle_x |1\rangle_y∣1⟩x​∣1⟩y​, the number of CNOT gates on the same pair of physical qubits should be no more than 10210^2102 in one error-correction period, or else the computation cannot be implemented reliably. This conclusion can help to determine the number of CNOT operations between coding and decoding in one error-correction period in fault-tolerant quantum computation.
Realisation of controlled-NOT gate (CNOT) operation between two individual ions as a universal two-bit quantum gate based from the proposals of Cirac and Zoller, is presented. Two single /sup 40/Ca/sup +/ ions are held in a linear Paul trap and are individually addressed with focused laser beams. Superpositions of the S/sub 1/2 /and the D/sub 5/2/ long lived electronic states represent a qubit. A sequence of well suited laser pulses flips the state of the target ion conditional to the state of the control ion.
The universality theorem in quantum computing states that any quantum computational task can be decomposed into a finite set of logic gates operating on one and two qubits. However, the process of such decomposition is generally inefficient, often leading to exponentially many gates to realize an arbitrary computational task. Practical processor designs benefit greatly from availability of multi-qubit gates that operate on more than two qubits to implement the desired circuit. In 1995, Cirac and Zoller proposed a method to realize native multi-qubit controlled-ZZZ gates in trapped ion systems, which has a stringent requirement on ground-state cooling of the motional modes utilized by the gate. An alternative approach, the Mølmer-Sørensen gate, is robust against residual motional excitation and has been a foundation for many high-fidelity gate demonstrations. This gate does not scale well beyond two qubits, incurring additional overhead when used to construct many target algorithms. Here, we take advantage of novel performance benefits of long ion chains to realize fully programmable and scalable high-fidelity Cirac-Zoller gates.
In this paper we describe one of the most promising platforms for the construction of a universal quantum computer, which consists of a chain of NNN ions trapped in a harmonic potential, whose internal states work out as qubits, and are coupled to collective vibrational modes of the chain. From such coupling, it is possible to build interactions between different ions of the chain, that is, qubit-qubit interactions that, together with individual operations on the ions, allow building a quantum computer as first proposed by Cirac and Zoller in the 1990s [Phys. Rev. Lett. 74, 4091 (1995)]. Here we discuss from the physics involved in trapping ions in electromagnetic potentials to the Hamiltonian engineering needed to generate a universal set of logic gates, fundamental for the execution of more complex quantum algorithms. Finally, we present the current state of the art of quantum computing in trapped ion systems, highlighting recent advances made by companies and government projects that use such architecture, such as IonQ and AQTION.